Difference Algebra (Algebra and Applications)

Difference Algebra

Algebra and Applications

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Volume 8

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Difference Algebra

Alexander Levin

ABC

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Alexander LevinT he Catholic University of America

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Preface

Difference algebra as a separate area of mathematics was born in the 1930s whenJ. F. Ritt (1893 - 1951) developed the algebraic approach to the study of systemsof difference equations over functional fields. In a series of papers publishedduring the decade from 1929 to 1939, Ritt worked out the foundations of bothdifferential and difference algebra, the theories of abstract algebraic structureswith operators that reflect the algebraic properties of derivatives and shifts ofarguments of analytic functions, respectively. One can say that differential anddifference algebra grew out of the study of algebraic differential and differenceequations with coefficients from functional fields in much the same way as theclassical algebraic geometry arose from the study of polynomial equations withnumerical coefficients.

Ritt’s research in differential algebra was continued and extended byH. Raudenbuch, H. Levi, A. Seidenberg, A. Rosenfeld, P. Cassidy, J. Johnson,W. Keigher, W. Sit and many other mathematicians, but the most importantrole in this area was played by E. Kolchin who recast the whole subject inthe style of modern algebraic geometry with the additional presence of deriva-tion operators. In particular, E. Kolchin developed the contemporary theoryof differential fields and created the differential Galois theory where finite di-mensional algebraic groups played the same role as finite groups play in thetheory of algebraic equations. Kolchin’s monograph [105] is the most deep andcomplete book on the subject, it contains a lot of ideas that determined themain directions of research in differential algebra for the last thirty years.

The rate of development of difference algebra after Ritt’s pioneer works andworks by F. Herzog, H. Raudenbuch and W. Strodt published in the 1930s (see[82], [166], [172], and [173]) was much slower than the rate of expansion of itsdifferential counterpart. The situation began to change in the 1950s due to R.Cohn whose works [28] - [44] not only raised the difference algebra to the levelcomparable with the level of the development of differential algebra, but alsoclarified why many ideas that are fruitful in differential algebra cannot be suc-cessfully applied in the difference case, as well as many methods of differencealgebra cannot have differential analogs. R. Cohn’s book [41] hitherto remainsthe only fundamental monograph on difference algebra. Since 60th various prob-lems of difference algebra were developed by A. Babbitt [4], I. Balaba [5] - [7],I. Bentsen [10], A. Bialynicki-Birula [11], R. Cohn [45] - [53], P. Evanovich [60],[61], C. Franke [67] - [73], B. Greenspan [75], P. Hendrics [78] - [81], R. Infante

v

vi PREFACE

[86] - [91], M. Kondrateva [106] - [110], B. Lando [117], [118], A. Levin [109],[110] and [120] - [136], A. Mikhalev [109], [110], [131] - [136] and [139] - [141],E. Pankratev [106] - [110], [139] - [141] and [151] - [154], M. van der Put andM. Singer [159], and some other mathematicians. Nowadays, difference algebraappears as a rich theory with its own methods that are very useful in the studyof system of equations in finite differences, functional equations, differentialequations with delay, algebraic structures with operators, group and semigrouprings. A number of interesting applications of difference algebra in the theoryof discrete-time nonlinear systems can be found in the works by M. Fliess [62] -[66], E. Aranda-Bricaire, U. Kotta and C. Moog [2], and some other authors.

This book contains a systematic study of both ordinary and partial differ-ence algebraic structures and their applications. R. Cohn’s monograph [41] waslimited to ordinary cases that seemed to be reasonable because just a few workson partial difference algebra had been published by 1965. Nowadays, due toefforts of I. Balaba, I. Bentsen, R. Cohn, P. Evanovich, A. Levin, A. Mikhalev,E. Pankratev and some other mathematicians, partial difference algebra pos-sesses a body of results comparable to the main stem of ordinary difference alge-bra. Furthermore, various applications of the results on partial difference fields,modules and algebras (such as the theory of strength of systems of differenceequations or the dimension theory of algebraic structures with operators) showthe importance of partial difference algebra for the related areas of mathematics.

This monograph is intended to help researchers in the area of differencealgebra and algebraic structures with operators; it could also serve as a workingtextbook for a graduate course in difference algebra. The book is almost entirelyself-contained: the reader only needs to be familiar with basic ring-theoretic andgroup-theoretic concepts and elementary properties of rings, fields and modules.All more or less advanced results in the ring theory, commutative algebra andcombinatorics, as well as the basic notation and conventions, can be found inChapter 1. This chapter contains many results (especially in the field theory,theory of numerical polynomials, and theory of graded and filtered algebraicobjects) that cannot be found in any particular textbook.

Chapter 2 introduces the main objects of Difference Algebra and discussestheir basic properties. Most of the constructions and results are presented inmaximal possible generality, however, the chapter includes some results on ordi-nary difference structures that cannot be generalized to the partial case (or theexistence of such generalizations is an open problem). One of the central topicsof Chapter 2 is the study of rings of difference and inversive difference polyno-mials. We introduce the concept of reduction of such polynomials and presentthe theory of difference characteristic sets in the spirit of the correspondingRitt-Kolchin scheme for the rings of differential polynomials. This chapter isalso devoted to the study of perfect difference ideals and rings that satisfy theascending chain condition for such ideals. Another important part of Chapter 2is the investigation of Ritt difference rings, that is, difference rings satisfying theascending chain condition for perfect difference ideals. In particular, we provethe Ritt-Raudenbush basis theorem for partial difference polynomial rings anddecomposition theorems for perfect difference ideals and difference varieties.

PREFACE vii

Chapter 3 is concerned with properties of difference and inversive differencemodules and their applications in the theory of linear difference equations. Weprove difference versions of Hilbert and Hilbert-Samuel theorems on character-istic polynomials of graded and filtered modules, introduce the concept of adifference dimension polynomial, and describe invariants of such a polynomial.In this chapter we also present a generalization of the classical Grobner basismethod that allows one to compute multivariable difference dimension polyno-mials of difference and inversive difference modules. In addition to the applica-tions in difference algebra, our construction of generalized Grobner bases, whichinvolves several term orderings, gives a tool for computing dimension polynomi-als in many other cases. In particular, it can be used for computation of Hilbertpolynomials associated with multi-filtered modules over polynomial rings, ringsof differential and difference-differential operators, and group algebras.

Chapter 4 is devoted to the theory of difference fields. The main concepts andobjects studied in this chapter are difference transcendence bases, the differencetranscendence degree of a difference field extension, dimension polynomials asso-ciated with finitely generated difference and inversive difference field extensions,and limit degree of a difference filed extension. The last concept, introduced byR. Cohn [39] for the ordinary difference fields, is generalized here and used inthe proof of the fundamental fact that a subextension of a finitely generatedpartial difference field extension is finitely generated. The chapter also containssome specific results on ordinary difference field extensions and discussion ofdifference algebras.

Chapter 5 treats the problems of compatibility, replicability, and monadicityof difference field extensions, as well as properties of difference specializations.The main results of this chapter are the fundamental compatibility and replica-bility theorems, the stepwise compatibility condition for partial difference fields,Babbitt’s decomposition theorem and its applications to the study of finitelygenerated pathological difference field extensions. The chapter also contains ba-sic notions and results of the theory of difference kernels over ordinary differencefields, as well as the study of prolongations of difference specializations.

In Chapter 6 we introduce the concept of a difference kernel over a partialdifference field and study properties of such difference kernels and their pro-longations. In particular, we discuss generic prolongations and realizations ofpartial difference kernels. In this chapter we also consider difference valuationrings, difference places and related problems of extensions of difference special-izations.

Chapter 7 is devoted to the study of varieties of difference polynomials, al-gebraic difference equations, and systems of such equations. One of the centralresults of this chapter is the R. Cohn’s existence theorem stating that everynontrivial ordinary algebraically irreducible difference polynomial has an ab-stract solution. We then discuss some consequences of this theorem and somerelated statements for partial difference polynomials. (As it is shown in [10],the Cohn’s theorem cannot be directly extended to the partial case, however,some versions of the existence theorem can be obtained in this case, as well.)Most of the results on varieties of ordinary difference polynomials were obtained

viii PREFACE

in R. Cohn’s works [28] - [37]; many of them have quite technical proofs whichhitherto have not been simplified. Trying not to overload the reader with techni-cal details, we devote a section of Chapter 7 to the review of such results whichare formulated without proofs. The other parts of this chapter contain the dis-cussion of Greenspan’s and Jacobi’s bounds for systems of ordinary differencepolynomials and the concept of the strength of a system of partial differenceequations. The last concept arises as a difference version of the notion of thestrength of a system of differential equations studied by A. Einstein [55]. Wedefine the strength of a system of equations in finite differences and treat itfrom the point of view of the theory of difference dimension polynomials. Wealso give some examples where we determine the strength of some well-knownsystems of difference equations.

The last chapter of the book gives some overview of difference Galois the-ory. In the first section we consider Galois correspondence for difference fieldextensions and related problems of compatibility and monadicity. The other twosections are devoted to the review of the classical Picard-Vissiot theory of linearhomogeneous difference equations and some results on Picard-Vessiot rings. Wedo not consider the Galois theory of difference equations based on Picard-Vessiotrings; it is perfectly covered in the book by M. van der Put and M. Singer [159],and we would highly recommend this monograph to the reader who is interestedin the subject.

I am very grateful to Professor Richard Cohn for his support, encouragementand help. I wish to thank Professor Alexander V. Mikhalev, who introducedme into the subject, and my colleagues P. Cassidy, R. Churchill, W. Keigher,M. Kondrateva, J. Kovacic, S. Morrison, E. Pankratev, W. Sit and all partici-pants of the Kolchin Seminar in Differential Algebra at the City University ofNew York for many fruitful discussions and advices.

Contents

1 Preliminaries 1

1.1 Basic Terminology and Background Material . . . . . . . . . . . 11.2 Elements of the Theory of Commutative Rings . . . . . . . . . . 151.3 Graded and Filtered Rings and Modules . . . . . . . . . . . . . . 371.4 Numerical Polynomials . . . . . . . . . . . . . . . . . . . . . . . . 471.5 Dimension Polynomials of Sets of m-tuples . . . . . . . . . . . . 531.6 Basic Facts of the Field Theory . . . . . . . . . . . . . . . . . . . 641.7 Derivations and Modules of Differentials . . . . . . . . . . . . . . 891.8 Grobner Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

2 Basic Concepts of Difference Algebra 103

2.1 Difference and Inversive Difference Rings . . . . . . . . . . . . . . 1032.2 Rings of Difference and Inversive Difference Polynomials . . . . . 1152.3 Difference Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . 1212.4 Autoreduced Sets of Difference and Inversive Difference

Polynomials. Characteristic Sets . . . . . . . . . . . . . . . . . . 1282.5 Ritt Difference Rings . . . . . . . . . . . . . . . . . . . . . . . . . 1412.6 Varieties of Difference Polynomials . . . . . . . . . . . . . . . . . 149

3 Difference Modules 155

3.1 Ring of Difference Operators. Difference Modules . . . . . . . . . 1553.2 Dimension Polynomials of Difference Modules . . . . . . . . . . . 1573.3 Grobner Bases with Respect to Several Orderings

and Multivariable Dimension Polynomials of DifferenceModules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

3.4 Inversive Difference Modules . . . . . . . . . . . . . . . . . . . . . 1853.5 σ∗-Dimension Polynomials and their Invariants . . . . . . . . . . 1953.6 Dimension of General Difference Modules . . . . . . . . . . . . . 232

ix

x CONTENTS

4 Difference Field Extensions 2454.1 Transformal Dependence. Difference Transcendental Bases

and Difference Transcendental Degree . . . . . . . . . . . . . . . 2454.2 Dimension Polynomials of Difference and Inversive Difference

Field Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 2554.3 Limit Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2744.4 The Fundamental Theorem on Finitely Generated Difference

Field Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 2924.5 Some Results on Ordinary Difference Field Extensions . . . . . . 2954.6 Difference Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 300

5 Compatibility, Replicability, and Monadicity 3115.1 Compatible and Incompatible Difference Field Extensions . . . . 3115.2 Difference Kernels over Ordinary Difference Fields . . . . . . . . 3195.3 Difference Specializations . . . . . . . . . . . . . . . . . . . . . . 3285.4 Babbitt’s Decomposition. Criterion of Compatibility . . . . . . . 3325.5 Replicability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3525.6 Monadicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

6 Difference Kernels over Partial Difference Fields. DifferenceValuation Rings 3716.1 Difference Kernels over Partial Difference Fields

and their Prolongations . . . . . . . . . . . . . . . . . . . . . . . 3716.2 Realizations of Difference Kernels over Partial Difference Fields . 3766.3 Difference Valuation Rings and Extensions of Difference

Specializations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385

7 Systems of Algebraic Difference Equations 3937.1 Solutions of Ordinary Difference Polynomials . . . . . . . . . . . 3937.2 Existence Theorem for Ordinary Algebraic Difference

Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4027.3 Existence of Solutions of Difference Polynomials in the Case

of Two Translations . . . . . . . . . . . . . . . . . . . . . . . . . 4127.4 Singular and Multiple Realizations . . . . . . . . . . . . . . . . . 4207.5 Review of Further Results on Varieties of Ordinary Difference

Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4257.6 Ritt’s Number. Greenspan’s and Jacobi’s Bounds . . . . . . . . . 4337.7 Dimension Polynomials and the Strength of a System of Algebraic

Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . . 4407.8 Computation of Difference Dimension Polynomials in the Case

of Two Translations . . . . . . . . . . . . . . . . . . . . . . . . . 455

8 Elements of the Difference Galois Theory 4638.1 Galois Correspondence for Difference Field Extensions . . . . . . 4638.2 Picard-Vessiot Theory of Linear Homogeneous Difference

Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472

CONTENTS xi

8.3 Picard-Vessiot Rings and the Galois Theory of DifferenceEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486

Bibliography 495

Index 507

Chapter 1

Preliminaries

1.1 Basic Terminology and BackgroundMaterial

1. Sets, relations, and mappings. Throughout the book we keep the stan-dard notation and conventions of the set theory. The union, intersection, anddifference of two sets A and B are denoted by A

⋃B, A

⋂B, and A\B, respec-

tively. If A is a subset of a set B, we write A ⊆ B or B ⊇ A; if the inclusion isproper (that is, A ⊆ B, A = B), we use the notation A ⊂ B (or A B) andA ⊃ B (or B A). If A is not a subset of B, we write A B or B A. Asusual, the empty set is denoted by ∅.

If F is a family of sets, then the union and intersection of all sets of thefamily are denoted by

⋃X |X ∈ F (or⋃

X∈F X) and⋂X |X ∈ F (or⋂

X∈F X), respectively. The power set of a set X, that is, the set of all subsetsof X is denoted by P(X).

The set of elements of a set X satisfying a condition φ(x) is denoted byx ∈ X |φ(x). If a set consists of finitely many elements x1, . . . , xk, it is de-noted by x1, . . . , xk (sometimes we abuse the notation and write x for the setx consisting of one element). Throughout the book Z, N, N+, Q, R, and Cwill denote the sets of integers, non-negative integers, positive integers, ratio-nal numbers, real numbers, and complex numbers respectively. For any positiveinteger m, the set 1, . . . , m will be denoted by Nm.

In what follows, the cardinality of a set X is denoted by CardX or |X| (thelast notation is convenient when one considers relationships involving cardinali-ties of several sets). In particular, if X is finite, CardX (or |X|) will denote thenumber of elements of X. We shall often use the following principle of inclusionand exclusion (its proof can be found, for example in [17, Section 6.1].

Theorem 1.1.1 Let P1, . . . , Pm be m properties referring to the elements of afinite set S, let Ai = x ∈ S |x has property Pi and let Ai = S \Ai = x ∈ S |xdoes not have property Pi (i = 1, . . . , m). Then the number of elements of Swhich have none of the properties P1, . . . , Pm is given by

1

2 CHAPTER 1. PRELIMINARIES

|A1 ∩ A2 · · · ∩ Am| = |S| −m∑

i=1

|Ai| +∑

1≤i<j≤m

|Ai ∩ Aj |

+∑

1≤i<j<k≤m

|Ai ∩ Aj ∩ Ak| + · · · + (−1)m|A1 ∩ A2 ∩ · · · ∩ Am| . (1.1.1)

As a consequence of formula (1.1.1) we obtain that if M1, . . . ,Mq are finitesets, then

|M1 ∩ · · · ∩ Mq| =q∑

i=1

|Mi| −∑

1≤i<j≤q

|Mi ∩ Mj |

+∑

1≤i<j<k≤m

|Mi ∩ Mj ∩ Mk| − · · · + (−1)q+1|M1 ∩ · · · ∩ Mq|.

Given sets A and B, we define their Cartesian product A × B to be the setof all ordered pairs (a, b) where a ∈ A, b ∈ B. (An order pair (a, b) is formallydefined as collection of sets a, a, b; a and b are said to be the first and thesecond coordinates of (a, b). Two ordered pairs (a, b) and (c, d) are equal if andonly if a = c and b = d.) A subset R of a Cartesian product A × B is calleda (binary) relation from A to B. We usually write aRb instead of (a, b) ∈ Rand say that a is in the relation R to b. The domain of a relation R is the setof all first coordinates of members of R, and its range is the set of all secondcoordinates.

A relation f ⊆ A × B is called a function or a mapping from A to B if forevery a ∈ A there exists a unique element b ∈ B such that (a, b) ∈ f . In thiscase we write f :A → B or A

f−→ B, and f(a) = b or f → b if (a, b) ∈ f . Theelement f(a) is called the image of a under the mapping f or the value of f at a.If A0 ⊆ A, then the set f(a) | a ∈ A0 is said to be the image of A0 under f ;it is denoted by f(A) or Imf . If B0 ⊆ B, then the set and a ∈ A | f(a) ∈ B0is called the inverse image of f ; it is denoted by f−1(B0).

A mapping f :A → B is said to be injective (or one-to-one) if for any a1, a2 ∈A, the equality f(a1) = f(a2) implies a1 = a2. If f(A) = B, the mapping iscalled surjective (or a mapping of A onto B). An injective and surjective mappingis called bijective. The identity mapping of a set A into itself is denoted by idA.If f : A → B and g : B → C, then the composition of these mappings is denotedby g f or gf . (Thus, g f(a) = g(f(a)) for every a ∈ A.)

If f : A → B is a mapping and X ⊆ A, then the restriction of f on X isdenoted by f |X . (This is the mapping from X to B defined by f |X(a) = f(a)for every a ∈ A.)

If I is a nonempty set and there is a function that associates with everyi ∈ I some set Ai, we say that we have a family of sets Aii∈I (also denotedby Ai | i ∈ I) indexed by the set I. The set I is called the index set of thefamily. The union and intersection of all sets of a family Aii∈I are denoted,respectively, by

⋃i∈I Ai (or

⋃Ai |i ∈ I) and⋂

i∈I Ai (or⋂Ai |i ∈ I). If

I = 1, . . . , m for some positive integer m, we also write⋃m

i=1 Ai and⋂m

i=1 Ai

for the union and intersection of the sets A1, . . . , Am, respectively.

1.1. BASIC TERMINOLOGY AND BACKGROUND MATERIAL 3

A family of sets is said to be (pairwise) disjoint if for any two sets A and Bof this family, either A

⋂B = ∅ or A = B. In particular, two different sets A

and B are disjoint if A⋂

B = ∅. A family Aii∈I of nonempty subsets of a setA is said to be a partition of A if it is disjoint and

⋃i∈I Ai = A.

A Cartesian product of a family of sets Aii∈I , that is, the set of all func-tions f from I to

⋃i∈I Ai such that f(i) ∈ Ai for all i ∈ I, is denoted by

∏i∈I Ai.

For any f ∈ ∏i∈I Ai, the element f(i) (i ∈ I) is called the ith coordinate of f ;

it is also denoted by fi, and the element f is written as (fi)i∈I or fi | i ∈ I. IfAi = A for every i ∈ I, we write AI for

∏i∈I Ai.

If I = N+, the Cartesian product of the sets of the sequence A1, A2, . . . is de-noted by

∏∞i=1 Ai. The Cartesian product of a finite family of sets A1, . . . , Am

is denoted by∏m

i=1 Ai. Elements of this product are called m-tuples. If A1 =· · · = Am = A, the elements of

∏mi=1 Ai are called m-tuples over the set A.

Let M and I be two sets. By an indexing in M with the index set I we meanan element a = (ai)i∈I of the Cartesian product

∏i∈I Mi where Mi = M for

all i ∈ I. Thus, an indexing in M with the index set I is a function a : I → M ;the image of an element i ∈ I is denoted by ai and called the ith coordinate ofthe indexing. If J ⊆ I, then the restriction of the function a on J is called asubindexing of the indexing a; it is denoted by (ai)i∈J . A finite indexing a withan index set 1, . . . , m is also referred to as an m-tuple a = (a1, . . . , am).

If A is a set and R ⊆ A × A, we say that R is a relation on A. It is called- reflexive if aRa for every a ∈ A;- symmetric if aRb implies bRa for every a, b ∈ A;- antisymmetric if for any elements a, b ∈ A, the conditions aRb and bRa

imply a = b;- transitive if aRb and bRc imply aRc for every a, b, c ∈ A. (A transitive

relation on a set A is also called a preorder on A.)If R is reflexive, symmetric and transitive, it is called an equivalence relation

on A. For every a ∈ A, the set [a] = x ∈ A |xRa is called the (R-) equivalenceclass of the element a. The fundamental result on equivalence relations statesthat the family [a] ∈ P(A) | a ∈ A of all equivalence classes forms a partitionof A and, conversely, every partition of A determines an equivalence relationR on A such that xRy if and only if x and y belong to the same set of thepartition. (In this case every equivalence class [a] (a ∈ A) coincides with the setof the partition containing a.)

Let R be a relation on a set A. The reflexive closure of R is a relationR′ on A such that aR′b (a, b ∈ A) if and only if aRb or a = b (the lastequality means that a and b denote the same element of A). A relation R′′ on A,such that aR′′b if and only if aRb or bRa, is called the symmetric closure of R.A relation R+ on A is said to be a transitive closure of the relation R if itsatisfies the following condition: aR+b (a, b ∈ A) if and only if there exist fi-nitely many elements a1, . . . , an ∈ A such that aRa1, . . . , anRb. It is easy tosee that R′, R′′, and R+ are the reflexive, symmetric, and transitive relationson A, respectively. Combining the foregoing constructions, one arrives at theconcepts of reflexive-symmetric, reflexive-transitive, symmetric-transitive, andreflexive-symmetric-transitive closures of a relation R. For example, a reflexive-


symmetric closure R∗ of R is defined by the following condition: aR∗b if andonly if aR′b or bR′a (that is, aRb or bRa or a = b).

A relation R on a set A is called a partial order on A if it is reflexive, tran-sitive, and antisymmetric. Partial orders are usually denoted by ≤ (sometimeswe shall also use symbols or ). If a ≤ b, we also write this as b ≥ a; if a ≤ band a = b, we write a < b or a > b.

If ≤ is a partial order on a set A, the ordered pair (A,≤) is called a partiallyordered set. We also say that A is a partially ordered set with respect to the or-der ≤. If A0 ⊆ A, we usually treat A0 as an ordered set with the same partialorder ≤.

An element m of a partially ordered set (A,≤) is called minimal if for everyx ∈ A the condition x ≤ m implies x = m. Similarly, and element s ∈ A is calledmaximal if for every x ∈ A, the condition s ≤ x implies x = s. An element a ∈ Ais the smallest element (least element or first element) in A if a ≤ x for everyx ∈ X, and b ∈ A is the greatest element (largest element or last element) in Aif x ≤ b for every x ∈ X. Obviously, a partially ordered set A has at most onegreatest and one smallest element, and the smallest (the greatest) element of A,if it exists, is the only minimal (respectively, maximal) element of A.

Let (A,≤) be a partially ordered set and B ⊆ A. An element a ∈ A is calledan upper bound for B if b ≤ a for every b ∈ B. Also, a is called a least upperbound (or supremum) for B if a is an upper bound for B and a ≤ x for everyupper bound x for B. Similarly, c ∈ A is a lower bound for B if c ≤ b for everyb ∈ B. If, in addition, for every lower bound y for B we have y ≤ c, then theelement c is called a greatest lower bound (or infimum) for B. We write sup(B)and inf(B) to denote the supremum and infimum of B, respectively.

If ≤ is a partial order on a set A and for every distinct elements a, b ∈ Aeither a ≤ b or b ≤ a, then ≤ is called a linear or total order on A. In this casewe say that the set A is linearly (or totally) ordered by ≤ (or with respect to≤). A linearly ordered set is also called a chain. Clearly, if (A,≤) is a linearlyordered set, then every minimal element in A is the smallest element and everymaximal element in A is the greatest element. In particular, linearly orderedsets can have at most one maximal element and at most one minimal element.

If (A,≤) is a partially ordered set and for every a, b ∈ A there is an elementc ∈ A such that a ≤ c and b ≤ c, then A is said to be a directed set.

The proof of the following well-known fact can be found, for example, in[102, Chapter 0].

Proposition 1.1.2 Let (A,≤) be a partially ordered set. Then the followingconditions are equivalent.

(i) (The condition of minimality.) Every nonempty subset of A contains aminimal element.

(ii) (The induction condition.) Suppose that all minimal elements of A havesome property Φ, and for every a ∈ A the fact that all elements x ∈ A, x < ahave the property Φ implies that the element a has this property as well. Thenall elements of the set A have the property Φ.


(iii) (Descending chain condition) If a1 ≥ a2 ≥ . . . is any chain of elementsof A, then there exists n ∈ N+ such that an = an+1 = . . . .

A linearly ordered set (A,≤) is called well-ordered if it satisfies the equivalentconditions of the last proposition. In particular, a linearly ordered set A is well-ordered if every nonempty subset of A contains a smallest element.

Throughout the book we shall use the axiom of choice and its alternativeversions contained in the following proposition whose proof can be found, forexample, in [102, Chapter 0]).

Proposition 1.1.3 The following statements are equivalent.(i) (Axiom of Choice.) Let A be a set, let Ω be a collection of nonempty

subsets of a set B, and let φ be a function from A to Ω. Then there is a functionf : A → B such that f(a) ∈ φ(a) for every a ∈ A.

(ii) (Zermelo Postulate) If Ω is a disjoint family of nonempty sets, then thereis a set C such that A ∩ C consists of a single element for every A ∈ Ω.(iii) (Hausdorff Maximal Principle.) If Ω is a family of sets and Σ is a chain

in Ω (with respect to inclusion as a partial order), then there is a maximal chainin Ω which contains Σ.(iv) (Kuratowski Lemma) Every chain in a partially ordered set is contained

in a maximal chain.(v) (Zorn Lemma) If every chain in a partially ordered set has an upper bound,

then there is a maximal element of the set.(vi) (Well-ordering Principle) Every set can be well-ordered.

2. Dependence relations. Let X be a nonempty set and ∆ ⊆ X ×P(X)a binary relation from X to the power set of X. We write x ≺ S and say that xis dependent on S if (x, S) ∈ ∆. Otherwise we write x ⊀ S and say that x doesnot depend on S or that x is independent of S. If S and T are two subsets of X,we write S ≺ T and say that the set S is dependent on T if s ≺ T for all s ∈ S.

Definition 1.1.4 With the above notation, a relation ∆ ⊆ X × P(X) is saidto be a dependence relation if it satisfies the following properties.

(i) S ≺ S for any set S ⊆ X.(ii) If x ∈ X and x ≺ S, then there exists a finite set S0 ⊆ S such that x ≺ S0.(iii) Let S, T , and U be subsets of X such that S ≺ T and T ≺ U . Then S ≺ U .(iv) Let S ⊆ X and s ∈ S. If x is an element of X such that x ≺ S, x ⊀ S\s,

then s ≺ (S \ s)⋃x.

In what follows we shall often abuse the language and refer to ≺ as a relationon X.


Definition 1.1.5 Let S be a subset of a set X. A set S is called dependentif there exists s ∈ S such that s ≺ S \ s (equivalenly, if S ≺ S \ s). Ifs ⊀ S \ s for all s ∈ S, the set S is called independent. (In particular, theempty set is independent.)

The proof of the following two propositions can be found in [167, Chapter 4].

Proposition 1.1.6 Let S and U be subsets of a set X and let T ⊆ U .(i) If S ≺ T , then S ≺ U .(ii) If T is dependent, so is U .(iii) If U is independent, so is T .(iv) If S is dependent, then some finite subset S0 of S is dependent. Equiva-

lently, if every finite subset of S is independent, then S is independent.(v) Let S be independent and let x be an element of X such that x ⊀ S. Then

S⋃x is independent.(vi) Let S be finite and dependent, and let S′ be an independent subset of S.

Then there exists s ∈ S \ S′ such that S ≺ S \ s.

Definition 1.1.7 If X is a nonempty set, then a set B ⊆ X is called a basisfor X if B is independent and X ≺ B.

Proposition 1.1.8 Let X be a nonempty set with a dependence relation ≺.(i) B ⊆ X is a basis for X if and only if it is a maximal independent set

in X.(ii) B ⊆ X is a basis for X if and only if B is minimal with respect to the

property X ≺ B.(iii) Let S ⊆ T ⊆ X where S is an independent set (possibly empty) and

X ≺ T . Then there is a basis B for X such that S ⊆ B ⊆ T .(iv) Any two bases for X have the same cardinality.

3. Categories and functors. A category C consists of a class of objects,obj C, together with sets of morphisms which arise as follows. There is a functionMor which assigns to every pair A,B ∈ obj C a set of morphisms Mor(A,B)(also written as MorC(A,B)). Elements of Mor(A,B) are called morphismsfrom A to B, and the inclusion f ∈ Mor(A,B) is also indicated as f : A → B

or Af−→ B. The sets Mor(A,B) and Mor(C,D) are disjoint unless A = C and

B = D in which case they coincide. Furthermore, for any three objects A,B,C ∈obj C there is a mapping (called a composition) Mor(B,C) × Mor(A,B) →Mor(A,C), (g, f) → gf , with the following properties:

1) The composition is associative. That is, if f ∈ Mor(C,D), g ∈ Mor(B,C),and h ∈ Mor(A,B), then (fg)h = f(gh).

2) For every object A, there exists a morphism 1A ∈ Mor(A,A) (called anidentity morphism) such that f = f1A = 1Bf for any object B and for anymorphism f ∈ Mor(A,B).


If f : A → B is a morphism in a category C, we say that the objects A andB are the domain and codomain (or range) of f , respectively. The morphism fis called an equivalence (or an isomorphism or an invertible morphism) if thereexists a morphism g ∈ Mor(B,A) such that gf = 1A and fg = 1B . (Then g isunique; it is called the inverse of f).

A category C is said to be a subcategory of a category D if (i) every objectof C is an object of D; (ii) for every objects A and B in C, MorC(A,B) ⊆MorD(A,B); (iii) the composite of two morphisms in C is the same as theircomposite in D; (iv) for every object A in C, the identity morphism 1A is thesame in D as it is in C.

We now list several examples of categories.

(a) The category Set: the class of objects is the class of all sets, the morphismsare functions (with the usual composition).

(b) The category Grp: the class of objects is the class of all groups andmorphisms are group homomorphisms.

(c) The category Ab where the class of objects is the class of all Abeliangroups, and morphisms are group homomorphisms.

(d) The category Ring: the class of objects is the class of all rings, and themorphisms are ring homomorphisms.

(e) The category RMod of left modules over a ring R: the class of objectsis the class of all left R-modules, the morphisms are module homomorphisms.The category ModR of right modules over R is defined in the same way.

(f) The category Top: the class of objects is the class of all topological spaces,the morphisms are continuous functions between them.

It is easy to see that the categories in examples (b) - (f) are subcategories ofSet, Ab is a subcategory of Grp, etc. In what follows, we adopt the standardnotation of the categories of algebraic objects and denote the sets of morphismsMor(A,B) in categories Grp, Ab and Ring by Hom(A,B); the correspondingnotation in the categories RMod, and ModR is HomR(A,B).

Let C and D be two categories. A covariant functor F from C to D is a pairof mappings (denoted by the same letter F ): the mapping that assigns to everyobject A of C an object F (A) in D, and the mapping defined on morphisms ofC which assigns to every morphism α : A → B in C a morphism F (α) : F (A) →F (B) in D. This pair of mappings should satisfy the following two conditions:

(a) F (1A) = 1F (A) for every object A in C;(b) If α and β are two morphisms in C such that the composition αβ is defined,

then F (αβ) = F (α)F (β).

If F and G are two covariant functors from a category C to a category D (inthis case we write F : C → D and G : C → D), then the natural transformationof functors h : F → G is a mappping that assigns to every object A in C amorphism h(A) : F (A) → G(A) in D such that for every morphism α : A → Bin C, we have the following commutative diagram in the category D:


F (A) h(A)G(A)

F (B) h(B)G(B)

F (α) G(α)

If in addition each h(A) is an equivalence, we say that h is a natural isomor-phism.

A contravariant functor G from a category C to a category D consists ofan object function, which assigns to each object A in C an object G(A) in D,and a mapping function which assigns to each morphism α : A → B in C amorphism G(α) : G(B) → G(A) in D (both object and mapping functions aredenoted by the same letter G). This pair of function must satisfy the followingtwo conditions: G(1A) = 1G(A) for any object A in C, and G(αβ) = G(β)G(α)whenever the composition of two morphisms α and β in C is defined. A naturaltransformation and natural isomorphism of contravariant functors are definedin the same way as in the case of their covariant counterparts.

An object A in a category C is called an initial object (respectively, a terminalor final object) if for any object B in C, Mor(A,B) (respectively, Mor(B,A))consists of a single morphism. If A is both an initial object and a terminalobject, it is called a zero (or null) object in C. For example, ∅ is an initial objectin Sets, while the trivial group is a zero object in Grp. If a category C has azero object 0, then a morphism f ∈ Mor(A,B) in C is called a zero morphismif there exists morphisms g : A → 0 and h : 0 → B such that f = hg. This zeromorphism is denoted by 0AB or 0. It is easy to see, that if C is a category witha zero object, then there is exactly one zero morphism from each object A toeach object B.

A morphism f : A → B in a category C is called monomorphism (we alsosay that f is monic) if whenever fg = fh for some morphisms g, h ∈ Mor(C,A)(C is an object in C), then g = h. A morphism f : A → B in C is calledan epimorphism (we also say that f is epi) if whenever gf = hf for somemorphisms g, h ∈ Mor(B,D) (D is an object in C), then g = h. f : A → B iscalled a bimorphism if f is both monic and epi.

Two monomorphisms f : A → C and g : B → C in a category C arecalled equivalent if there exists an isomorphism τ : A → B such that f = gτ .An equivalence class of monomorphisms with codomain C is called a subobjectof the object C. The dual notion is the notion of quotient (or factor object)of C. Note that in the categories Grp, Ab, Ring, RMod, and ModR (R is aring) the monomorphisms and epimorphisms are simply injective and surjectivehomomorphisms, respectively. Bimorphisms in these categories are precisely theequivalences (that is, isomorphisms of the corresponding structures); the conceptof a subobject of an object becomes the concept of a subgroup or a subring ora submodule in the categories of groups or rings or modules, respectively.

Let I be a set and let Aii∈I be a family of objects in a category C. A productof this family is a pair (P, pii∈I) where P is an object of C and pi (i ∈ I) are


morphisms in Mor(P,Ai) satisfying the following condition. If Q is an objectof C and for every i ∈ I there is a morphism qi : Q → Ai, then there existsa unique morphism η : Q → P such that qi = piη for all i ∈ I. Similarly, acoproduct of a family Aii∈I of objects in C is defined as a pair (S, sii∈I)where S is an object of C and si (i ∈ I) are morphisms in Mor(Ai, S) with thefollowing property. If T is an object of C and for every i ∈ I there is a morphismti : Ai → T , then there exists a unique morphism ζ : S → T such that ti = ζsi

for all i ∈ I.The product and coproduct of a family of objects Aii∈I are denoted by∏

i∈I Ai and∐

i∈I Ai, respectively. It is easy to see that if a product (or coprod-uct) of a family of objects exists, then it is unique up to an isomorphism. Noticethat the concept of a coproduct in the categories Grp, Ab, Ring, RMod, andModR coincides with the concept of a direct sum of the corresponding algebraicstructures. In this case we use the symbol

⊕rather than

∐.

A category C is called preadditive if for every pair A,B of its objects,Mor(A,B) is an Abelian group satisfying the following axioms:

(a) The composition of morphisms Mor(B,C)×Mor(A,B) → Mor(A,C) isbilinear, that is, f(g1 + g2) = fg1 + fg2 and (f1 + f2)g = f1g + f2g wheneverthe compositions are defined;

(b) C contains a zero object 0.

If C is a preadditive category, then the following conditions are equivalent:(1) C has an initial object; (2) C has a terminal object; (3) C has a zero object.The proof of this fact, as well as the proof of the following statement, can befound, for example, in [15, Vol. 2, Section 1.2].

Proposition 1.1.9 Given two objects A and B in a preadditive category C, thefollowing conditions are equivalent.

(i) The product (P, pA, pB) exists.(ii) The coproduct (S, sA, sB) exists.(iii) There exists an object P and morphisms pA : P → A, pB : P → B,

sA : A → P , and sB : B → P such that pAsA = 1A, pBsB = 1B, pAsB = 0,pBsA = 0, and sApB + sBpA = 1P .

If A and B are two objects in a preadditive category C, then a quintuple(P, pA, pB , sA, sB) satisfying condition (iii) of the last proposition is called abiproduct of A and B.

A preadditive category which satisfies one of the equivalent conditions of thelast proposition is called additive. A functor F from an additive category C toan additive category D is called additive if for any two objects A and B in Cand for any two morphisms f, g ∈ Mor(A,B), F (f + g) = F (f) + F (g).

Let C be a category with a zero object and let f : A → B be a morphism inC. We say that a monomorphism α : M → A (or a pair (M,α)) is a kernel of fif (i) fα = 0 and (ii) whenever fν = 0 for some morphism ν : N → A, thenthere exists a unique morphism τ : N → M such that ν = ατ . In this case M iscalled the kernel object; it is often referred to as the kernel of f as well. Dually,


a cokernel of a morphism f : A → B is an epimorphism β : B → C (or a pair(C, β)) such that βf = 0, and for any morphism γ : B → D with γf = 0 thereexists a unique morphism µ : C → D such that γ = µf . The object C is calleda cokernel object; it is often referred to as the cokernel of f as well.

An additive category C is called Abelian if it satisfies the following conditions.

(a) Every morphism in C has a kernel and cokernel.(b) Every monomorphism in C is the kernel of its cokernel.(c) Every epimorphism in C is the cokernel of its kernel.

It can be shown (see [15, Vol. II, Proposition 1.5.1]) that a morphism in anAbelian category is a bimorphism if and only if it is an isomorphism.

As it follows from the classical theories of groups, rings and modules, thecategories Grp, Ab, Ring, RMod, and ModR (R is a fixed ring) are Abelian.The concepts of kernel and cokernel of a morphism in each of these categoriescoincide with the concepts of kernel and cokernel of a homomorphism. Indeed,consider, for example, the category of all left modules over a ring R, wherea kernel and a cokernel of a homomorphism f : A → B are defined as leftR-modules N = a ∈ A | f(a) = 0 and P = B/f(A), respectively. Thenthe embedding N → A and the canonical epimorphism B → B/f(A) can benaturally treated as kernel and cokernel of the morphism f in RMod (in thesense of the category theory).

It is easy to prove (see, for example, [19, Propositions 5.11, 5.12]) that ifA is an object of an Abelian category C, (A′, i) is a subobject of A (that is,i : A′ → A is a representative of an equivalence class of monomorphisms) and(A′′, j) is a factor object defined by the cokernel of i, then Ker j = (A′, i).Dually, if (A′′, j) is a factor object of A (that is, j : A → A′′ is a representativeof an equivalence class of epimorphisms) and (A′, i) is a subobject of A definedby the kernel of j, then Coker i = (A′′, j). These two statements give the one-to-one correspondence between the classes of all subobjects and all factor objectsof an object A. In what follows, the factor object corresponding to a subobjectA′ of A will be denoted by A/A′. (This notation agrees with the usual notationfor algebraic structures; for example, if A is an object of the category RMod,then A/A′ is the factor module of a module A by its submodule A′.) The proofof the following classical result can be found, for example, in [83, Chapter 2,Section 9].

Proposition 1.1.10 In an Abelian category C, every morphism f : A → Bhas a factorization f = me, with m : N → B monic and e : A → N epi;moreover, m = Ker(Coker f) and e = Coker(Ker f). Furthermore, given any

other factorization f ′ = m′e′ (A′ e′−→ N ′ m′

−−→ B′) with m′ monic and e′ epi, andgiven two morphisms g : A → A′ and h : B → B′ such that f ′g = hf , thereexists a unique morphism k : N → N ′ such that e′g = ke and m′k = hm.

An object P in a category C is called projective if for any epimorphismπ : A → B and for any morphism f : P → B in C, there exists a morphismh : P → A such that f = πh. An object E in C is called injective if for any


monomorphism i : A → B and for any morphism g : A → E, there existsa morphism λ : B → E such that g = λi. An Abelian category C is said tohave enough projective (injective) objects if for every its object A there existsan epimorphism P → A (respectively, a monomorphism A → Q) where Pis a projective object in C (respectively, Q is an injective object in C). Theimportance of Abelian categories with enough projective and injective objectsis determined by the fact that one can develop the homological algebra in suchcategories (one can refer to [15 ] or [19])). Furthermore, ( see, for example, [149,Chapter 2]), every module is a factor module of a projective module and everymodule can be embedded into an injective module, so RMod and ModR areAbelian categories with enough projective and injective objects.

Let C be an Abelian category. By a (descending) filtration of an object A ofC we mean a family (An)n∈Z of subobjects of A such that An ⊇ Am wheneverm ≥ n. The subobjects An are said to be the components of the filtration. Anobject A together with some filtration of this object is said to be a filtered objector an object with filtration. (While considering filtered objects in categories ofrings and modules we shall often consider ascending filtration which are definedin a dual way.)

If B is another filtered objects in C with a filtration (An)n∈Z, then a mor-phism f : A → B is said to be a morphism of filtered objects if f(An) ⊆ Bn

for every n ∈ Z. It is easy to see that filtered objects of C and their morphismsform an additive (but not necessarily Abelian) category. An associated gradedobject for a filtered object A with a filtration (An)n∈Z is a family grnAn∈Z

where grnA = An/An+1.A spectral sequence in C is a system of the form E = (Ep,q

r , En), p, q, r ∈Z, r ≥ 2 and the family of morphisms between these objects described as follows.

(i) Ep,qr | p, q, r ∈ Z, r ≥ 2 is a family of objects of C.

(ii) For every p, q, r ∈ Z, r ≥ 2, there is a morphism dp,qr : Ep,q

r → Ep+r,q−r+1r .

Furthermore, dp+r,q−r+1r dp,q

r = 0.(iii) For every p, q, r ∈ Z, r ≥ 2, there is an isomorphism αp,q

r : Ker(dp,qr )/Im

(dp−r,q+r−1r ) → Ep,q

r+1.(iv) En |n ∈ Z is a family of filtered objects in C.(v) For every fixed pair (p, q) ∈ Z2, dp,q

r = 0 and dp−r,q+r−1r = 0 for all

sufficiently large r ∈ Z (that is, there exists an integer r0 such that the lastequalities hold for all r ≥ r0). It follows that for sufficiently large r, an objectEp,q

r (p, q ∈ Z) does not depend on r; this object is denoted by Ep,q∞ .

(vi) For every n ∈ Z, the components Eni of the filtration of En are equal to

0 for all sufficiently large i ∈ Z; also Eni = E for all sufficiently small i.

(vii) There are isomorphisms βp,q : Ep,q∞ → grpE

p+q.

The family Enn∈Z (without filtrations) is said to be the limit of the spec-tral sequence E. A morphism of a spectral sequence E = (Ep,q

r , En) to a spectralsequence F = (F p,q

r , Fn) is defined as a family of morphisms up,qr : Ep,q

r → F p,qr

and filtered morphisms un : En → Fn which commute with the morphisms dp,qr ,

αp,qr , and βp,q. The category of spectral sequences in an Abelian category form


an additive (but not Abelian) category. An additive functor from an Abeliancategory to a category of spectral sequences is called a spectral functor. A spec-tral sequence is called cohomological if Ep,q

r = 0 whenever at least one of theintegers p, q is negative. In this case, Ep,q

r = Ep,q∞ for r > maxp, q +1, En = 0

for n < 0, and the m-th component of the filtration of En is 0, if m > 0, andEn, if m ≤ 0.

4. Algebraic structures. Throughout the book, by a ring we alwaysmean an associative ring with an identity (usually denoted by 1). Every ringhomomorphism is unitary (maps an identity onto an identity), every subring ofa ring contains the identity of the ring, every module, as well as every algebraover a commutative ring, is unitary (the multiplication by the identity of thering is the identity mapping of the module or algebra). By a proper ideal of aring R we mean an ideal I of R such that I = R.

An injective (respectively, surjective or bijective) homomorphism of rings,modules or algebras is called monomorphism (respectively, epimorphism or iso-morphism) of the corresponding algebraic structures. An isomorphism of anytwo algebraic structures is denoted by ∼= (if A is isomorphic to B, we writeA ∼= B). The kernel, image, and cokernel of a homomorphism f : A → B aredenoted by Ker f , Imf (or f(A)), and Coker f , respectively. (This terminologyagrees with the corresponding terminology for objects in Abelian categories).

If A and B are (left or right) modules over a ring R or if they are algebras overa commutative ring R, the set of all homomorphisms from A to B is denoted byHomR(A,B). This set is naturally considered as an Abelian group with respectto addition (f, g) → f +g defined by (f +g)(x) = f(x)+g(x) (x ∈ A). As usual,a homomorphism of an algebraic structure into itself is called an endomorphism,and a bijective endomorphism is called an automorphism.

A pair of homomorphisms of modules (rings, algebras) Af−→ B

g−→ C is said tobe exact at B if Ker g = Imf . A sequence (finite or infinite) of homomorphisms

. . .fn−1−−−→ An−1

fn−→ Anfn+1−−−→ An+1 → . . . is exact if it is exact at each An; i.

e., for each successive pair fn, fn+1, Imfn = Ker fn+1. For example, every

homomorphism f : A → B produces the exact sequence 0 → Ker fi−→ A

f−→B

π−→ Coker f → 0 where i is the inclusion mapping (also called an embedding)and π is the natural epimorphism B → B/Imf . An exact sequence of the form

0 → Af−→ B

g−→ C → 0 is called a short exact sequence.The direct product and direct sum of a family of (left or right) modules

(Mi)i∈I over a ring R are denoted by∏

i∈I Mi and⊕

i∈I Mi, respectively. (Ifthe index set I is finite, I = 1, . . . , n, we also write

∏ni=1 Mi and

⊕ni=1 Mi; if

I = N+, we write∏∞

i=1 Mi and⊕∞

i=1 Mi, respectively.)Let R and S be two rings. A functor F : RMod → SMod is called left

exact if whenever 0 → Ai−→ B

j−→ C → 0 is an exact sequence in RMod, then

the sequence 0 → F (A)F (i)−−−→ F (B)

F (j)−−−→ F (C) is exact, if F is covariant, or

the sequence 0 → F (C)F (j)−−−→ F (B)

F (i)−−−→ F (A) is exact if F is contravariant.Similarly, a functor G : RMod → SMod is right exact if whenever 0 → A

i−→


Bj−→ C → 0 is an exact sequence in RMod, then the sequence F (A)

F (i)−−−→F (B)

F (j)−−−→ F (C) → 0 is exact, if F is covariant, or the sequence F (C)F (j)−−−→

F (B)F (i)−−−→ F (A) → 0 is exact if F is contravariant. The exactness of functors

between categories of right modules is defined similarly.A chain complex A is sequence of R-modules and homomorphisms · · · →

An+1dn+1−−−→ An

dn−→ An−1 → . . . (n ∈ Z) such that dndn+1 = 0 for each n. Thesubmodule Ker dn of An is denoted by Zn(A); its elements are called cycles.The submodule Imdn+1 of An is denoted by Bn(A); its elements are calledboundaries. Bn(A) ⊆ Zn(A) since dndn+1 = 0, and Hn(A) = Zn(A)/Bn(A) iscalled the nth homology module of A. The chain complex A is called exact if itis exact at each An (in this case Hn(A) = 0 for all n).

As we have mentioned, if R is a ring then RMod and ModR (as well asAb) are Abelian categories. The projective and injective objects of these cate-gories are called, respectively, projective and injective (right or left) R-modules.The following two propositions summarize basic properties of these concepts.The statements, whose proofs can be found, for example, in [176, Chapter 3],are formulated for left R-modules called “R-modules”; the results for right R-modules are similar.

Proposition 1.1.11 Let R be a ring. Then(i) An R-module P is projective if and only if P is a direct summand of a

free R-module. In particular, every free R-module is projective.(ii) A direct sum of R-modules

⊕j∈J Pj is projective if and only if every Pj

(j ∈ J) is a projective R-module.(iii) An R-module P is projective if and only if the covariant functor

HomR(P, ·) : RMod → Ab is exact, that is, every exact sequence of R-modules

0 → Ai−→ B

j−→ C → 0 induces the exact sequence of Abelian groups 0 →HomR(P,A) i∗−→ HomR(P,B)

j∗−→ HomR(P,C) → 0 (i∗(φ) = iφ, j∗(ψ) = jψ

for any φ ∈ HomR(P,A), ψ ∈ HomR(P,B)).(iv) Every projective module is flat, that is, P

⊗R · is an exact functor from

RMod to Ab. (In the case of right R-modules, the flatness means the exactnessof the functor ·⊗R P : ModR → Ab.)

(v) Any R-module A has a projective resolution, that is, there exists an

exact sequence P : · · · → Pn+1dn+1−−−→ Pn

dn−→ Pn−1 → . . .d1−→ P0

ε−→ A → 0 inwhich every Pn is a projective R-module.

Proposition 1.1.12 Let R be a ring. Then(i) (Baer’s Criterion) A left (respectively, right) R-module M is injective if

and only if for any left (respectively, right) ideal I of R every R-homomorphismI → M can be extended to an R-homomorphism R → M .

(ii) A direct product of R-modules∏

j∈J Ej is injective if and only if every Ej

(j ∈ J) is an injective R-module.


(iii) An R-module E is injective if and only if the contravariant functorHomR(·, E) : RMod → Ab is exact.(iv) Every R-module can be embedded into an injective R-module.(v) Any R-module M has an injective resolution, that is, there exists an

exact sequence E : 0 → Mη−→ E0

d0−→ E1d1−→ · · · → En−1

dn−1−−−→ Endn−→

En+1 → . . . in which every En is an injective R-module.

Let R and S be two rings and F : RMod → SMod a covariant functor.

Let · · · → Pn+1dn+1−−−→ Pn

dn−→ Pn−1 → . . .d1−→ P0

ε−→ M → 0 be a projectiveresolution of a leftR-module M (we shall denote it by (P, dn)). If we apply Fto this resolution and delete F (M), we obtain a chain complex (F (P), F (dn)) :

· · · → F (Pn+1)F (dn+1)−−−−−→ F (Pn)

F (dn)−−−−→ F (Pn−1) → . . .F (d1)−−−−→ P0

F (d0)−−−−→ 0 (withd0 = 0). It can be shown (see, for example, [176, Chapter 5]) that every homologyof the last complex is independent (up to isomorphism) of the choice of a projec-tive resolution of M . We set LnF (M) = Hn(F (P)) = Ker F (dn)/Im F (dn+1).Also, if N is another left R-module and (P′, d′n) is any its projective resolution,then every R-homomorphism φ : M → N induces R-homomorphisms of ho-mologies Lnφ : Hn(F (P)) → Hn(F (P′)). We obtain a functor LnF called thenth left derived functor of F .

If we apply F to an injective resolution (E, dn) of an R-module M and dropF (M), then the n-th homology of the result complex is, up to isomorphism, in-dependent of the injective resolution. This homology is denoted by RnF (M). Asin the case of projective resolutions, every homomorphism of left R-modules φ :M → N induces R-homomorphisms of homologies Rnφ : RnF (M) → RnF (N),so we obtain a functor RnF called the nth right derived functor of F . Theconcepts of left and right derived functors for a contravariant functor can beintroduced in a similar way (see [176, Chapter 5] for details).

The following result will be used in Chapter 3 where we consider the categoryof inversive difference modules.

Theorem 1.1.13 Let C = AMod, C′ = BMod, and C′′ = CMod be the cate-gories of left modules over rings A, B, and C, respectively. Let F : C → C′ andG : C′ → C′′ be two covariant functors such that G is left exact and F mapsevery injective A-module M to a B-module F (M) annulled by every right derivedfunctor RqG (q > 0) of the functor G. Then for every left A-module N , thereexists a spectral sequence in the category C′′ which converges to Rp+q(GF )(N)(p, q ∈ N) and whose second term is of the form Ep,q

2 = RpG(RqF (N)).

If R is a commutative ring and Σ ⊆ R, then (Σ) will denote the ideal of Rgenerated by the set Σ, that is, the smallest ideal of R containing Σ. If R0 is asubring of R, we say that R is a ring extension or an overring of R0. In such acase, if B ⊆ R, then the smallest subring of R containing R0 and B is denotedby R0[B]; if B is finite, B = b1, . . . , bm, this smallest subring is also denotedby R0[b1, . . . , bm]. If R = R0[B], we say that B is the set of generators of R overR0 or that R is generated over R0 by the set B. In this case every element of

1.2. ELEMENTS OF THE THEORY OF COMMUTATIVE RINGS 15

R can be written as a finite linear combination of power products of the formbk11 . . . bkn

n (b1, . . . , bn ∈ B) with coefficients in R0.If K is a subfield of a field L, we say that L is a field extension or an overfield

of K. We also say that we have a filed extension L/K (or “K ⊆ L is a fieldextension”). If B ⊆ L, then K(B) will denote the smallest subfield of L con-taining K and B (if B is finite, B = b1, . . . , bm, we also write K(b1, . . . , bm) ).If L = K(B), the set B is called a set of generators of L over K. Clearly, K(B)coincides with the quotient field of the ring K[B].

If R[X1, . . . , Xn] is an algebra of polynomials in n variables X1, . . . , Xn overa ring R, then the power products M = Xk1

1 . . . Xknn (k1, . . . , kn ∈ N) will be

called monomials. The degree (or total degree) of such a monomial is the sumk1 + · · ·+kn. The number ki is called the degree of the monomial with respect toXi (or relative to Xi). If f is a polynomial in R[X1, . . . , Xn], then f has a uniquerepresentation as a linear combination of distinct monomials M1, . . . , Mr withnonzero coefficients. The maximum of degrees of these monomials is called thedegree (or total degree) of the polynomial f ; it is denoted by deg f . The maximalof degrees of the monomials Mk (1 ≤ k ≤ r) with respect to Xi (1 ≤ i ≤ n) iscalled the degree of f with respect to (or relative to) Xi; it is denoted by degXi

f .(If f = 0, we set deg f = −1 and degXi

f = −1 for i = 1, . . . , n.) If f is apolynomial in one variable t over a ring R and deg f < m (m ∈ N), we writef = o(tm).

1.2 Elements of the Theory of CommutativeRings

In this section we review some classical results of commutative algebra that playan important role in the theory of difference rings and fields. As we shall seelater, many ideas and constructions in difference algebra have their roots in thetheory of commutative rings, so the material of this section should be considerednot only as a source of references that makes the book self-contained, but alsoas an exposition of these roots. The proofs of the statements can be found intexts on commutative algebra such as [3], [57], [115] and [138].

Multiplicative sets. Prime and maximal ideals. Let A be a commu-tative ring. A set S ⊆ A is said to be multiplicatively closed or multiplicative if1 ∈ S and st ∈ S whenever s ∈ S and t ∈ S. A multiplicatively closed set S iscalled saturated if for any a, b ∈ A, the inclusion ab ∈ S implies that a ∈ S andb ∈ S.

A proper ideal P of A is said to be prime if it satisfies one of the followingequivalent conditions:

(i) The set A \ P is multiplicatively closed;(ii) For any two elements a, b ∈ P , the inclusion ab ∈ P implies a ∈ P or

b ∈ P .


(iii) For any two ideals I and J in A, the inclusion IJ ⊆ P implies I ⊆ P orJ ⊆ P . (As usual, the product IJ of two ideals is defined as the ideal generatedby all products ij with i ∈ I and j ∈ J .)(iv) The factor ring A/P is an integral domain.

The following proposition summarizes some basic properties of prime ideals.

Proposition 1.2.1 Let P,P1, . . . , Pn be prime ideals of a commutative ring A.(i) If I1, . . . , In are ideals in A such that

⋂ni=1 Ii ⊆ P , then Ik ⊆ P for some

k. Furthermore, if⋂n

i=1 Ii = P , then P = Ik for some k (1 ≤ k ≤ n).(ii) If I is an ideal in A and I ⊆ ⋃n

i=1 Pi, then I ⊆ Pi for some i (1 ≤ i ≤ n).(iii) Let f be a homomorphism of A to some other commutative ring B and

let Q be a prime ideal in B. Then f−1(Q) is a prime ideal in A. Furthermore,there is a one-to-one correspondence between the prime ideals of the ring f(A)and prime ideals of A containing Ker f .(iv) Let S be a multiplicative set in A such that 0 /∈ S. Let I be the ideal in A

such that I⋂

S = ∅ and I is maximal among all ideals of A whose intersectionwith S is empty. Then the ideal I is prime.

(v) Let M be an A-module and let I be an ideal in A that is maximal amongall annihilators of non-zero elements of M . (Here and below we assume that theset of ideals of A is partially ordered by inclusion.) Then I is prime.(vi) Let I be an ideal in A. Suppose that I is not finitely generated and is

maximal among all ideals that are not finitely generated. Then I is prime.(vii) Let Pii∈I be a descending chain of prime ideals in A. Then

⋂i∈I Pi is

a prime ideal of A. It follows that every prime ideal of A contains a minimal(with respect to inclusion) prime ideal.

An ideal M of a commutative ring A is said to be maximal if M is themaximal member (with respect to inclusion) of the set of proper ideals of A.(By Zorn’s lemma, at least one such an ideal always exist. Moreover, every idealof A is contained in a maximal ideal.) Clearly, M is a maximal ideal of A if andonly if A/M is a field. (Therefore, every maximal ideal is prime.)

Exercises 1.2.2 Let A be a commutative ring.1. Prove that S ⊆ A is a saturated multiplicative subset of A if and only if

A \ S is a union of (possibly empty) family of prime ideals of A.2. Let S be a multiplicative subset of A and let S′ be the intersection of all

saturated subsets of A containing S. Show that S′ is a saturated multiplicativeset. (Thus, S′ is the smallest saturated multiplicative set containing S; it iscalled the saturation of S.)

3. Let S be a multiplicative subset of A and S′ a saturation of S. Prove thatA \ S′ is the union of all prime ideals Q of A such that Q ∩ S = ∅.

4. Let S = 1. Show that the saturation S′ consists of all units of the ring.5. Prove that the set of all non-zero-divisors in A is a saturated multiplica-

tive set.


6. An element p ∈ A is said to be a principal prime if the principal ideal(p) is prime and non-zero. Show that if A is an integral domain, then the setof all elements in A expressible as a product of principal primes constitute asaturated multiplicative set.

7. Let A = K[X1, . . . , Xn] be a polynomial ring in n variables X1, . . . , Xn

over a field K and let f ∈ A. Prove that the principal ideal (f) is prime if andonly if the polynomial f is irreducible.

8. Let A[X] denote the ring of polynomials in one variable X over A. Provethat if P is a prime ideal of A, then P [X] is a prime ideal of A[X]. (P [X] denotesthe set of all polynomials with coefficients in P .)

9. Show that if A is a principal integral domain (that is an integral domainwhere every ideal is principal), then every prime ideal of A is maximal.

10. Let I be maximal among all non-principal ideals of A. Prove that I isprime.

11. Let I be maximal among all ideals of A that are not countably generated.Show that I is prime.

12. Let Pii∈I be an ascending chain of prime ideals in A. Prove that⋃

i∈I Pi

is a prime ideal of A.13. Let I be an ideal in A and let P be a prime ideal of A containing I.

Show that P contains a minimal prime ideal of A containing I. [Hint: Applythe the Zorn’s lemma to the set of all prime ideals P ⊇ I ordered by the reverseinclusion: P1 ≤ P2 if and only if P1 ⊇ P2.]

The last exercise justifies the following definition: if I is an ideal of a com-mutative ring A, then a prime ideal P of A is said to be minimal over I (or aminimal or an isolated prime ideal of I) if P is minimal among all prime idealscontaining I.

A commutative ring A with exactly one maximal ideal m is called a localring. In this case the field k = A/m is said to be the residue field of A.

Examples 1.2.3 1. Let K[[X]] be the ring of formal power series in a variableX over a field K. It is easy to see that an element f =

∑∞n=0 anXn ∈ K[[X]]

has an inverse in K[[X]] if and only if a0 = 0. (In this case f = a0(1+Xg) withsome g ∈ K[[X]] and f−1 = a−1

0 (1 − Xg + X2g2 − . . . .) It follows that K[[X]]is a local ring with the maximal ideal (X).

2. Let K be the field R or C. It is known from real (and complex) analysisthat a power series f =

∑∞n=0 anXn in one variable X over K has a positive

radius of convergence if and only if lim sup log|an|n < ∞. In this case f is con-

vergent on a disc around 0, and can be viewed as an analytic function near 0.Let KX be the set of power series with positive radii of convergence. As itfollows from elementary properties of analytic functions, for every f ∈ KX,the function f−1 = 1/f is an analytic function represented by a convergentpower series if and only if a0 = 0. Thus, KX is a local ring with the maximalideal (X).


Exercises 1.2.4 Let A be a commutative ring.1. Prove that A is local if and only if the set of all non-units of A is an ideal.2. Let M be a maximal ideal of A such that every element 1+x with x ∈ M

is a unit. Show that A is a local ring.

The set of all prime ideals of a commutative ring A is called the spectrumof A; it is denoted by SpecA. The set of all maximal ideals of A is called themaximum spectrum of A, and written m-SpecA. For every ideal I of A, letV (I) = P ∈ SpecA |P ⊇ I. It is easy to show that if J is another idealof A, then V (I)

⋃V (J) = V (I

⋂J) = V (IJ) and for any family Iλ |λ ∈ Λ

of ideals of A we have⋂

λ∈Λ V (Iλ) = V (∑

λ∈Λ Iλ). It follows that the familyF = V (I) | I is an ideal of A is closed under finite unions and arbitraryintersections, so that there is a topology on SpecA for which F is the set ofclosed sets. Any homomorphism f : A → B of A to some other commutative ringB induces a mapping f∗ : SpecB → SpecA defined by taking a prime ideal Qof B into f−1(Q). Clearly, (f∗)−1(V (I)) = V (f(I)B) for any ideal I in A, so f∗

is continuous. Also, if g is a homomorphism of B into some other commutativering C, then (gf)∗ = f∗g∗, so that the correspondence A → SpecA, f → f∗,defines a contravariant functor from the category of rings to the category oftopological spaces.

If I is an ideal of a commutative ring A, then the intersection of all primeideals of A containing I is called the radical of I; it is denoted by r(I). The idealr((0)) is called the nilradical of the ring A; it is denoted by rad(A). Theintersection of all maximal ideals of a commutative ring A is called the Jacobsonradical of A. It is denoted by J(A). The proofs of the following properties of theradical can be found, for example, in [3, Chapter 1].

Proposition 1.2.5 Let A be a commutative ring. Then(i) If I is an ideal of A, then r(I) = x ∈ A |xn ∈ I for some n ∈ N. In

particular, rad(A) is the set of all nilpotents of A (that is, the set of all elementsx ∈ A such that xn = 0 for some n ∈ N).

(ii) J(A) = a ∈ A | for any x ∈ A, 1 + ax is a unit in A.(iii) (The Nakayama lemma) Let M be a finitely generated A-module and I an

ideal of A such that IM = M . Then (1+x)M = 0 for some x ∈ I. Furthermore,if I ⊆ J(A), then M = 0.(iv) Let I be an ideal of A contained in J(A), M an A-module, and N an

A-submodule of M such that the A-module M/N is finitely generated and N +IM = M . Then M = N .

An ideal J of a commutative ring A is called radical if r(J) = J , that is,the inclusion xn ∈ J (x ∈ A,n ∈ N) implies that x ∈ J . (It is easy to seethat an ideal J is radical if the inclusion x2 ∈ J (x ∈ A) implies x ∈ J .) Thedefinition of the radical immediately implies that every radical ideal J in a com-mutative ring A is the intersection of the set of all prime ideals of A containing J .


Two ideals I and J of a commutative ring A are called coprime if I +J = A.Ideals I1, . . . , In are called pairwise coprime (or coprime in pairs) if Ik + Il = Awhenever 1 ≤ k, l ≤ n and k = l.

Exercises 1.2.6 Let I, I1 and I2 be ideals of a commutative ring A.1. Prove that r(I1+I2) = r(r(I1)+r(I2)), r(I1I2) = r(I1∩I2) = r(I1)∩r(I2),

and r(r(I)) = r(I) (so that r(I) is a radical ideal). Also, show that r(I) = A ifand only if I = A.

2. Show that if the ideal I is prime, then r(In) = I for every n ∈ N+.3. Prove that if f : A → B is a ring homomorphism and L an ideal of B,

then r(f−1(L)) = f−1(r(L)).4. Show that the intersection of any family of radical ideals is a radical ideal.5. Prove that if Q1 ⊆ Q2 ⊆ . . . is a chain of radical ideals of A, then the

union of all ideals of this chain is a radical ideal.6. For any set S ⊆ A, let S denote the smallest radical ideal of A contain-

ing S (in other words, S is the intersection of all radical ideals of A contain-ing S). Prove that if S, T ⊆ A and ST denote the set st | s ∈ S, t ∈ T, then

(a) S = r((S)) (as usual, (S) denote the ideal of A generated by S).(b) ST ⊆ ST;(c) S⋂T = ST.6. Show that if Q is a radical ideal of A and S ⊆ A, then the ideal Q : S =

a ∈ A | as ∈ Q for all s ∈ S is radical.7. Let A[X] be the ring of polynomials in one variable X over A. Prove the

following two statements.(a) J(A[X]) = rad(A[X]).(b) If I is a radical ideal of A, then I[X] is a radical ideal of A[X].8. Suppose that the ideals I1 and I, as well as the ideals I2 and I, are coprime.

Prove that the ideals I1I2 and I are also coprime.9. Let J1, . . . , Jn (n ≥ 2) be pairwise coprime ideals of A.

(a) Prove that J1J2 . . . Jn = J1

⋂J2

⋂ · · ·⋂ Jn and the ideals Jn andJ1 . . . Jn−1 are coprime.

(b) Consider a ring homomorphism f : A → (A/J1)⊗ · · ·⊗(A/Jn) given by

f(a) = (a + J1, . . . , a + Jn). Prove that Ker f = J1

⋂J2

⋂ · · ·⋂ Jn, so thatthe rings A/J1

⋂J2

⋂ · · ·⋂ Jn and (A/J1)⊗

. . . (A/Jn) are isomorphic. (Thisresult is known as the Chinese Remainder Theorem.)

The proof of the following result can be found in [99, Chapter II].

Theorem 1.2.7 Let L be a field, K a subfield of L, B the ring of polynomialsin a (possibly infinite) set of variables X over L, and A the ring of polynomialsin the same set of variables X over K. Let P be an ideal of A, J the ideal of Bgenerated by P , and I = r(J). Then

(i) If the ideal P is radical, then I⋂

A = P .


(ii) Suppose that P is a prime ideal of A and that ab ∈ I for some a ∈ A,b ∈ B. Then either a ∈ P or b ∈ I.(iii) Suppose that L is a field of zero characteristic and P A. Let x ∈ X and

let c ∈ L \ K. Then x − c /∈ I.

Rings and modules of fractions. Localization. If S is a multiplicativesubset of a commutative ring A, then one can construct a ring of fractions ofA with respect to S. This ring (denoted by S−1A) appears as follows. Considera relation ∼ on A × S such that (a, s) ∼ (b, t) if and only if u(ta − sb) = 0 forsome u ∈ S. It is easy to check that ∼ is an equivalence relation. Let us denotethe equivalence class of a pair (a, s) ∈ A × S by

a

s, and let S−1A denote the

set of all equivalence classes of ∼. Then S−1A becomes a ring if one defines the

addition and multiplication bya

s+

b

t=

ta + sb

stand

a

s

b

t=

ab

st, respectively.

The fact that the operations are well-defined and determine the ring structureon S−1A can be easily verified. The ring homomorphism f : A → S−1A givenby f(a) =

a

1is referred to as the natural homomorphism.

The ring of fractions S−1A, together with the natural homomorphism f :A → S−1A, has the following universal property. If B is another commutativering and g : A → B is a ring homomorphism such that g(s) is a unit of B forevery s ∈ S, then there is a unique homomorphism h : S−1A → B such thatg = hf . (In fact, h is defined by h(

a

s) = g(a)g(s)−1.) We leave the proof of this

statement to the reader as an exercise.It is easy to see that if A is an integral domain and S = A \ 0, then S−1A

is the standard field of fractions of A (also called the quotient field of A).

The construction of a ring of fractions can be easily generalized to the caseof modules. If M is a module over a commutative ring A and S a multiplicativesubset of A, one can consider a relation ∼ on M ×S such that (m, s) ∼ (m′, t) ifand only if u(tm− sm′) = 0 for some u ∈ S. As in the construction of a ring offractions, one can check that ∼ is an equivalence relation. The equivalence classof a pair (m, s) ∈ M ×S is denoted by

m

s. The set of all such equivalence classes

can be naturally considered as an S−1A-module if we setm

s+

m′

t=

tm + sm′

st

anda

s

m′

t=

ab

stfor any elements

m

s,m′

t∈ S−1M ,

a

s∈ S−1A. (The fact that the

operations are well-defined can be easily verified. We leave the verification tothe reader as an exercise.) This S−1A-module is called a module of fractions ofM with respect to S; it is denoted by S−1M . It is easy to see that the mappingg : M → S−1M defined by g(m) =

m

1is a homomorphism of A-modules

(S−1M is naturally treated as an A-module where am

s=

a

1m

s=

am

s). Clearly,

Ker g = m ∈ M | sm = 0 for some s ∈ S.An important example of a ring of fractions arises when a multiplicative

subset is a complement of a prime ideal. Let P be a prime ideal of a commutative


ring A and S = A \P . Then the ring of fractions S−1A is called the localizationof A at P ; it is denoted by AP . If M is an A-module, then the module of fractionsS−1M is denoted by MP and called the localization of M at P . The transitionfrom the ring A to the local ring AP (or from an A-module M to MP ) is alsocalled a localization of the ring A at P (or a localization of the A-module M atP , respectively).

If S is a multiplicative set in a commutative ring A, then every homomor-phism of A-modules f : M → N induces a homomorphism of S−1A-modules

S−1f : S−1M → S−1N such that S−1f(m

s) =

f(m)s

for anym

s∈ S−1M . The

following proposition gives some properties of such induced homomorphisms(the proofs can be found, for example, in [3, Chapter 3]).

Proposition 1.2.8 Let A be a commutative ring and S a multiplicative set in A.(i) If M1 and M2 are A-submodules of an A-module M , then S−1(M1+M2) =

S−1M1 + S−1M2, S−1(M1 ∩ M2) = S−1M1

⋂S−1M2, and the S−1A-modules

S−1(M/M1) and S−1M/S−1M1 are isomorphic.(ii) Let f : M → N and g : M → P be homomorphisms of A-modules. Then

(a) S−1(fg) = (S−1f)(S−1g);

(b) If the sequence 0 → Mf−→ N

g−→ P → 0 is exact, then the corresponding

sequence 0 → S−1MS−1f−−−→ S−1N

S−1g−−−→ S−1P → 0 is also exact.(c) There exists a unique isomorphism of S−1A-modules φ : S−1A

⊗A

M → S−1M such that φ(a

s

⊗m) =

am

sfor every

a

s∈ S−1A,m ∈ M . (This

property, in particular, shows that S−1A is a flat A-module.)

(iii) If M and N are two A-modules, then there exists a unique isomorphism

ψ : S−1M⊗

S−1A S−1N → S−1(M⊗

A N) such that ψ(m

s

⊗D

n

t) =

m⊗

n

stfor any

m

s∈ S−1M ,

n

t∈ S−1N . In particular, if P is a prime ideal of A, then

MP

⊗AP

NP∼= (M

⊗N)P .

(iv) Let f : A → S−1A be the natural homomorphism and let for any ideal Iin A, S−1I denote the ideal of S−1A generated by f(I). Then

(a) If I is an ideal of A and J is an ideal of S−1A, then f−1(S−1I) ⊇ Iand S−1(f−1(J)) = J . Furthermore, S−1r(I) = r(S−1I) and f−1(r(J)) =r(f−1(J)).

(b) If I1, I2 are ideals of A and J1, J2 are ideals of S−1A, then S−1(I1

⋂I2) =

S−1I1

⋂S−1I2,S−1(I1+I2)=S−1I1+S−1I2,f−1(J1

⋂J2)=f−1(J1)

⋂f−1(J2),

and f−1(J1 + J2) = f−1(J1) + f−1(J2).(c) If P is a prime ideal of A and P ∩ S = ∅, then S−1P is a prime ideal of

S−1A and f−1(S−1P ) = P . Thus, the mapping P → S−1P gives a one-to-onecorrespondence between the set of all prime ideals P of A such that P

⋂S = ∅

and the set of all prime ideals of S−1A. (The inverse mapping is Q → f−1(Q)).Moreover, this correspondence preserves the inclusion of the ideals.


Noetherian and Artinian rings and modules. Let A be a commutativering and M an A-module. We say that M satisfies and ascending (respectively,descending) chain condition for submodules if any sequence of its A-submodulesM1 ⊆ M2 ⊆ . . . (respectively, M1 ⊇ M2 ⊇ . . . ) terminates (that is, there is aninteger n such that Mn = Mn+1 = . . . ). An A-module M is said to satisfy themaximum (respectively, minimum) condition if every non-empty family of sub-modules of M , ordered by inclusion, contains a maximal (respectively, minimal)element.

A module M over a commutative ring A is called Noetherian if it satisfiesone of the following three equivalent conditions.

(1) M satisfies the ascending chain condition.(2) M satisfies the maximum condition.(3) Every submodule of M (including M itself) is finitely generated over A.

An A-module M is said to be Artinian if it satisfies one of the following twoequivalent conditions.(1′) M satisfies the descending chain condition.(2′) M satisfies the minimum condition.

Exercise 1.2.9 Prove the equivalence of the conditions (1)−(3) and the equiv-alence of the conditions (1′) and (2′).

A ring is called Noetherian (respectively, Artinian) if it is a Noetherian (re-spectively, Artinian) A-module (with ideals as A-submodules).

Exercise 1.2.10 Prove that a commutative ring A is Noetherian if and onlyif every its prime ideal is finitely generated. [Hint: Show that if the set of non-finitely generated prime ideals of a commutative ring is not empty, then this setcontains a maximal element with respect to inclusion, and it is a prime ideal.]

The following two theorems summarize some basic properties of Noetherianand Artinian rings and modules.

Theorem 1.2.11 Let A be a commutative ring.(i) If A is Noetherian (Artinian), then every finitely generated A-module is

Noetherian (respectively, Artinian).(ii) Let 0 → M ′ → M → M ′′ → 0 be an exact sequence of A-modules. Then

the module M is Noetherian (Artinian) if and only if both M ′ and M ′′ areNoetherian (respectively, Artinian). In particular a submodule and a factor mod-ule of a Noetherian (Artinian) module are Noetherian (respectively, Artinian).(iii) A homomorphic image of a Noetherian (Artinian) ring is also Noetherian

(respectively, Artinian). In particular, if the ring A is Noetherian (Artinian)and I is an ideal of A, then the factor ring A/I is Noetherian (respectively,Artinian).


(iv) The direct sum⊕n

i=1 Mi of Noetherian (Artinian) A-modules M1, . . . , Mn

is a Noetherian (respectively, Artinian) A-module if and only if the modulesM1, . . . ,Mn are all Noetherian (respectively, Artinian).

(v) If M is Noetherian (Artinian) A-module and S is a multiplicative set inA, then the S−1A-module S−1M is also Noetherian (respectively, Artinian). Inparticular, if A is a Noetherian (Artinian) ring, then the ring of fractions S−1Ais also Noetherian (respectively, Artinian).(vi) (Hilbert basis theorem) Let the ring A be Noetherian and let A[X1, . . . , Xn]

be the ring of polynomials in n variables X1, . . . , Xn over A. Then the ringA[X1, . . . , Xn] is Noetherian.(vii) If the ring A is Noetherian, then every finitely generated A-algebra is alsoNoetherian.

Theorem 1.2.12 Let A be an Artinian commutative ring. Then(i) There is only finitely many maximal ideals in A.(ii) Every prime ideal of A is maximal. (Hence J(A) = radA.)(iii) The ring A is Noetherian.(iv) A is isomorphic to a direct sum of finitely many Artinian local rings.

Exercises 1.2.13 Let A be a Noetherian commutative ring.1. Prove that the ring of formal power series A[[X]] is Noetherian.2. Show that any ideal of A contains a power of its radical. In particular, the

nilradical of A is nilpotent.3. Prove that if I is an ideal of A and J =

⋂∞n=1 In, then J = IJ .

Exercises 1.2.14 Let A be an Artinian commutative ring.1. Suppose that the ring A is local and M is its maximal ideal. Prove that

the following statements are equivalent.(i) Every ideal of A is principal.(ii) The maximal ideal M is principal.(iii) If one considers M/M2 as a vector space over the field k = A/M , then

dimkM/M2 ≤ 1.2. Prove that there exist Artinian local rings A1, . . . , An such that A is iso-

morphic to the direct product ring∏n

i=1 Ai.

Theorem 1.2.15 Let A be a Noetherian ring.(i) (The Krull intersection theorem.) Let I be an ideal of A, M a finitely

generated A-module, and N =⋂∞

n=1 InM . Then IN = N and there is anelement r ∈ I such that (1 − r)N = 0.

(ii) If A is an integral domain or a local ring and I is a proper ideal of A,then

⋂∞n=1 In = 0.

Let M be a module over a commutative ring A. By a descending chainof submodules of M we mean a sequence of its submodules M = M0


M1 · · · Mn. The number n is said to be the length of the chain. Thechain is said to be a composition (or Jordan-Holder) series if Mn = 0 and everyA-module Mi/Mi−1 (1 ≤ i ≤ n) is simple. (Recall that an A-module S is saidto be simple if it has no submodules other than 0 and itself. For any 0 = x ∈ Swe then have S = Ax ∼= A/Q where Q = a ∈ A | ax = 0, the annihilatorof x, is a maximal ideal of A.) Equivalently, a composition series is a maximaldescending chain of submodules of M . The proof of the following theorem canbe found, for example, in [57, Section 2.4]. The third statement of the theoremshows that if an A-module M has a composition series, its length is an invariantof M independent of the choice of composition series. This invariant is calledthe length of M and written l(M) or lA(M). If M does not have compositionseries, we set l(M) = ∞.

Theorem 1.2.16 Let A be a commutative ring and let M be an A-module.(i) M has a composition series if and only if M is Noetherian and Artinian.(ii) If M has a composition series of length n, then every descending chain of

A-submodules of M has length ≤ n, and can be refined to a composition series.(iii) (Jordan-Holder Theorem.) If M = M0 M1 · · · Mn = 0 and

M = M0 M1 · · · Mm = 0 are two composition series of M , thenm = n and there exists a permutation π of the set 0, 1, . . . , n − 1 such thatMi/Mi+1

∼= Mπ(i)/Mπ(i)+1 for i = 0, 1, . . . , n − 1.(iv) The sum of the natural localization maps M → MP , for P a prime ideal,

gives an isomorphism of A-modules M ∼= ⊕P MP where the sum is taken over

all maximal ideals P such that some Mi/Mi+1 in a composition series M =M0 M1 · · · Mn = 0 is isomorphic to A/P . The number of Mi/Mi+1

isomorphic to A/P is the length of the AP -module MP , and is thus independentof the composition series chosen.

Exercises 1.2.17 Let A be a commutative ring.1. Let M be an A-module and P a maximal ideal of A. Prove that M = MP

if and only if M is annihilated by some power of P . [Hint: Apply the laststatement of Theorem 1.2.16.]

2. Let 0 → M ′ → M → M ′′ → 0 be a short exact sequence of A-modules.Prove that M is of finite length if and and only if both M ′ and M ′′ are of finitelength.

3. Let 0 → M1 → M2 → · · · → Mn → 0 be an exact sequence of A-moduleswhere each Mi has finite length. Prove that

∑ni=1(−1)ilA(Mi) = 0.

Primary decomposition. An ideal Q of a commutative ring A is calledprimary if Q = A and for any x, y ∈ A, the inclusion xy ∈ Q implies that eitherx ∈ Q or y ∈ r(Q) (that is, yn ∈ Q for some n ∈ N).

It is easy to see that if Q is a primary ideal, then the ideal P = r(Q) isprime. In this case we say that Q is a P -prime ideal of A.

The following exercise contains some properties of primary ideals.


Exercises 1.2.18 Let A be a commutative ring and Q a proper ideal of A.1. Show that if the ideal r(Q) is maximal, then Q is primary.2. Prove that if there is a maximal ideal M such that Mn ⊆ Q for some

n ∈ N, then the ideal Q is M -primary.3. Give an example of a non-primary ideal whose radical is a prime ideal.4. Let P be a prime ideal of A and let Q1, . . . , Qn be a finite family of

P -primary ideals of A. Prove that the ideal⋂n

i=1 Qi is also P -primary.5. Let Q be a primary ideal, P = r(Q), and x ∈ A. Prove that

Q : x =

⎧⎨⎩A, if x ∈ Q,Q, if x /∈ P,a P -primary ideal, if x ∈ P \ Q

(Recall that Q : x = a ∈ A | ax ∈ Q.)

Let I be an ideal of a commutative ring A. A representation of I as an inter-section of finitely many primary ideals of A is called a primary decompositionof I. If I =

⋂ni=1 Qi is a primary decomposition of I such that r(Qk) = r(Ql)

for k = l (1 ≤ k, l ≤ n) and Qi ⋂

j =i Qi for i = 1, . . . , n, then the pri-mary decomposition of I is said to be minimal or irredundant. It is clear (seeExercise 1.2.18.4) that if an ideal I has a primary decomposition (such an idealis called decomposable), it has a minimal primary decomposition as well. Thefollowing two theorems show that minimal primary decompositions have certainuniqueness properties.

Theorem 1.2.19 Let I be a decomposable ideal in a commutative ring A.(i) Suppose that I =

⋂ni=1 Qi is a minimal primary decomposition of I and

Pi = r(Qi), 1 ≤ i ≤ n. If P is a prime ideal of A, then the following statementsare equivalent:

(a) P = Pi for some i, 1 ≤ i ≤ n;(b) there exists a ∈ P such that I : a is a P -primary ideal;(c) there exists a ∈ P such that r(I : a) = P .

(ii) (The First Uniqueness Theorem for Primary Decomposition) Let I =⋂ni=1 Qi with r(Qi) = Pi (1 ≤ i ≤ n) and I =

⋂mj=1 Q′

j with r(Q′j) = P ′

j

(1 ≤ j ≤ m) be two minimal primary decompositions of I. Then m = n andP1, . . . , Pn = P ′

1, . . . , P′m.

Statement (ii) of Theorem 1.2.19 shows that the number of terms in aminimal primary decomposition of I does not depend on the choice of such adecomposition, and the set of prime ideals which occur as the radicals of theseprimary terms does not depend on the choice of the decomposition either. Thelast set is called the set of associated prime ideals of I and denoted by Ass I orAssAI. The members of Ass I are referred to as the associate prime ideals orassociated primes of I, and are said to belong to I.


Theorem 1.2.20 Let I be a decomposable ideal in a commutative ring A.(i) Let P be a prime ideal of A. Then P is minimal over I if and only if P

is a minimal element of Ass I (with respect to inclusion). Thus, the set ΣI ofall minimal prime ideals of I is a subset of Ass I; the prime ideals of the setAss I \ ΣI are called the embedded prime ideals of I.

(ii) (The Second Uniqueness Theorem for Primary Decomposition) LetAss I = P1, . . . , Pn and let I =

⋂ni=1 Qi and I =

⋂ni=1 Q′

i be two mini-mal primary decompositions of I with r(Qi) = r(Q′

i) = Pi (i = 1, . . . , n). Thenfor every i, 1 ≤ i ≤ n, for which Pi is a minimal prime of I, we have Q′

i = Qi.(In other words, in a minimal primary decomposition of I, the primary termcorresponding to a minimal prime ideal of I is uniquely determined by I and isindependent of the choice of minimal primary decomposition.)

Exercises 1.2.21 Let I =⋂n

i=1 Qi be a minimal primary decomposition of anideal I in a commutative ring A. Let Ass I = P1, . . . , Pn where Pi = r(Qi)(1 ≤ i ≤ n).

1. Prove that⋃n

i=1 Pi = x ∈ A | I : x = I. In particular, if the zero idealof A is decomposable, then the set of all zero divisors of A is the union of theprime ideals belonging to (0).

2. A set of prime ideals Σ ⊂ Ass I is said to be isolated if for any P ∈ Σ,P ′ ∈ Ass I, the inclusion P ′ ⊆ P implies P ′ ∈ Σ. Prove that if Σ = Pi1 , . . . , Pim

is an isolated set of prime ideals of I, then⋂m

k=1 Qikis independent of the choice

of minimal prime decomposition of I.

An ideal I of a commutative ring A is called irreducible if I = A and I cannotbe represented as an intersection of two proper ideals of A strictly containing I.If A is Noetherian, we have the following well-known result (see, for example,[3, Chapter 7]).

Theorem 1.2.22 Let A be a Noetherian commutative ring, Then(i) Every proper ideal of A can be represented as an intersection of finitely

many irredicible ideals.(ii) Every irreducible ideal of A is primary. (Thus, every proper ideal of A has

a primary decomposition and a minimal primary decomposition.)(iii) Let I be a proper ideal of A. Then AssA is precisely the set of all prime

ideals of the form I : a where a ∈ A.

Exercise 1.2.23 Let A be a commutative ring and A[X] a ring of polynomialsin one variable X over A. Prove that if I =

⋂ni=1 Qi is a minimal primary

decomposition of an ideal I in A, then I[X] =⋂n

i=1 Qi[X] is a minimal primarydecomposition of the ideal I[X] in A[X]. Furthermore, show that if P is aminimal prime ideal of I in A, then P [X] is a minimal prime ideal of I[X] inA[X].

Theorems 1.2.22 and 1.2.19 imply the following result on a representation ofa radical ideal as an intersection of prime ideals.


Theorem 1.2.24 Let A be a Noetherian commutative ring. Then(i) Every radical ideal of A can be represented as an intersection of finitely

many prime ideals. Moreover, a radical ideal I of A can be represented in theform I =

⋂ni=1 Pi where P1, . . . , Pn are prime ideals and Pi Pj if i = j (such

a representation of I as an intersection of prime ideals is called irredundant).(ii) The prime ideals in the irreduntant representation of a radical ideal of A

are uniquely determined.

Dimension of commutative rings. Let A be a commutative ring. Thesupremum of the lengths r taken over all strictly decreasing chains P0 P1 · · · Pr of prime ideals of A is called the Krull dimension, or simply thedimension of A, and denoted dimA. If P is a prime ideal of A, then the supre-mum of the length s taken over all strictly decreasing chains of prime idealsP = P0 P1 · · · Ps starting from P , is called the height of P , and denotedhtP . The supremum of the length t taken over all strictly increasing chainsof prime ideals P = P0 P1 · · · Pt starting from P , is called the co-height of P , and denoted coht P . Obviously, htP = dimAP , coht P = dim A/P ,and htP + coht P ≤ dimA. (Note that some authors call htP and coht P thecodimension and dimension of a prime ideal P , respectively.)

The height of an arbitrary ideal I of A is defined as the infimum of theheights of the prime ideals containing I. As in the case of a prime ideal, thecoheight of I is defined as dim A/I.

The following theorem summarizes basic properties of Krull dimension.

Theorem 1.2.25 Let A be a commutative ring. Then(i) The ring A is Artinian if and only if A is Noetherian and dimA = 0.(ii) Suppose that A is an integral domain which is finitely generated over a field

K. Then dimA = trdegKA. (The transcendence degree of A over K is definedas the transcendence degree of the quotient field of A over K, see Section 1.6for the definition and properties of the transcendence degree of a field exten-sion.) In particular, if A = K[X1, . . . , Xn] is a ring of polynomials in variablesX1, . . . , Xn over K, then dimA = n.(iii) (Krull’s Theorem). If A is Noetherian, then every prime ideal of A has

finite height. In fact, if P is a minimal prime ideal over an ideal I with rgenerators, then htP ≤ r.(iv) Let A be Noetherian and P a prime ideal of A of height d. Then there

exist elements x1, . . . , xd ∈ P such that P is a minimal prime ideal over theideal (x1, . . . , xd) and ht(x1, . . . , xk) = k for k = 1, . . . , d.

(v) If A is a Noetherian local ring with a maximal ideal m and k = A/m isthe corresponding residue field, then dimA ≤ dimkm/m2.

Exercises 1.2.26 Let A be a Noetherian commutative ring.1. Let I be an ideal of A. Show that ht I < ∞, and if x ∈ I is not a zero

divisor, then ht I/(x) = ht I − 1.2. Let A be a local ring with a maximal ideal M . Show that if x ∈ M is not

a zero divisor, then htA/(x) = htA − 1.


3. Let A[X] be the ring of polynomials in one variable X over A. Let Q bea prime ideal of A[X] and P = Q

⋂A. Prove the following statements:

(a) If Q = PA[X], then htQ = htP .(b) If Q = PA[X], then htQ = htP + 1.

The following theorem shows the relationships between the dimension ofa ring A and dimensions of polynomial rings over A. As before, A[X] andA[X1, . . . , Xn] denote, respectively, the ring of polynomials in one variable Xand variables X1, . . . , Xn over A.

Theorem 1.2.27 Let A be a commutative ring. Then(i) dimA + 1 ≤ dimA[X] ≤ 2dimA + 1.(ii) If the ring A is Noetherian, then dimA[X1, . . . , Xn] = dimA + n.(iii) (Macaulay’s Theorem). Let A be a field and let I be an ideal in

A[X1, . . . , Xn]. If I is generated by d elements and ht I = d, then all primeideals associated with I are also of height d.(iv) Let A be a field and let P1 and P2 be two prime ideals in A[X1, . . . , Xn].

If P is a minimal prime ideal of P1 + P2, then htP ≤ htP1 + htP2.

Integral extensions. Let A be a subring of a commutative ring B. Anelement b ∈ B is said to be integral over A if b is a root of a monic polynomialwith coefficients in A, that is, there exist elements a0, . . . , an−1 ∈ A (n ∈ N+)such that bn + an−1b

n−1 + · · · + a1b + a0 = 0. If every element of B is integralover A, we say that B is integral over A or B is an integral extension of A. Inthis case we also say that we have an integral extension B/A. The following twotheorems, summarizes some basic properties of integral elements and integralextensions.

Theorem 1.2.28 Let B be a commutative ring and A a subring of B.(i) If b ∈ B, then the following conditions are equivalent:(a) b is integral over A;(b) the ring A[b] is finitely generated as an A-module;(c) A[b] is contained in a subring of B which is finitely generated as an

A-module;(d) there exists a finitely generated A-module M ⊆ B such that bM ⊆ M and

for any c ∈ A[b], the equality cM = 0 implies c = 0.

(ii) The set A of all elements of B integral over A is a subring of B. (Thissubring is called the integral closure of A in B; if A = A, we say that A isintergally closed in B.)(iii) If elements b1, . . . , bn ∈ B are integral over A, then A[b1, . . . , bn] is a

finitely generated A-module.(iv) Let R be a subring of B containing A. Then the extension B/A is integral

if and only if R/A and B/R are integral.


(v) If B/A is integral and φ : B → C is a ring homomorphism, then φ(B) isintegral over φ(A).(vi) If B/A is integral and S is a multiplicative subset of A, then S−1B is

integral over S−1A.

An integral domain is said to be integrally closed if it is integrally closed inits quotient field.

Theorem 1.2.29 Let B be an integral extension of a commutative ring A.(i) If Q is a prime ideal of B, then Q is maximal if and only if the ideal

Q⋂

A of the ring A is maximal.(ii) If Q1 and Q1 are two prime ideals of B such that Q1 ⊆ Q2 and Q1

⋂A =

Q2

⋂A, then Q1 = Q2.

(iii) If P is a prime ideal of A, then there exists a prime ideal Q of the ring Bsuch that P = Q

⋂A. (In this case we say that Q lies over P .) More generally,

for every ideal I of B such that I⋂

A ⊆ P , there exists a prime ideal Q of Bwhich contains I and lies over P .(iv) (“Going up” Theorem) Let P1 ⊆ P2 ⊆ · · · ⊆ Pm ⊆ · · · ⊆ Pn be a chain

of prime ideals of A and let Q1 ⊆ Q2 ⊆ · · · ⊆ Qm (0 ≤ m < n) be a chainof prime ideals of B such that Qi

⋂A = Pi for i = 1, . . . , m. Then there exists

an extension of the last chain to a chain of prime ideals of B of the formQ1 ⊆ Q2 ⊆ · · · ⊆ Qm ⊆ · · · ⊆ Qn such that Qi

⋂A = Pi for i = 1, . . . , n.

(v) (“Going down” Theorem) Let A be an integrally closed domain no elementof which is a zero divisor in B (that is, if a ∈ A and b ∈ B are two nonzeroelements, then ab = 0). Let P1 ⊆ P2 ⊆ · · · ⊆ Pm ⊆ · · · ⊆ Pn (0 ≤ m < n) be achain of prime ideals of A and let Qm ⊆ · · · ⊆ Qn be a chain of prime ideals ofB such that Qi

⋂A = Pi for i = m, . . . , n. Then there exists an extension of the

last chain to a chain of prime ideals of B of the form Q1 ⊆ Q2 ⊆ · · · ⊆ Qm ⊆· · · ⊆ Qn such that Qi

⋂A = Pi for i = 1, . . . , n.

(vi) dimA = dimB.

Exercises 1.2.30 Let B be an integral extension of a commutative ring A.1. Prove that if I is an ideal of B, then B/I is an integral extension of

A/I⋂

A.2. Let B be an integral domain. Show that if every nonzero prime ideal of

A is maximal, then every nonzero prime ideal of B is maximal.3. Let I be an ideal of A. Prove that r(IB) can be described a set of all

elements b ∈ B which are roots of monic polynomials with coefficients in I.

Exercises 1.2.31 1. Prove that if A is an integrally closed domain and S amultiplicative subset of A, then S−1A is an integrally closed domain.

2. Show that a unique factorization domain is integrally closed.3. Show that if B is an integral extension of a commutative ring A, then

every homomorphism of A into an algebraically closed field can be extendedto B.


4. Use the result of the previous exercise to prove that every homomorphismof a field K into an algebraically closed field can be extended to every finitelygenerated ring extension of K.

5. Let R and S be subrings of a commutative ring A with R ⊆ S ⊆ A.Suppose that the ring R is Noetherian, A is finitely generated as an R-algebraand either A is finitely generated as an S-module or A is integral over S. Provethat S is finitely generated as an R-algebra.

Affine algebraic varieties and affine algebras. Let K be a field andL a field extension of K. In what follows, K[X1, . . . , Xn] will denote the ringof polynomials in variables X1, . . . , Xn over K, and An

K(L) will denote the n-dimensional affine space over L, that is, the set of all n-tuples (a1, . . . , an) withai ∈ L (i = 1, . . . , n). Such n-tuples will be also called points.

A subset V ⊆ AnK(L) is said to be an affine algebraic K-variety (or simply a

K-variety or a variety over K[X1, . . . , Xn]) if there are polynomials f1, . . . , fm

such that V is a solution of the system of equations fi(X1, . . . , Xn) = 0(i = 1, . . . , m), i.e., the set of all those a = (a1, . . . , an) ∈ Ln for whichfi(a1, . . . , an) = 0 (1 ≤ i ≤ m). The system of polynomial equations is said tobe a system of defining equations of V , K is said to be a field of definition, andL is called the coordinate field. (Notice that many authors define the conceptof an affine algebraic K-variety in the case L = K or when L is an algebraicclosure of K. We prefer not to specify the field L at the early stage.)

Since the ring K[X1, . . . , Xn] is Noetherian, the set of zeros of any set S ⊆K[X1, . . . , Xn] (that is, the set (a1, . . . , an) ∈ Ln | f(a1, . . . , an) = 0 for everyf ∈ S) is a K-variety. This K-variety is denoted by V (S). (If the set S consistsof a single element f , we write V (f) instead of V (f).) On the other hand, itis easy to see that if V ⊆ An

K(L), then the set f ∈ K[X1, . . . , Xn] | f(a) = 0for all a ∈ V is an ideal of K[X1, . . . , Xn]. This ideal is called the ideal (or thevanishing ideal) of V ; it is denoted by J(V ).

A K-variety V is called irreducible if it cannot be represented as a unionof two nonempty K-varieties strictly contained in V . The following propositiongives some properties of the correspondences S → V (S) (S ⊆ K[X1, . . . , Xn])and V → J(V ) (V ⊆ An

K(L)) described above. We leave the proof of theseproperties to the reader as an exercise.

Proposition 1.2.32 (i) J(∅) = K[X1, . . . , Xn]. Furthermore, if the field L isinfinite, then J(An

K(L)) = (0).(ii) For any V ⊆ An

K(L), J(V ) is a radical ideal of K[X1, . . . , Xn].(iii) If V ⊆ An

K(L) is a variety, then V (J(V )) = V .(iv) Let V1 and V2 be K-varieties. Then V1 ⊆ V2 if and only if J(V1) ⊇ J(V2).

Furthermore, V1 V2 if and only if J(V1) J(V2).(v) If V1 and V2 are K-varieties, then J(V1 ∪ V2) = J(V1)

⋂J(V2) and

V1

⋃V2 = V (J(V1)J(V2)). (It follows that a union of two K-varieties is a

K-variety and the same is true for the union of any finite family of K-varieties.)(vi) If Vii∈I is any family of K-varieties, then

⋂i∈I Vi = V (

∑i∈I J(Vi)).

Therefore, the intersection of any family of K-varieties is a K-variety.


(vii) A K-variety V ⊆ AnK(L) is irreducible if and only if the ideal J(V ) is

prime.

The last statement of Proposition 1.2.32 implies that there is a one-to-onecorrespondence between prime ideals in the polynomial ring K[X1, . . . , Xn] andirreducible varieties in An

K(L).

Exercises 1.2.33 1. With the above notation, a K-variety defined by a singlepolynomial equation is said to be a K-hypersurface. Prove that if the field Lis infinite and n ≥ 1, then outside any K-hypersurface V ⊆ An

K(L) there areinfinitely many points of An

K(L).2. Prove that if the field L is algebraically closed and n ≥ 2, then any

K-hypersurface in AnK(L) contains infinitely many points.

3. Prove that if H ⊆ AnK(L) is a K-hypersurface defined by an equation

f(X1, . . . , Xn) = 0 and f = afk11 . . . fkr

r is a decomposition of f into a productof powers of pairwise unassociated irreducible factors fi (a ∈ K\0), then J(H)is a principal ideal of K[X1, . . . , Xn] generated by the polynomial g = f1 . . . fr.

4. Show that a K-hypersurface H ⊆ AnK(L) is irreducible if and only if

H = V (f) where f is an irreducible polynomial in K[X1, . . . , Xn].5. Prove that every K-hypersurface H ⊆ An

K(L) has a unique (up to theorder of terms) representation H = H1

⋃ · · ·⋃Hm where Hi (1 ≤ i ≤ m) areirreducible hypersurfaces.

Statements (v) and (vi) of Proposition 1.2.32 show that the K-varieties forma family of closed sets in a topology on An

K(L) called the Zariski topology. IfV ⊆ An

K(L), then V carries the relative topology called the Zariski topologyon V . Since the polynomial ring K[X1, . . . , Xn] is Noetherian, Proposition 1.2.32implies that every decreasing chain V1 ⊇ V2 ⊇ . . . of K-varieties in An

K(L)terminates. Thus, every K-variety V (in particular, An

K(L)) is a Noetheriantopological space in the Zariski topology. (Recall that a topological space iscalled Noetherian if every descending chain of its closed subsets terminates.)A topological space X is called irredicible if X cannot be represented as a unionof two its proper closed subsets. A subset Y ⊆ X is called irreducible if it isirreducible as a topological subspace of X with the induced topology. A maximalirreducible subset of X is called an irreducible component of X. Obviously, anaffine algebraic K-variety V ⊆ An

K(L) is irreducible if it is irreducible as atopological space with the Zariski topology.

The following exercises contain some basic facts about irreducible andNoetherian topological spaces.

Exercises 1.2.34 1. Show that for a subset Y of a topological space X thefollowing conditions are equivalent.

(a) Y is irreducible.(b) If U1 and U2 are open subsets of X such that Ui

⋂Y = ∅ (i = 1, 2), then

U1

⋂U2

⋂Y = ∅.

(c) The closure Y of Y is irreducible.


2. Prove that any irreducible subset of a topological space X is contained inan irreducible component of X.

3. Show that every topological space is the union of its irreducible compo-nents.

4. Let X be a Noetherian topological space. Prove that X has only finitelymany irreducible components. Furthermore, show that none of these irreduciblecomponents is contained in the union of the others.

As a consequence of the statement of the last exercise we obtain the followingresult.

Proposition 1.2.35 Let V ⊆ AnK(L) be a K-variety. Then V has only finitely

many irreducible components (that is, maximal irreducible K-varieties containedin V ). If V1, . . . , Vr are all irreducible components of V , then V =

⋃ri=1 Vi and

Vi ⋃

i=j Vi for i = 1, . . . , r. (Thus, no Vi is superfluous in the representationV =

⋃ri=1 Vi).

The proof of the following classical result and its corollary can be found, forexample, in [115, Chapter 1, Section 3]. (The terminology and basic concepts ofthe field theory are reviewed in Section 1.6.)

Theorem 1.2.36 (Hilbert’s Nullstellenzats) Let K be a field and L an alge-braically closed field extension of K (for example, L might be an algebraic closureof K). Then the correspondence V → J(V ) defines a bijection of the set of allK varieties V ⊆ An

K(L) onto the set of all radical ideals of K[X1, . . . , Xn]. Forany ideal I of K[X1, . . . , Xn], J(V (I)) = r(I). (In other words, a polynomialf ∈ K[X1, . . . , Xn] vanishes at every common zero of polynomials of I if andonly if f ∈ r(I).)

Corollary 1.2.37 Let K be a field, L an algebraically closed field extension ofK, and I a proper ideal in the polynomial ring K[X1, . . . , Xn]. Then

(i) V (I) is a nonempty K-variety.(ii) If R is a field extension of K and R is finitely generated as a K-algebra

(that is, R = K[u1, . . . , um] for some elements u1, . . . , um ∈ R), then the fieldextension R/K is algebraic.(iii) Let I1 and I2 be two ideals in K[X1, . . . , Xn]. Then V (I1) = V (I2) if and

only if r(I1) = r(I2).(iv) Let M be a maximal ideal of K[X1, . . . , Xn]. Then K[X1, . . . , Xn]/M is a

finite field extension of K. Furthermore, if F is any field extension of K, thenM has at most finitely many zeros in Fn.

(v) If K is algebraically closed and M is a maximal ideal of K[X1, . . . , Xn],then there exist elements a1, . . . , an ∈ K such that M = (X1−a1, . . . , Xn −an).

Let I be an ideal in K[X1, . . . , Xn]. An n-tuple a with coordinates in somefield extension of K is called a generic zero of I if a is a zero of every polynomialin I and f(a) = 0 for any polynomial f /∈ I. In this case I is called a defining


ideal of a. Note that if I has a generic zero with coordinates in some overfield ofK, then I has a generic zero in the universal field U over K defined as an alge-braic closure of the field K(u1, u2, . . . ) obtained from K by adjoining infinitelymany indeterminates ui (i ∈ N+). It follows from the easily verified fact thatfor any finitely generated field extension L of K, there is an K-isomorphism ofL into U .

Remark 1.2.38 Clearly, if an ideal in a polynomial ring over a field has ageneric zero, the ideal is prime. Conversely, every prime ideal P in K[X1, . . . , Xn]has a generic zero: it is sufficient to consider the n-tuple (X1, . . . ,Xn) whereXi is the canonical image of Xi (1 ≤ i ≤ n) in the quotient field of the factorring K[X1, . . . , Xn]/P . Furthermore, if a and b are two generic zeros of a primeideal P of K[X1, . . . , Xn], then they are equivalent in the sense that there existsa K-isomorphism of K(a) onto K(b).

If V is an irreducible K-variety (here and below we assume that the coordi-nate field is the universal field U), then a generic zero of the prime ideal J(V )in K[X1, . . . , Xn] is called a generic zero of the variety V .

The dimension of an irreducible K-variety V is defined as trdegKK(a) wherea is a generic zero of V (We remind the definition and basic properties of thetranscendental degree in Section 1.6.) It follows from the last remark that thisdefinition does not depend on the choice of a generic zero of V . We denote thedimension of an irreducible variety V by dimV .

Remark 1.2.39 By Theorem 1.2.25, we have dimV = coht J(V ) = dim(K[X1, . . . , Xn]/J(V )). It follows that if V1 and V2 are irreducible K-varieties andV1 ⊆ V2, then dimV1 ≤ dimV2 with equality if and only if V1 = V2.

Theorem 1.2.40 (see [119, Chapter II, Corollary to Theorem 11]) Let Kbe a field and let V1 and V2 be two varieties over K[X1, . . . , Xn] such thatV1 ∩ V2 = ∅. Then the irreducible components of V1 ∩ V2 have dimensions notless than dimV1 + dimV1 − n.

Let V be an irreducible K-variety and J(V ) the corresponding prime idealin the polynomial ring K[X1, . . . , Xn]. By a complete set of parameters of V wemean a maximal subset Xi1 , . . . , Xik

of X1, . . . , Xn (1≤ i1 < . . . < ik ≤n)such that J(V )

⋂K[Xi1 , . . . , Xik

] = (0). (Equivalently, J(V ) contains nononzero polynomial in Xi1 , . . . , Xik

, but for any i /∈ i1, . . . , ik, J(V ) containsa nonzero polynomial in Xi1 , . . . , Xik

, Xi.) In this case, if a = (a1, . . . , an) is ageneric zero of V , then ai1 , . . . , aik

is a maximal algebraically independent setover K whence k = trdegKK(a1, . . . , an) (see Theorem 1.6.30 below). Thus,the dimension of V is equal to the number of elements in any complete set ofparameters of V .

Exercise 1.2.41 With the above notation, show that the (n − 1)-dimensionalirreducible varieties are precisely the varieties of the principal ideals of the ringK[X1, . . . , Xn] generated by irreducible polynomials. [Hint: Use the result ofExercises 1.2.33.4.]


In the rest of this section, for any subset S of a ring A, the ideal and radicalideal of A generated by S will be denoted by (S)A and rA(S), respectively.A proof of the following theorem can be found in [41, Introduction, Section 10].

Theorem 1.2.42 Let L be a field, R = L[X1, . . . , Xn] the ring of polynomialsin variables X1, . . . , Xn over L and K a subfield of L. Furthermore, let A denotethe polynomial subring K[X1, . . . , Xn] of R. Then

(i) If I1, . . . , Im are ideals of A, then (I1 . . . Im)R = (I1)R . . . (Im)R.(ii) If P1, . . . , Pk are essential prime divisors of an ideal I of A, then all

essential prime divisors of (I)R in R are contained in the union of the setsof essential prime divisors of the ideals (Pi)R (i = 1, . . . , k).(iii) If J is a radical ideal of A, then rR(J)

⋂A = J . If J is prime and ab ∈

rR(J), where a ∈ A, b ∈ R, then either a ∈ J or b ∈ rR(J).(iv) If P is a prime ideal of A and Q is an essential prime divisor of (P )R in

the ring R, then Q⋂

A = P .(v) Let P be a prime ideal in A which has a generic zero a such that the field

extensions K(a)/K and L/K are quasi-linearly disjoint (see Definition 1.6.42below). Then rR(P ) is a prime ideal of R and its coheight in the ring R is equalto the coheight of P in A, that is, dim(R/rR(P )) = dim(A/P ) . Furthermore, ifK(a)/K and L/K are linearly disjoint (see Theorem 1.6.24 and the definitionbefore the theorem), then (P )R is a prime ideal of R.(vi) If L is a perfect closure of K (this concept is introduced after Theorem

1.6.18 below) and P is a prime ideal of the ring A, then rR(P ) is a prime idealof R. Furthermore, in this case every generic zero of P in an overfield of L isa generic zero of rR(P ), and dim(A/P ) = dim(R/rR(P )).(vii) Let the extension L/K be primary (see Proposition 1.6.47 and the defini-

tion before the proposition) and let P be a prime ideal of A. Then rR(P ) is aprime ideal of R and dim(A/P ) = dim(R/rR(P )). If Char K = 0, then (P )R

is a prime ideal of R.(viii) Let P be a prime ideal of A and Q an essential prime divisor of (P )R inR. Then dim(A/P ) = dim(R/Q) and every complete set of parameters of P isa complete set of parameters of Q.

While considering singular solutions of ordinary algebraic difference equa-tions we will need the following three theorems about solutions of ideals of apolynomial ring in its power series overring. The proof of the theorems can befound in [41, Introduction, Section 12]. As usual, by a formal power series inindeterminates (or parameters) t1, . . . , tm over an integral domain K we meana formal expression of the form

∑(i1,...,im)∈Nm ai1...im

ti11 . . . timm where all coeffi-

cients ai1...imlie in K. The set of all formal power series over K (also referred

to as power series over K) form an integral domain under the obvious definitionof addition and multiplication; this domain is denoted by K[[t1, . . . , tm]]. It canbe naturally considered as an overring of K if one identifies an element a ∈ Kwith the power series whose only term is a. (Note that every ti is transcendentalover the quotient field of K[[t1, . . . , ti−1, ti+1, . . . , tm]].)


Theorem 1.2.43 Let K be a field of zero characteristic and let K[X1, . . . , Xn]be the ring of polynomials in indeterminates X1, . . . , Xn over K.

(i) Let L be an overfield of K, Σ a subset of the polynomial ring L[X1, . . . , Xn],and L[[t1, . . . , tm]] the ring of power series in parameters t1, . . . , tm over L.Furthermore, suppose that the set Σ has a solution of the form (a1+f1, . . . , an +fn) with coordinates in L[[t1, . . . , tm]] such that ai ∈ L and fi is either 0 ora power series whose terms are all of positive degree in the parameters (i =1, . . . , n). Then (a1, . . . , an) is also a solution of Σ.

(ii) Let V be an irreducible variety over K[X1, . . . , Xn] such that dimV > 0,let (a1, . . . , an) ∈ V , and let g ∈ K[X1, . . . , Xn] \J(V ) (as before, J(V ) denotesthe vanishing ideal of V ). Then there is a solution of J(V ), not annulling g,which has the form (a1 + f1, . . . , an + fn) where fi is either 0 or a power seriesin one parameter t with coefficients in some overfield of K(a1, . . . , an) whoseterms are all of positive degree in t (i = 1, . . . , n).

Theorem 1.2.44 Let K be a field of zero characteristic, K[X1, . . . , Xn] thering of polynomials in indeterminates X1, . . . , Xn over K, and P a prime idealof K[X1, . . . , Xn] with dimP > 0. Furthermore, let Xk+1, . . . , Xn be a completeset of parameters of P . Then there exists and element D ∈ K[X1, . . . , Xn] \P such that if (a1, . . . , an) is a solution of P not annulling D, then P has aunique solution of the form (a1 +f1, . . . , ak +fk, ak+1 + tk+1, . . . , an + tn) wheretk+1, . . . , tn are parameters and f1, . . . , fk are power series in positive powers ofthese parameters with coefficients in K(a1, . . . , an).

Theorem 1.2.45 Let K be a field of zero characteristic and let B be a polyno-mial in one indeterminate X whose coefficients are formal power series in pa-rameters t1, . . . , tk with coefficients in K. Let C and D be polynomials obtainedfrom B and the formal partial derivative ∂B/∂X, respectively, by substitutingti = 0 (i = 1, . . . , k). Furthermore, let a be an element in some overfield of Kwhich is a solution of C but not of D. Then B has at most one solution of theform X = a + f where f is a series in positive powers of the parameters withcoefficients in an overfield of K(a).

If K is a field, then a finitely generated K-algebra is called an affine K-algebra. By Theorem 1.2.11(vii), an affine K-algebra is a Noetherian ring. Fur-thermore (see Corollary 1.2.37 (ii)), if an affine algebra R over a field K is afield, then R/K is a finite algebraic extension. One of the main results on affinealgebras over a field is the following Noether Normalization Theorem (its proofcan be found in [115, Chapter 2, Section 3]).

Theorem 1.2.46 Let A be an affine algebra over a field K and let I be a properideal of A. Then there exist two numbers d,m ∈ N, 0 ≤ d ≤ m, and elementsY1, . . . , Ym ∈ A such that

(a) Y1, . . . , Ym are algebraically independent over K. (It means that thereis no nonzero polynomial f in m variables with coefficients in K such thatf(Y1, . . . , Ym) = 0.)


(b) A is integral over K[Y1, . . . , Ym]. (Equivalently, A is a finitely generatedK[Y1, . . . , Ym]-module).

(c) I ∩ K[Y1, . . . , Ym] = (Yd+1, . . . , Ym).If the field K is infinite and A = K[x1, . . . , xn], then every Yi (1 ≤ i ≤ d) is

of the form Yi =∑n

k=1 aikxk where aik ∈ K (1 ≤ i ≤ d, 1 ≤ k ≤ n).

We conclude this section with a theorem by B. Lando [118] that gives somebound for dimensions of irreducible components of a variety. The theorem willuse the following notation. If A = (rij) is an n × n-matrix with nonnegativeinteger entries, then any sum r1j1 + r2j2 + · · · + rnjn

, where j1, . . . , jn is apermutation of 1, . . . , n, is said to be a diagonal sum of A. If A is an m × n-matrix and s = minm,n, then a diagonal sum of any s × s-submatrix of A(that is, a matrix obtained by choosing s rows and s columns of A) is called a adiagonal sum of A. The maximal diagonal sum of A is called the Jacobi numberof this matrix; it is denoted by J (A).

Theorem 1.2.47 Let K[X1, . . . , Xn] be the ring of polynomials in indetermi-nates X1, . . . , Xn over a field K and let f1, . . . , fm be polynomials in this ring.Let A = (rij) be the m × n-matrix with rij = 1 if Xj appears in the polyno-mial Ai and rij = 0 if not. Furthermore, let V denote the variety of the familyf1, . . . , fm over K. Then, if the variety V is not empty, the dimension ofevery irreducible component of V is not less than s − J (A).

VALUATIONS AND VALUATION RINGS

Definition 1.2.48 An integral domain R is called a valuation ring if for everyelement x of its quotient field K, one has either x ∈ R or x−1 ∈ R.

Theorem 1.2.49 Let R be a valuation ring and K its quotient field. Then(i) If I and J are two ideals of R, then either I ⊆ J or J ⊆ I.(ii) R is a local integrally closed ring.

Definition 1.2.50 A discrete valuation of a field K is a mapping v : K → Z∪ ∞such that

(i) v(ab) = v(a) + v(b),(ii) v(a + b) ≥ minv(a), v(b) for every a, b ∈ K, and(iii) v(a) = ∞ if and only if a = 0.

The valuation ring of v is the set of all a ∈ K with v(a) ≥ 0.

Proposition 1.2.51 Let v be a nontrivial discrete valuation of a field K (thatis, v(a) = 0 for some a ∈ K). Then the valuation ring of v is a valuation ringin the sense of Definition 1.2.47. This ring is a regular local integral domainof dimension 1 (that is a local ring whose maximal ideal is principal), and anysuch integral domain is the valuation ring of a discrete valuation of its quotientfield.

More information about discrete valuations and valuation rings can be found,for example, in [3, Chapters 5, 9] or [138, Chapter 4].

1.3. GRADED AND FILTERED RINGS AND MODULES 37

1.3 Graded and Filtered Rings and Modules

A graded ring is a ring A together with a direct sum decomposition A =⊕n∈Z A(n) where A(n) (n ∈ Z) are subgroups of the additive group of A such

that A(m)A(n) ⊆ A(m+n) for any m,n ∈ Z. This decomposition is said to be agradation of A. The Abelian groups A(n) are called the homogeneous componentsof the graded ring A; the elements of A(n) are said to be homogeneous elementsof degree n (if a ∈ A(n), we write deg a = n). Every nonzero element of A hasa unique representation as a finite sum a1 + · · · + ak of nonzero homogeneouselements; these elements are called the homogeneous components of a.

A subring (or a left, right or two-sided ideal) I of A is said to be a gradedor homogeneous if I =

⊕n∈Z(I

⋂A(n)). If J is a two-sided graded ideal of

A, then the factor ring A/J can be naturally treated as a graded ring withthe homogeneous components (A(n) + J)/J , A/J =

⊕n∈Z(A(n) + J)/J ∼=⊕

n∈Z A(n)/J (n) where J (n) = J⋂

A(n).If B =

⊕n∈Z B(n) is another graded ring, then a homomorphism of rings f :

A → B is called homogeneous (or a homomorphism of graded rings) if it respectsthe grading structures on A and B, that is, if f(A(n)) ⊆ B(n) for all n ∈ Z. Ifthere is an integer r such that f(A(n)) ⊆ B(n+r) for all n ∈ Z, we say that f isa homomorphism of graded rings of degree r or a graded ring homomorphism ofdegree r. (Thus, a homogeneous homomorphism is a homomorphism of gradedrings of degree 0.)

If A(n) = 0 for n < 0, we say that A is a positively graded ring; in this caseA+ =

⊕∞n=1 A(n) is a two-sided ideal of A and A/A+

∼= A(0).

Examples 1.3.1 1. Every ring A can be treated as a “trivially” graded ringif one considers A as a direct sum

⊕n∈Z A(n) with A(0) = A and A(n) = 0 for

all n = 0.2. Let A = K[X1, . . . , Xm] be a polynomial ring in m variables X1, . . . , Xm

over a field K. For every n ∈ N, let A(n) denote the vector K-space generated byall monomials Xk1

1 . . . Xkmm of total degree n, that is, the set of all homogeneous

polynomials of total degree n. Then the ring A becomes a positively gradedring: A =

⊕n∈N A(n). In what follows, while considering a polynomial ring as a

graded one, we shall always mean that its homogeneous components are of thistype.

3. Let Q = K(X1, . . . , Xm) be a ring of rational fractions in m variablesX1, . . . , Xm over a field K (this is the quotient field of K[X1, . . . , Xm]). Let

F be a subfield of Q consisting of all rational fractionsf

gwhere f and g are

homogeneous polynomials in K[X1, . . . , Xm]. Then F can be treated as a graded

ring if we define the degree of a fractionf

g∈ F as deg f

g = deg f − deg g and

consider F as a direct sum F =⊕

n∈Z F (n) where F (n) consists of rationalfractions of degree n. Graded rings of this type arise as homogeneous coordinaterings and function fields of projective curves in Algebraic Geometry.


Let A =⊕

n∈Z A(n) be a graded ring. A graded left A-module is a leftA-module M together with a direct sum decomposition (also called a gradation)M =

⊕n∈Z M (n) where M (n) (n ∈ Z) are subgroups of the additive group of M

such that A(m)M (n) ⊆ M (m+n) for any m,n ∈ Z. The graded right A-moduleis defined in the same way. (In what follows, unless otherwise is indicated, byan A-module we mean a left A-module; the treatment of right A-modules issimilar. Of course, if the ring A is commutative, so that there is no differencebetween left and right modules, one does not need this remark.) The Abeliangroups M (n) are said to be the homogeneous components of M , elements of M (n)

are called homogeneous elements of degree n (if x ∈ M (n), we write deg x =n). Every nonzero element u ∈ M can be uniquely represented as a sum ofnonzero homogeneous elements called the homogeneous components of u. An A-submodule N of a graded A-module M =

⊕n∈Z M (n) is said to be a graded or

homogeneous submodule of M if it can be generated by homogeneous elements.This condition is equivalent to either of the following two:

(i) For any x ∈ M , if x ∈ N , then each homogeneous component of x is in N ;(ii) N =

⊕n∈Z(N

⋂M (n)).

If N is a homogeneous submodule of M , then M/N can be also treatedas a graded submodule with the homogeneous components (M (n) + N)/N ∼=M (n)/(M (n)

⋂N) (n ∈ Z). Furthermore, for any r ∈ Z, one can consider a

gradation of M whose n-th homogeneous component is M (n+r) (n ∈ Z). Weobtain a graded A-module M(r) =

⊕n∈Z M (n+r) called the r-th suspension

of M .Let M =

⊕n∈Z M (n) and P =

⊕n∈Z P (n) be graded modules over a graded

ring A and r ∈ Z. A homomorphism of A-modules φ : M → P is said to bea graded homomorphism of degree r (or homomorphism of graded modules ofdegree r) if φ(M (n)) ⊆ P (n+r) for all n ∈ Z. The restriction φ(n) : M (n) →P (n+r) of φ on M (n) (n ∈ Z) is called the nth component of φ. A homomor-phism of graded modules of degree 0 will be referred to as a homomorphismof graded modules (or graded homomorphism). By an exact sequence of graded

A-modules we mean an exact sequence . . .φi−1−−−→ Li−1

φi−→ Liφi+1−−−→ Li+1 → . . .

where . . . , Li−1, Li, Li+1, . . . are graded A-modules and . . . , φi−1, φi, φi+1, . . .are graded homomorphisms.

If M and N are two graded modules over a graded ring A, then for everyp ∈ Z, the graded homomorphisms of degree p from M to N form a sub-group of HomA(M,N), the Abelian group of all A-homomorphisms from M

to N . This subgroup is denoted by Hom(p)grA(M,N). Let HomgrA(M,N) de-

note the set of all graded homomorphisms of various degrees from M to N .Then HomgrA(M,N) =

⊕p∈Z Hom

(p)grA(M,N) can be considered as a graded

Abelian group with homogeneous components Hom(p)grA(M,N). (By a graded

Abelian group we mean a graded Z-module when Z is treated as a triviallygraded ring.)


Exercises 1.3.2 Let A =⊕

n∈Z A(n) be a graded ring.1. Suppose that M =

⊕n∈Z M (n) is a graded A-module with homogeneous

generators x1, . . . , xs, xi ∈ M (ki) for some integers k1, . . . , ks. Prove that M (n) =∑si=1 A(n−ki)xi for every n ∈ Z.2. Let M and N be two graded A-modules. Prove that if M is finitely gen-

erated over A, then HomA(M,N) = HomgrA(M,N).3. Show that graded left modules over a graded ring, together with graded

morphisms (of degree 0) form an Abelian category with enough injective andprojective objects. This category is denoted by grAMod. Define the categoryModgrA of all graded right A-modules and the category grAMod-ModgrAof all A-B-bimodules (B is another graded ring) and show that these categorieshave the same property.

4. Show that a graded A-module P is a projective object in grAMod if andonly if P is a projective A-module in the regular sense, that is, M is a projectiveobject in AMod.

5. Prove that if a graded A-module Q is injective in AMod, then it is injec-tive in grAMod. Show that the converse statement is not true. [Hint: Considerthe graded ring R = K[X,X−1] of Laurent polynomials in one variable X overa field K, that is, the group K-algebra of the free group on a single generator X.The homogeneous components of this ring are R(n) = aXn | a ∈ K (n ∈ Z).Prove that R is injective in grRMod, but not in RMod.]

6. Let Mii∈I be a directed family of graded submodules of a gradedA-module M (that is, for every i, j ∈ I, there exists k ∈ I such that Mi ⊆ Mk

and Mj ⊆ Mk). Prove that if N is any graded A-submodule of M , then(∑

i∈I Mi)⋂

N =∑

i∈I(Mi

⋂N). (An Abelian category with this property for

objects is said to satisfy the Grothendieck’s axiom A5, see [76, Section 1.5]).

Let A =⊕

n∈Z A(n) be a graded ring. By a free graded A-module we meanan A-module F which has a basis consisting of homogeneous elements. It is easyto show (we leave the proof to the reader as an exercise) that this condition isequivalent to the following: there exists a family of integers mii∈I such thatF is isomorphic (as a graded A-module) to the graded A-module

⊕n∈Z A

(n)I

where A(n)I =

⊕i∈I A(n−mi) for all n ∈ Z.

Suppose that the graded ring A is left Noetherian and a graded (left)A-module M is generated by a finite set of homogeneous elements x1, . . . , xk

where xi ∈ M (ri) for some integers r1, . . . , rk. Let us take a free A-module withfree generators f1, . . . , fk and produce a free graded A-module F1 =

⊕n∈Z F

(n)1

with homogeneous components F(n)1 =

∑ki=1 A(n−ri)f1. Let φ1 : F1 → M be the

natural epimorphism of graded A-modules which maps each fi to xi (1 ≤ i ≤ k).Since the ring A is left Noetherian, N1 = Ker φ1 is a graded A-submodule ofF1 which has a finite set of homogeneous generators. As before, we can find afree graded A-module F2 and an epimorphism of graded modules ψ1 : F2 → N1.Setting φ2 = ψ1α, where α is the embedding N1 → F1, we obtain an exactsequence of graded modules F2

φ2−→ F1φ1−→ M → 0. Considering the graded


A-submodule N2 = Ker φ2 ⊆ F2 one can repeat the same procedure, etc. If theprojective dimension of the A-module M is finite, this process terminates andwe obtain a finite free resolution of the graded A-module M .

Exercises 1.3.3 Let A be a graded ring.1. Prove that a free graded A-module is projective (i. e., it is, a projective

object in grAMod).2. Prove that a graded A-module M is projective if and only if M is a direct

summand of a free graded A-module.

A graded module M =⊕

n∈Z M (n) over a graded ring is said to be leftlimited (respectively, right limited) if there is n0 ∈ Z such that M (i) = 0 for alli < n0 (respectively, for all i > n0). The same terminology is applied to gradedrings. As in the case of rings, a graded module M =

⊕n∈Z M (n) M is said to

be positively graded if M (i) = 0 for all i < 0.

Exercises 1.3.4 Let A be a left limited graded ring and M a graded leftA-module.

1. Prove the following statements.(i) If M is finitely generated over A, then M is left limited.(ii) If M is left limited, then there exists a free graded left-limited module F

and a graded epimorphism F → M .2. Let A be positively graded and let A+ =

⊕∞n=1 A(n) (as we have seen,

A+ is an ideal of A). Prove that if the graded module M is left-limited, thenA+M = M if and only if M = 0.

Let M =⊕

n∈Z M (n) and N =⊕

n∈Z N (n) be two graded modules over agraded ring A =

⊕n∈Z A(n). Then the Abelian group M

⊗A N can be con-

sidered as a graded Z-module whose nth homogeneous component (n ∈ Z)is the additive subgroup of M

⊗A N generated by the elements x

⊗A y with

x ∈ M (i), y ∈ N (j) and i + j = n . (In this case Z is considered as a triviallygraded ring.)

Our definition of the tensor product of graded modules produces a functor⊗A : ModgrA × grAMod → grZMod. Obviously, if we fix M ∈ grAMod,

then the functor⊗

A M : ModgrA → grZMod will be right exact.

Exercises 1.3.5 1. Let M =⊕

n∈Z M (n) and N =⊕

n∈Z N (n) be gradedmodules over a graded ring A, and for any r ∈ Z, let M(r) denote the rthsuspension of M . Show that M(r)

⊗A N(s) = (M

⊗A N)(r+s) for any r, s ∈ Z.

2. Let A and B be two graded rings and let M ∈ ModgrA, P ∈ ModgrB,and N ∈ grAMod-ModgrB.

(a) Show that M⊗

A N is a graded right B-module and there is a naturalisomorphism HomgrB(M

⊗A N,P ) ∼= HomgrA(M,HomgrB(N,P )).


(b) Show that there is a homomorphism of Abelian groups φ : M⊗

A

HomgrB(N,P ) → HomgrB(HomgrA(M,N), P ) defined by φ(m⊗

f)(g) =(fg)(m) for every m ∈ M,f ∈ HomgrB(N,P ), and g ∈ HomgrA(M,N).

3. Prove that a graded left module M over a graded ring A is flat in grAModif and only if M is flat in AMod.

In what follows we concentrate on graded modules over Noetherian rings.

Proposition 1.3.6 A positively graded commutative ring A =⊕

n∈N A(n) isNoetherian if and only if A(0) is Noetherian and A is finitely generated as aring over A(0).

Let A =⊕

n∈N A(n) be a positively graded commutative Noetherian ringand let M =

⊕n∈N M (n) be a finitely generated positively graded A-module.

Then M can be generated by a finite number of homogeneous elements: M =∑si=1 Axi where xi ∈ M (ei) for some nonnegative integers e1, . . . , es. For every

n ∈ N we have M (n) =∑s

i=1 A(n−ei)xi (where A(j) = 0 for j < 0), hence M (n)

is a finitely generated A(0)-module. In particular, if the ring A is Artinian, thenl(M (n)) < ∞ where l denotes the length of an A(0)-module. (If A(0) is a field,then l(M (n)) is equal to the dimension of M (n) as a vector A(0)-space.) In thiscase the power series P (M, t) =

∑∞n=0 l(M (n))tn ∈ Z[[t]] is called the Hilbert

series of M .

Theorem 1.3.7 (Hilbert-Serre) Let A =⊕

n∈N A(n) be a positively gradedcommutative Noetherian ring and let M =

⊕n∈Z M (n) be a finitely generated

positively graded A-module. Furthermore, suppose that the ring A(0) is Artinianand A = A(0)[x1, . . . , xs] where xi is a homogeneous element of A of degree di

(1 ≤ i ≤ s). Then(i) P (M, t) = f(t)/

∏si=1(1 − tdi) where f(t) is a polynomial with integer

coefficients.(ii) If di = 1 for i = 1, . . . , s, then P (M, t) = f(t)/(1 − t)d where f(t) ∈ Z[t]

and d ≥ 0. If d > 0, then f(1) = 0. Furthermore, in this case there exists apolynomial φM (t) of degree d − 1 with rational coefficients such that l(M (n)) =φM (n) for all n ≥ r + 1 − d where r is the degree of the polynomial f(t) =(1 − t)dP (M, t).

The polynomial φM (X), whose existence is stated in the second part of thelast theorem, is called the Hilbert polynomial of the graded module M . Thenumerical function l(M (n)) is called the Hilbert function of M .

Example 1.3.8 Let A = K[X1, . . . , Xm] be the ring of polynomials in mvariables X1, . . . , Xm over an Artinian commutative ring K. Consider A as apositively graded ring whose r-th homogeneous component A(r) consists of allhomogeneous polynomials of total degree r (r ∈ N) (in this case we say thatthe polynomial ring is equipped with the standard gradation). Since the number


of monomials of degree r is(

m + r − 1m − 1

), l(A(r)) = l(K)

(m + r − 1

m − 1

)for all

r ∈ N, and the right-hand side of the last equality is φA(r). Thus, the Hilbertpolynomial φA(t) is of the form

φA(t) = l(K)(

t + m − 1m − 1

)=

l(K)(m − 1)!

(t + m − 1)(t + m − 2) . . . (t + 1).

Example 1.3.9 Let K be a field, A = K[X1, . . . , Xm] the ring of polynomialsin m variables X1, . . . , Xm over K, and f ∈ A a homogeneous polynomial ofsome positive degree p. If A is considered as a graded ring with the standardgradation, then its principal ideal (f) is homogeneous and one can consider thegraded factor ring B = A/(f) with the homogeneous components B(r) = A(r) +(f)/(f), r ∈ N (we use the notation of the previous example). Then l(B(r)) =(

r + m − 1m − 1

)−

(r + m − p − 1

m − 1

)for all r ≥ p whence the Hilbert polynomial

of the graded ring B is of the form φB(t) =(

t + m − 1m − 1

)−

(t + m − p − 1

m − 1

)=

p

(m − 2)!tm−2 + o(tm−2).

Let K be a field and A = K[X1, . . . , Xm] the ring of polynomials in mvariables X1, . . . , Xm considered as a graded ring with the standard gradation.Let F be a finitely generated free graded A-module and let f1, . . . , fs be asystem of its free homogeneous generators. If di = deg fi (1 ≤ i ≤ s), thenthe r-th homogeneous component of F is of the form F (r) =

∑si=1 A(r−di)fi.

Applying the result of Example 1.3.8 one obtains the Hilbert polynomial of F :

φF (t) =s∑

i=1

(t + m − di − 1

m − 1

). (1.3.2)

Therefore, if a graded A-module M has a finite free resolution, the Hilbertpolynomial φM (t) can be obtained as an alternate sum of the polynomial of theform (1.3.2).

Theorem 1.3.10 (Hilbert Syzygy Theorem) Let K be a field and let A =K[X1, . . . , Xm] be the ring of polynomials in m variables X1, . . . , Xm consid-ered as a graded ring with the standard gradation. Then every finitely generatedgraded A-module has a finite free resolution of length at most m whose termsare finitely generated free A-modules.

A filtered ring is a ring A together with an ascending chain (An)n∈Z ofadditive subgroups of A such that 1 ∈ A0 and AmAn ⊆ Am+n for every m,n ∈Z. The family of these subgroups is called the (ascending) filtration of A; asubgroup An (n ∈ Z) is said to be the nth component of the filtration. Note thatthe definition implies that A0 is a subring of A.


Remark 1.3.11 Our definition of a filtered ring can be naturally convertedinto the definition of a filtered ring with a descending filtration. This is a ring Btogether with a descending chain (Bn)n∈Z of additive subgroups of B such that1 ∈ B0 and BmBn ⊆ Bm+n for all m,n ∈ Z. A classical example of a descendingfiltration on a ring A is the I-adic filtration A = I0 ⊇ I ⊇ I2 ⊇ . . . defined byan ideal I of A. In what follows, by a filtration we mean an ascending filtration(otherwise, we shall use the adjective “descending”).

Examples 1.3.12 1. Any ring A can be treated as a filtered ring with thetrivial filtration (An)n∈Z such that An = 0 if n < 0 and An = A if n ≥ 0.

2. Let R = A[X1, . . . , Xm] be the polynomial ring in m variables X1, . . . , Xm

over a ring A. Then R can be considered as a filtered ring with the filtration(Rn)n∈Z such that Rn = 0 if n < 0, and for any n ≥ 0, Rn consists of all poly-nomials of degree ≤ n. This filtration of the polynomial ring is called standard.

3. If A =⊕

n∈Z A(n) is a graded ring, then A can be also considered as afiltered ring with the filtration (Ar)n∈Z where Ar =

∑n≤r A(n). We say that

this filtration is generated by the given gradation of A.

Let A be a filtered ring with a filtration (An)n∈Z. A left A-module M is saidto be a (left) filtered A-module if there exists an ascending chain (Mn)n∈Z ofadditive subgroups of M such that AmMn ⊆ Mm+n for every m,n ∈ Z. Thefamily of these subgroups is called a filtration of M ; a subgroup Mn (n ∈ Z)is said to be the nth component of the filtration. This definition of a filteredA-module can be naturally adjusted to the case of descending filtrations.

Let A be a filtered ring with a filtration (An)n∈Z and let M be a filteredA-module with a filtration (Mn)n∈Z. The filtration of M is called exhaustive(respectively, separated) if

⋃n∈Z Mn = M (respectively,

⋂n∈Z Mn = 0). The

filtration (Mn)n∈Z is said to be discrete if Mn = 0 for all sufficiently smalln ∈ Z, that is, if there exists n0 ∈ Z such that Mn = 0 for all n ≤ n0. If Mn = 0for all n < 0, we say that the filtration of M is positive. Clearly, if the filtrationof the ring A is trivial, then any ascending chain (Mn)n∈Z of additive subgroupsof an A-module M is a filtration of M . If Mn = 0 for all n < 0 and Mn = Mfor all n ≥ 0, the filtration of M is said to be trivial.

Example 1.3.13 If A is a filtered ring with a filtration (An)n∈Z and an A-module M is generated by a set S ⊆ M , then M can be naturally treated asa filtered A-module with a filtration (Mn)n∈Z such that Mn =

∑x∈S Anx for

every n ∈ Z. We say that this filtration is associated with the set of generatorsS. More general, one can assign an integer nx to every element x ∈ S andconsider M as a filtered module with the filtration (

∑x∈S An−nx

x)n∈Z (in thiscase nx is said to be the weight of a generator x ∈ S).

Let A be a filtered ring and let M and N be filtered A-modules with thefiltrations (Mn)n∈Z and (Nn)n∈Z, respectively. A homomorphism of A-modulesf : M → N is said to be a homomorphism of filtered modules of degree p(p ∈ Z) if f(Mn) ⊆ Nn+p for all n ∈ Z. A homomorphism of filtered mod-ules of degree 0 is referred to as just a homomorphism of filtered modules.


The set of all homomorphisms of filtered modules M → N of some degreeform an additive subgroup of the Abelian group HomA(M,N) of all A-modulehomomorphisms from M to N ; this subgroup is denoted by HomFA(M,N).It is easy to see that for every p ∈ Z, the homomorphisms from M to N ofdegree p form an additive subgroup of HomFA(M,N); it will be denoted byHomFA(M,N)p. Clearly, HomFA(M,N)p ⊆ HomFA(M,N)q whenever p ≤ q,and

⋃p∈Z HomFA(M,N)p = HomFA(M,N).

If f : M → N is a homomorphism of filtered A-modules with filtrations(Mn)n∈Z and (Nn)n∈Z, respectively, then Ker f and Coker f can be natu-rally treated as filtered A-modules. The n-th components of their filtrations are(Ker f)n = Ker f ∩ Mn and (Coker f)n = f(M) + Nn/f(M), respectively. Asin the case of graded modules, one can consider a shift of the filtration (Mn)n∈Z

of a filtered A-module M . In other words, for any r ∈ Z, one can consider Mtogether with the filtration (Mn+r)n∈Z; the resulting filtered module is denotedby M(r).

Exercise 1.3.14 Let Mi | i ∈ I be a family of filtered modules over a filteredring A. Consider the A-modules

⊕i∈I Mi and

∏i∈I Mi as filtered A-modules

where the nth components of the filtrations are, respectively, the direct sum anddirect product of the nth components of Mi (i ∈ I). Prove that if the filtration ofevery Mi (i ∈ I) is exhaustive, so is the filtration of

⊕i∈I Mi. Give an example

showing that a similar statement is not true for the direct product.

It is easy to see that filtered left modules over a filtered ring A and homomor-phisms of such modules form an additive category denoted by filtAMod. Thiscategory, however, is not Abelian, since not every bijection is an isomorphismin this category. Indeed, let M be a filtered module over a filtered ring A suchthat at least one component of the filtration of M is not 0. Let M ′ denote thesame A-module M treated as a filtered A-module with the zero filtration (allcomponents are 0). The the identity homomorphism M ′ → M is a bijection (itlies in HomFA(M ′,M)0) but not an isomorphism, since the identity mappingdoes not belong to HomFA(M,M ′).

Let A be a filtered ring with a filtration (An)n∈Z. For any n ∈ Z, letgrnA denote the factor group An/An−1 and let gr A denote the Abelian group⊕

n∈Z grnA. It is easy to check that if for any homogeneous elements x =x + Am−1 ∈ grmA and y = y + An−1 ∈ grnA, one defines their product asxy = xy+Am+n−1 ∈ grm+nA and extends the multiplication to the whole groupgr A by distributivity, then gr A becomes a graded ring with the homogeneouscomponents grnA (n ∈ Z). This graded ring is said to be an associated gradedring of the filtered ring A. If M is a filtered A-module with a filtration (Mn)n∈Z,then one can consider the graded gr A-module gr M =

⊕n∈Z grnM with ho-

mogeneous components grnM = Mn/Mn−1 (n ∈ Z). It is called the associatedgraded module of M . The gr A-module structure on the Abelian group gr Mis provided by the mapping gr A × gr M → gr M such that (a + Am−1, x +Mn−1) → ax+Mm+n−1 ∈ grm+nM for any homogeneous elements a+Am−1 ∈grmA, x + Mn−1 ∈ grnM .


Let A be a filtered ring with a filtration (An)n∈Z. If M is a filtered A-module with a filtration (Mn)n∈Z and x ∈ Mi \ Mi−1 for some i ∈ Z, wesay that x is an element of M of degree i and write deg x = i. The elementH(x) = x + Mi−1 ∈ griM is said to be the head of x. If L ⊆ M , the set⋃

x∈L H(x) is denoted by H(L).A filtered A-module F with a filtration (Fn)n∈Z is said to be a free filtered

A-module if F is a free A-module with a basis xii∈I and there exists a familyof integers nii∈I such that Fn =

∑i∈I An−ni

xi for all n ∈ Z. (It follows thatdeg xi = ni for all i ∈ I.) The set of pairs (xi, ni) | i ∈ I is said to be a filt-basisof F .

Exercises 1.3.15 Let A be a filtered ring with a filtration (An)n∈Z, M a fil-tered A-module with a filtration (Mn)n∈Z, and L an A-submodule of M withthe filtration (Ln = Mn ∩ L)n∈Z.

1. Prove that H(L) is a homogeneous gr A-submodule of gr M .2. Let π : M → M/L be the natural epimorphism of graded A-modules

(M/L is considered with the filtration (Mn + L/L)n∈Z). Prove that the gradedgr A-modules gr(M/L) and gr M/H(L) are isomorphic.

3. Prove that F is a free filtered A-module with a filt-basis (xi, ni) | i ∈ I ifand only if it is isomorphic to the filtered module

⊕i∈I A(−ni). (Recall that for

any m ∈ Z, A(m) denotes the filtered A-module with the filtration (An+m)n∈Z.)4. Let F be a free filtered A-module with a filt-basis (xi, ni) | i ∈ I and let

f be a mapping from xii∈I to M such that f(xi) ∈ Mni+p for every i ∈ I(p is a fixed integer). Prove that there exists a unique homomorphism of filteredA-modules f : F → M of degree p which extends f .

5. Prove that if the filtration of A is exhaustive (respectively, separated),then the filtration of any free filtered A-module is exhaustive (respectively, sep-arated).

6. Show that if F is a free filtered A-module with a filt-basis (xi, ni) | i ∈ I,then gr F is a free graded gr A-module with the homogeneous basis H(xi)i∈I .

7. Prove that if F is a free graded gr A-module, then there exists a freefiltered A-module F ′ such that F = gr F ′.

Let M and N be filtered modules over a filtered ring A equipped withfiltrations (Mn)n∈Z and (Nn)n∈Z, respectively. Then a homomorphism of fil-tered A-modules f : M → N induces a homomorphism of graded modulesgr f : gr M → gr N such that gr f(x + Mi−1) = f(x) + Ni−1 for any homo-geneous element x + Mi−1 ∈ griM (i ∈ Z). It easy to see that if g : N → Pis another homomorphism of filtered A-modules, then gr(gf) = (gr g)(gr f).We obtain a functor gr from the category of all filtered left A-modules tothe category of all left graded gr A-modules. The following proposition givessome properties of this functor. We leave the proof of the statements of thisproposition to the reader as an exercise.


Proposition 1.3.16 Let A be a filtered ring and let M be a filtered A-modulewith a filtration (Mn)n∈Z.

(i) If the filtration of M is exhaustive and separate, then M = 0 if and onlyif gr M = 0.

(ii) If the filtration of M is discrete, then the graded gr A-module gr M isleft-limited.

(iii) Let 0 → Nα−→ M

β−→ P → 0 be an exact sequence of filtered A-modules,where the filtrations of N and P are induced by the filtration of M (thatis, for every n ∈ Z, the nth components of the filtrations of N and P areα−1(Mn

⋂α(N)) and β(Mn) ∼= Mn + α(N)/α(N), respectively). Then the

induced sequence of graded gr A modules 0 → gr N → gr M → gr P → 0 isexact.(iv) The functor gr commutes with the direct sums and direct products.(v) Let gr M be a free graded gr A-module with a basis xii∈I where each xi

belongs to some homogeneous component grniM (ni ∈ Z). If the filtration of M

is discrete, then M is a free filtered A-module with the filt-basis (xi, ni) | i ∈ I.(vi) Let F be a free filtered A-module and let g : gr F → gr M be a homo-

morphism of graded gr A-modules of some degree p (p ∈ Z). Then there existsa homomorphism of filtered modules f : F → M such that g = gr f .(vii) If the filtration of M is exhaustive, then M has a free resolution infiltAMod. In other words, there exists an exact sequence of filtered A-modules . . .

φ2−→ F2φ1−→ F0

φ0−→ M → 0 where Fi (i = 0, 1, 2, . . . ) are freefiltered A-modules with filtrations (Fin)n∈Z such that φi(Fin) = Imφi

⋂Fi−1,n

(i = 1, 2, . . . ) and φ0(F0n) = Mn for all n ∈ Z. Furthermore, if the filtration ofA is discrete, one can choose the free resolution in such a way that the filtrationsof all Fi are discrete.

Remark 1.3.17 Let A be a filtered commutative ring with a positive filtration(An)n∈Z such that the ring A0 is Artinian, A1 is a finitely generated A0-moduleand AmAn = Am+n for any m,n ∈ N. Then the ring A is Noetherian and gr Ais a positively graded Noetherian ring whose zero component is an Artinianring. Let M be any filtered A-module with a positive filtration (Mn)n∈Z suchthat every Mn (N ∈ Z) is a finitely generated A0-module and there existsn0 ∈ Z such that AiMn = Mi+n for all i ∈ N and for all n ∈ Z, n ≥ n0. Thengr M is a positively graded finitely generated gr A-module. By Theorem 1.3.7,there exists a polynomial φ(t) in one variable t with rational coefficients suchthat φ(n) = l(grnM) for all sufficiently large n ∈ Z (l denotes the length of amodule over the ring gr0A = A0). Since l(Mn) =

∑ni=0 l(griM) for all n ∈ N,

there exists a polynomial ψ(t) ∈ Q[t] such that ψ(n) = l(Mn) for all sufficientlylarge n ∈ Z (see Corollary 1.4.7 in the next section). In Chapter 3 we presentmore general results of this type.

1.4. NUMERICAL POLYNOMIALS 47

1.4 Numerical Polynomials

In this section we consider properties of polynomials in several variables with ra-tional coefficients that take integer values for all sufficiently large integer valuesof arguments. Such polynomials will arise later when we study the dimensiontheory of difference algebraic structures.

Definition 1.4.1 A polynomial f(t1, . . . , tp) in p variables t1, . . . , tp (p ≥ 1)with rational coefficients is called numerical if f(t1, . . . , tp) ∈ Z for all suffi-ciently large (t1, . . . , tp) ∈ Np, i.e., there exists an element (s1, . . . , sp) ∈ Np

such that f(r1, . . . , rp) ∈ Z as soon as (r1, . . . , rp) ∈ Np and ri ≥ si for alli = 1, . . . , p.

It is clear that every polynomial in several variables with integer coefficientsis numerical. As an example of a numerical polynomial in p variables with

non-integer coefficients (p ∈ N+) one can consider the polynomialp∏

i=1

(timi

)(m1, . . . , mp ∈ Z), where(

t

k

)=

t(t − 1) . . . (t − k + 1)k!

(k ∈ N+);(

t

0

)= 1;

(t

k

)= 0 (k < 0)

(1.4.1)

In what follows we will often use the relationships between “binomial” numericalpolynomials

(tk

)that arise from well-known identities for binomial coefficients.

In particular, the classical identity(n+1m

)=

(nm

)+(

nm−1

)(n,m ∈ N, n ≥ m > 0)

implies the polynomial identity(t + 1m

)=

(t

m

)+

(t

m − 1

)(1.4.2)

The following proposition gives some other useful relationships between “bino-mial” numerical polynomials.

Proposition 1.4.2 Let n, p, and r be non-negative integers. Thenn∑

i=0

(t + i

i

)=

(t + n + 1

n

); (1.4.3)

n∑i=0

(t + i

r

)=

(t + n + 1

r + 1

)−

(t

r + 1

); (1.4.4)

n∑i=0

(t

i

)(k

n − i

)=

(t + k

n

); (1.4.5)

n∑i=0

2i

(n

i

)(t

i

)=

n∑i=0

(n

i

)(t + i

n

); (1.4.6)

n∑i=0

2i

(n

i

)(t

i

)=

n∑i=0

(−1)n−i2i

(n

i

)(t + i

i

). (1.4.7)


The proof of identities (1.4.3) - (1.4.7) can be found in [110, Section 2.1](and also in the union of books on combinatorics such as [17], [161], and [168]).

If f(t) is a polynomial in one variable t over a ring (in particular, if f(t)is a numerical polynomial), then by the first difference of this polynomial wemean the polynomial ∆f(t) = f(t+1)−f(t). The second, third, etc. differencesof f(t) are defined by induction: ∆kf(t) = ∆(∆k−1f(t)) for k = 2, 3, . . . . Bythe difference of zero order we mean the polynomial f(t) itself. For example, if

f(t) =(

tk

)(k ∈ N), then ∆

(t

k

)=

(t

k − 1

)(see formula (1.4.2)).

More general, if f(t1, . . . , tp) is a polynomial in p variables t1, . . . , tp over aring, then the first difference of f(t1, . . . , tp) relative to ti (1 ≤ i ≤ p) is defined asthe polynomial ∆if(t1, . . . , tp) = f(t1, . . . , ti−1, ti+1, ti+1, . . . , tp)−f(t1, . . . , tp).

If l1, . . . , lp are non-negative integers then the difference of order (l1, . . . , lp)of the polynomial f(t1, . . . , tp) (denoted by ∆l1

1 . . . ∆lpp f(t1, . . . tp)) is defined

by induction as follows: if li > 0 for some i = 1, . . . , p, then ∆l11 . . . ∆lp

p f =∆i(∆l1

1 . . . ∆li−1i−1 ∆li−1

i ∆li+1i+1 . . . ∆lp

p f). (Clearly, ∆i∆jf = ∆j∆if for any i, j =1, . . . , p, so ∆l1

1 . . . ∆lpp f is well-defined). By the difference of order (0, . . . , 0) of

a polynomial f(t1, . . . tp) we mean the polynomial f(t1, . . . tp) itself. It is easyto see that all differences of a numerical polynomial are numerical polynomialsas well.

Remark 1.4.3 In what follows we shall sometimes use another version ofthe first difference (and the differences of higher orders). For any polynomialf(t) in one variable t we define ∆′f(t) = f(t) − f(t − 1) (that is ∆′f(t) =(∆f)(t−1)), (∆′)2f(t) = ∆′(∆′f(t)), etc. In particular, for any k ∈ N, we have

∆′(

t

k

)=

(t − 1k − 1

), (∆′)2

(t

k

)=

(t − 2k − 2

), etc.

If f(t1, . . . , tp) is a polynomial in p variables t1, . . . , tp, then we set∆′

if(t1, . . . tp) = ∆if(t1, . . . , ti−1, ti − 1, ti+1, . . . , tp) (i = 1, . . . , p) and define(∆′

1)l1 . . . (∆′

p)lpf(t1, . . . tp) (l1, . . . , lp ∈ N) in the same way as we define

∆l11 . . . ∆lp

p f(t1, . . . tp) (using operators ∆′i instead of ∆i).

As usual, if f is a numerical polynomial in p variables (p > 1), then deg fand degti

f (1 ≤ i ≤ p) will denote the total degree of f and the degree of frelative to the variable ti, respectively. The following theorem gives a “canonical”representation of a numerical polynomial in several variables.

Theorem 1.4.4 Let f(t1, . . . , tp) be a numerical polynomial in p variablest1, . . . , tp, and let degti

f = mi (m1, . . . , mp ∈ N). Then the polynomialf(t1, . . . , tp) can be represented in the form

f(t1, . . . tp) =m1∑

i1=0

. . .

mp∑ip=0

ai1...ip

(t1 + i1

i1

). . .

(tp + ip

ip

)(1.4.8)

with integer coefficients ai1...ip(0 ≤ ik ≤ mk for k = 1, . . . , p). These coefficients

are uniquely defined by the numerical polynomial.


PROOF. If we divide each power tki (1 ≤ i ≤ p, k ∈ N) by(

ti + k

k

)in

the ring Q[ti], then divide the remainder of this division by(

ti + k − 1k − 1

)and

continue this process, we can represent tki as tki =k∑

j=0

cij

(ti + j

j

)where the

rational coefficients ci0, . . . , cik ∈ Q are defined uniquely. It follows that anyterm btk1

1 . . . tkpp (b = 0) that appears in f(t1, . . . , tp) can be uniquely writ-

ten ask1∑

i1=0

· · ·kp∑

ip=0

ci1...ip

(t1 + i1

i1

). . .

(tip

+ ipj

)(ci1...ip

∈ Q for 0 ≤ i1 ≤

k1, . . . , 1 ≤ ip ≤ kp), where ck1...,kp= k1! . . . kp! b. Thus, our numerical poly-

nomial f(t1, . . . tp) can be uniquely written in the form (1.4.8) with rationalcoefficients ai1...ip

, and it remains to show that all coefficients ai1...ipare in-

tegers. We shall prove this by induction on (m1, . . . , mp), where mi = degtif

(i = 1, . . . , p) and (m1, . . . , mp) is considered as an element of the well-orderedset Np provided with the lexicographic order <lex.

If (m1, . . . , mp) = (0, . . . , 0), our statement is obvious (in this case f(t1, . . . tp)∈ Z for all t1, . . . tp). Let (m1, . . . , mp) = (0, . . . , 0). Applying formula (1.4.3) we

obtain that for any r ∈ N, r ≥ 1, ∆i

(ti + r

r

)=

r−1∑ν=0

(ti + ν

ν

)(i = 1, . . . , p). It

follows that a finite difference ∆l11 . . . ∆lp

p f (0 ≤ li ≤ mi for i = 1, . . . , p) can be

writtenas∆l11 . . . ∆lp

p f(t1, . . . tp) =m1−l1∑j1=0

. . .

mp−lp∑jp=0

bj1...jp

(t1 + j1

j1

). . .

(tp + jp

jp

),

where bj1...jp∈ Q (0 ≤ jν ≤ mν for ν = 1, . . . , p) and bm1−l1,...,mp−lp = am1...mp

.In particular, ∆m1

1 . . . ∆mpp f = am1...mp

∈ Z, since ∆m11 . . . ∆mp

p f is a numericalpolynomial of zero degree.

Thus, g(t1, . . . , tp) = f(t1, . . . , tp) − am1...mp

(t1 + m1

m1

). . .

(tp + mp

mp

)is a

numerical polynomial such that (degt1g, . . . , degtpg) <lex (m1, . . . , mp). By the

inductive hypothesis, g(t1, . . . , tp) can be written in the form (1.4.8) with integercoefficients, therefore the same is true for f(t1, . . . , tp). Corollary 1.4.5 Let f(t) be a numerical polynomial in one variable t and letdeg f = d. Then the polynomial f(t) can be represented in the form

f(t) =d∑

i=0

ai

(t + i

i

)(1.4.9)

where a0, a1, . . . , ad are integers uniquely defined by f(t). In particular, a0 =∆df(t).

Since(t+nn

)(n ∈ N) is an integer for any integer value of t, Theorem 1.4.4

implies the following statement that allows one to define a numerical polynomialas a polynomial that takes integer values for all integer values of arguments.


Corollary 1.4.6 Let f(t1, . . . , tp) be a numerical polynomial in p variablest1, . . . , tp. Then f(s1, . . . , sp) ∈ Z for any element (s1, . . . , sp) ∈ Zp. Corollary 1.4.7 Let f(t) be a numerical polynomial of degree d in one variablet and let s0 be a positive integer. Then there exists a numerical polynomial g(t)with the following properties:

(i) g(s) =s∑

k=s0+1

f(k) for any s ∈ Z, s > s0;

(ii) deg g(t) = d + 1;

(iii) If f(t) =d∑

i=0

ai

(t + i

i

)is a representation of the polynomial f(t) in the

form (1.4.9) (a0, . . . , ad ∈ Z), then the polynomial g(t) can be written as g(t) =d+1∑i=0

bi

(t + i

i

)where bi = ai−1 for i = 1, . . . , d + 1 and b0 ∈ Z. In particular, the

leading coefficients cf =ad

d!and cg =

bd+1

(d + 1)!of the polynomials f(t) and g(t),

respectively, are connected with the equality cg =cf

d + 1.

PROOF. For anys ∈ Z, s > s0,we haves∑

k=s0+1

f(k)=d∑

i=0

ai

s∑k=s0+1

(i + k

i

)=

d∑i=0

ai

[(s + i + 1

i + 1

)−

(s0 + i + 1

i + 1

)](see formula (1.4.4)), so that

s∑k=s0+1

f(k) =

d∑i=0

ai

(s + i + 1

i + 1

)− C where C =

d∑i=0

ai

(s0 + i + 1

i + 1

)∈ Z. Thus, the numerical

polynomial g(t) =d∑

i=0

ai

(t + i + 1

i + 1

)− C satisfies condition (i). It is easy to see

that g(t) satisfies conditions (ii) and (iii) as well: deg g(t) = deg

(t + d + 1

d + 1

)=

d + 1 and g(t) =d+1∑i=0

bi

(t + i

i

)where bi = ai−1 for i = 1, . . . , d + 1 and

b0 = −C ∈ Z.

Numerical polynomials in one variable will be mostly written in the form(1.4.9), but we shall also use another form of such polynomials given by thefollowing statement.

Theorem 1.4.8 Let f(t) be a numerical polynomial in one variable t and letdeg f = d. Then the polynomial f(t) can be represented in the form

f(t) =d∑

i=0

[(t + i

i + 1

)−

(t + i − mi

i + 1

)](1.4.10)


where m0,m1, . . . , md are integers uniquely defined by f(t). Furthermore,the coefficients ai in the representation (1.4.9) of the polynomial f(t) canbe expressed in terms of m0,m1, . . . , md as follows: ad = md and

ai = mi +d−i∑j=1

(−1)j

(mi+j + 1

j + 1

)(1.4.11)

for i = 0, . . . , d − 1.

PROOF. First of all, we use induction on d to prove the existence anduniqueness of representation (1.4.10). If d = 0, then f(t) = a0 ∈ Z so that f(t)

can be written as f(t) =(

t + 11

)−(

t + 1 − a0

1

). Clearly, such a representation

of a0 in the form (1.4.10) is unique.

Let d > 0 and let f(t) =d∑

i=0

ai

(t + i

i

)for some a0, . . . , ad ∈ Z. By

(1.4.4),(

t + d

d + 1

)−

(t + d − ad

d + 1

)=

ad−1∑k=0

(t + d − ad + k

d

)= ad

(t + d

d

)+

ad−1∑k=0

[(t + d − ad + k

d

)−

(t + d

d

)]= ad

(t + d

d

)+

ad−1∑k=0

ad−1−k∑l=0(

t + d − ad + k + l

d − 1

). By the induction hypothesis, the numerical polyno-

mial g(t) = f(t) −[(

t + d

d + 1

)−

(t + d − ad

d + 1

)], whose degree does not exceed

d − 1, has a unique representation as g(t) =d−1∑i=o

[(t + i

i + 1

)−

(t + i − mi

i + 1

)]where m0, . . . , md−1 ∈ Z. It follows that the polynomial f(t) has a uniquerepresentation in the form (1.4.10) (with md = ad).

Now, let us use induction on d = deg f(t) to prove the second part of thetheorem, that is, to prove that the equality

d∑i=0

ai

(t + i

i

)=

d∑i=0

[(t + i

i + 1

)−

(t + i − mi

i + 1

)](1.4.12)

(a0, . . . , ad,m0, . . . , md ∈ Z) implies (1.4.11).Applying the operator ∆′ to the both sides of (1.4.12) we obtain (see Re-

mark 1.4.3) thatd−1∑i=0

ai+1

(t + i

i

)=

d−1∑i=0

[(t + i

i + 1

)−

(t + i − mi+1

i + 1

)]. By the

induction hypothesis, ad = md and ai+1 = mi+1 +d−(i+1)∑

j=1

(−1)j

(mi+1+j + 1

j + 1

)for i = 0, . . . , d − 2, so formula (1.4.11) is true for i = 1, . . . , d − 1. In order


to prove the formula for i = 0 (and to complete the proof of the theorem)one should just set t = −1 in identity (1.4.12) and obtain the relationship

a0 = m0 +d∑

i=1

[−(

i − 1 − mi

i + 1

)]= m0 +

d∑i=1

(−1)i

(mi + 1i + 1

), that is, formula

(1.4.11) for i = 0.

We conclude this section with some combinatorial results related to the prob-lems of computation of difference dimension polynomials considered in furtherchapters. The proof of these results, presented in Proposition 1.4.9 below, canbe found in [110, Section 2.1].

In what follows, for any positive integer m and non-negative integer r, weset

µ(m, r) = Card

(x1, . . . , xm) ∈ Nm|

m∑i=1

xi = r

µ+(m, r) = Card

(x1, . . . , xm) ∈ Nm|

m∑i=1

xi = r and xi > 0 for i = 1, . . . , m

µ(m, r) = Card

(x1, . . . , xm) ∈ Zm|

m∑i=1

|xi| = r

ρ(m, r) = Card

(x1, . . . , xm) ∈ Nm|

m∑i=1

xi ≤ r

ρ(m, r) = Card

(x1, . . . , xm) ∈ Nm|

m∑i=1

|xi| ≤ r

Furthermore, for any elements u = (u1, . . . , um), v = (v1, . . . , vm) ∈ Nm

such that u ≤P v, we set Cmr(u, v) = Card (x1, . . . , xm) ∈ Nm|m∑

i=1

xi = r and

ui ≤ xi ≤ vi for i = 1, . . . , m

Proposition 1.4.9 With the above notation,

µ(m, r) =(

m + r − 1m − 1

)(1.4.13)

µ+(m, r) =(

r − 1m − 1

)(1.4.14)

µ(m, r) =m∑

i=0

2i

(m

i

)(r − 1i − 1

)(1.4.15)

ρ(m, r) =(

r + m

m

)(1.4.16)

1.5. DIMENSION POLYNOMIALS OF SETS OF M-TUPLES 53

ρ(m, r) =m∑

i=0

2i

(m

i

)(r

i

)=

m∑i=0

(m

i

)(r + i

m

)=

m∑i=0

(−1)m−i2i

(m

i

)(r + i

i

)(1.4.17)

Cmr(u, v) =(

m + r − R − 1m − 1

)+

m∑k=1

(−1)k∑

1≤j1<···<jk≤mdj1+···+djk

≤r−R

(m + r − R − dj1 − · · · − djk

− 1m − 1

)(1.4.18)

where R = u1 + · · · + um and di = vi − ui + 1 (1 ≤ i ≤ m).

1.5 Dimension Polynomials of Sets of m-tuples

In this section we deal with subsets of the sets Nm and Zm where m is apositive integer. Elements of these sets will be called m-tuples, the addition andmultiplication of m-tuples are defined in the natural way: if a = (a1, . . . , am)and b = (b1, . . . , bm) are two elements of Zm, then a+b = (a1 +b1, . . . , am +bm)and ab = (a1b1, . . . , ambm).

Considering N and Z as ordered sets with respect to the natural order on Z,we will often treat Nm and Zm as partially ordered sets relative to the productorder ≤P such that (a1, . . . , am) ≤P (b1, . . . , bm) if and only if ai ≤ bi fori = 1, . . . , m (≤ denotes the natural order on Z). The direct products Nm ×Nk

and Zm × Nk (Nk denotes the set 1, . . . , k where k ∈ N, k ≥ 1) will bealso considered as partially ordered sets with respect to the product order ≤P

(in this case (a1, . . . , am, a) ≤P (b1, . . . , bm, b) means ai ≤ bi for i = 1, . . . , mand a ≤ b). Furthermore, we are going to consider the lexicographic order≤lex on the sets of the form Nm, Zm, Nm × Nk, and Zm × Nk, the gradedlexicographic order ≤grlex on Nm ((a1, . . . , am) ≤grlex (b1, . . . , bm) if and onlyif (

∑mi=1 ai, a1, . . . , am) ≤lex (

∑mi=1 bi, b1, . . . , bm) in Nm+1), and the order ≤0

on the sets of the form Nm × Nk such that (a1, . . . , am, a) ≤0 (b1, . . . , bm, b) ifand only if (

∑mi=1 ai, a, a1, . . . , am) ≤lex (

∑mi=1 bi, b, b1, . . . , bm) in Nm+2.

The following statement, whose proof can be found in [110, Section 2.2],gives some useful properties of the introduced orders.

Lemma 1.5.1 (i) Any infinite subset of Nm × Nk (m, k ∈ N, k ≥ 1) containsan infinite sequence strictly increasing with respect to the product order and suchthat the projections of all elements of the sequence onto Nk are equal.

(ii) The set Nm × Nk is well-ordered with respect to the order ≤0 and thefollowing two conditions hold:

(a) (a1, . . . , am, a) ≤0 (a1 + c1, . . . , am + cm, a) for any elements (a1, . . . ,am, a) ∈ Nm × Nk, (c1, . . . , cm) ∈ Nm.

(b) If (a1, . . . , am, a) ≤0 (b1, . . . , bm, b), then (a1 + c1, . . . , am + cm, a) ≤0

(b1 + c1, . . . , bm + cm, b) for any (c1, . . . , cm) ∈ Nm.


(iii) The set Nm×Nk is well-ordered with respect to any linear order satisfyingcondition (a).

By a partition of a set Nm = 1, . . . , m (m ∈ N+) we mean a representationof Nm as a union of disjoint non-empty subsets, that is, a representation of theform

Nm = σ1

⋃· · ·

⋃σp (1.5.1)

where p is a positive integer and σi

⋂σj = ∅ for i = j (1 ≤ i, j ≤ p). Such a

partition will be denoted by (σ1, . . . , σp).Let (σ1, . . . , σp) be a partition of the set Nm and let mi = Card σi (i =

1, . . . , p). Then for any set A ⊆ Nm and for any r1, . . . , rp ∈ N, A(r1, . . . , rp)will denote the set of all m-tuples (a1, . . . , am) ∈ A such that

∑i∈σk

ai ≤ mk

for k = 1, . . . , p.Furthermore, if A ⊆ Nm, then VA will denote the set of all m-tuples v =

(v1, . . . , vm) ∈ Nm that are not greater than or equal to any m-tuple from A withrespect to the product order ≤P . (Clearly, an element v = (v1, . . . , vm) ∈ Nm

belongs to VA if and only if for any element (a1, . . . , am) ∈ A there existsi ∈ N, 1 ≤ i ≤ m, such that ai > vi.)

The following result can be considered as a fundamental theorem on numer-ical polynomials associated with subsets of Nm. Its proof, as well as the proofsof the other results of this section, can be found in [110, Section 2.2].

Theorem 1.5.2 Let (σ1, . . . , σp) be a partition of the set Nm and let mi =Card σi (i = 1, . . . , p). Then for any set A ⊆ Nm, there exists a numericalpolynomial ωA(t1, . . . , tp) with the following properties.

(i) ωA(r1, . . . , rp) = CardVA(r1, . . . , rp) for all sufficiently large (r1, . . . , rp) ∈Np.

(ii) The total degree of the polynomial ωA does not exceed m and degtiωA ≤ mi

for i = 1, . . . , p.(iii) deg ωA = m if and only if the set A is empty. In this case ωA(t1, . . . , tp) =p∏

i=1

(ti + mi

mi

).

(iv) ωA is a zero polynomial if and only if (0, . . . , 0) ∈ A.

Definition 1.5.3 The polynomial ωA(t1, . . . , tp) whose existence is stated byTheorem 1.5.2 is called the dimension polynomial of the set A ⊆ Nm associatedwith the partition (σ1, . . . , σp) of Nm.

As a consequence of Theorems 1.5.2 (in the case p = 1) we obtain thefollowing result due to E. Kolchin.

Theorem 1.5.4 Let A be a subset of Nm (m ≥ 1). Then there exists a numer-ical polynomial ωA(t) such that

(i) ωA(r) = CardVA(r) for all sufficiently large r ∈ N. (In accordance withour notation, VA(r) = (x1, . . . , xm) ∈ VA|x1 + · · · + xm ≤ r.)


(ii) deg ωA ≤ m.

(iii) deg ωA = m if and only if A = ∅. In this case ωA(t) =(

t + m

m

).

(iv) ωA = 0 if and only if (0, . . . , 0) ∈ A.

Definition 1.5.5 The polynomial ωA(t), whose existence is stated by Theorem1.5.4, is called the Kolchin polynomial of the set A ⊆ Nm.

It is clear that if A is any subset of Nm and A′ is the set of all minimalelements of A with respect to the product order on Nm, then the set A′ isfinite and ωA(t1, . . . , tp) = ωA′(t1, . . . , tp). The following proposition reducesthe computation of the dimension polynomial of a finite set A in Nm to thecomputation of dimension polynomials of subsets of Nm with less than CardAelements.

Proposition 1.5.6 Let A = a1, . . . , an be a finite subset of Nm (m ≥ 1, n ≥1), (σ1, . . . , σp) a partition of the set Nm, and mi = Card σi (i = 1, . . . , p).Let ak = (ak1, . . . , akm) (1 ≤ k ≤ n) and for every k = 1, . . . , n − 1, leta′

k = (maxak1 − an1, 0 . . . ,maxakm − anm, 0). Furthermore, let A′ =a′

1, . . . , a′n−1 and A′′ = a1, . . . , an−1. Then (using the notation of Theo-

rem 1.5.2)

ωA(t1, . . . , tp) = ωA′′(t1, . . . , tp)−ωA′

(t1−

∑j∈σ1

anj , . . . , tp−∑j∈σp

anj

). (1.5.2)

Applying the last result one can obtain the following theorem (see [110,Section 2.2] for the proof).

Theorem 1.5.7 Let A = a1, . . . , an be a finite subset of Nm, where n isa positive integer, and m = m1 + · · · + mp for some nonnegative integersm1, . . . ,mp (p ≥ 1). Let ak = (ak1, . . . , akm) (1 ≤ k ≤ n) and for anyl ∈ N, 0 ≤ l ≤ n, let Γ(l, n) denote the set of all l-element subsets of theset Nn = 1, . . . , n. Furtermore, let a∅j = 0 and for any σ ∈ Γ(l, n), σ = ∅, letaσj = maxaνj |ν ∈ σ (1 ≤ j ≤ m). Finally, let bσi =

∑h∈σi

aσh (i = 1, . . . , p).

Then

ωA(t1, . . . , tp) =n∑

l=0

(−1)l∑

σ∈Γ(l,n)

p∏i=1

(ti + mi − bσi

mi

)(1.5.3)

Theorem 1.5.7 provides a method of computation of a numerical polynomialassociated with any subset of Nm (and with any given partition (σ1, . . . , σp) ofthe set Nm): one should first find the set of all minimal points of the subset andthen apply Theorem 1.5.7. The corresponding algorithm can be found in [110,Section 2.3].


Corollary 1.5.8 With the notation of Theorem 1.5.7, the Kolchin polynomialof a set A = a1, . . . , an ⊆ Nm can be found by the following formula:

ωA(t) =n∑

l=0

(−1)l∑

σ∈Γ(l,n)

(t + m −∑m

j=1 aσj

m

)(1.5.4)

Example 1.5.9 Let B = (1, 2), (3, 1) be a subset of N2 and let (σ1, σ2) be apartition of N2 such that σ1 = 1 and σ2 = 2. Then (using the notation ofTheorem 1.5.7) a1,21 = 3, a1,22 = 2, a11 = 1, a12 = 2, a21 = 3, a22 =1, b11 = a11 = 1, b12 = a12 = 2, b21 = a21 = 3, b22 = a22 = 1,b1,21 = 3, b1,22 = 2 whence ωB(t1, t2) = (t1 + 1)(t2 + 1) − [(t1 + 1 − 1)(t2 + 1 − 2) + (t1 + 1 − 3)(t2 + 1 − 1)] + (t1 + 1 − 3)(t2 + 1 − 3) = t1 + t2 + 3.

Now we are going to introduce some natural order on the set of all numericalpolynomials and show that the set of all Kolchin polynomials is well-orderedwith respect to this order.

Definition 1.5.10 Let f(t) and g(t) be two numerical polynomials in one vari-able t. We say that f(t) is less than or equal to g(t) and write f(t) g(t) iff(r) ≤ g(r) for all sufficiently large r ∈ Z. If f(t) g(t) and f(t) = g(t), wewrite f(t) ≺ g(t).

It is easy to see that ≺ satisfies the axioms of an order on the set of all numer-ical polynomials and if one represents numerical polynomials in the canonical

form (1.4.9), f(t) =m∑

i=0

ai

(t + i

i

)and g(t) =

m∑i=0

bi

(t + i

i

), then f(t) g(t) if

and only if (am, am−1, . . . , a0) is less than (bm, bm−1, . . . , b0) with respect to thelexicographic order on Zm.

Let W denote the set of all Kolchin polynomials ωE(t) where E is a subset ofsome set Nm (m = 1, 2, . . . ). The following theorem, whose proof can be foundin [110, Section 2.4], gives some properties of the set W . In particular, it showsthat the set W is well-ordered with respect to ≺ (this result is due to W. Sit[170]).

Theorem 1.5.11 (i) Let ω(t) =d∑

i=0

ai

(t + i

i

)be a numerical polynomial writ-

ten in the canonical form (1.4.9) (a0, . . . , ad ∈ Z). Then ω(t) ∈ W if and only

if ad > 0 and the polynomial v(t) = ω(t + d) −(

t + d + 1 + ad

d + 1

)+

(t + d + 1

d + 1

)belongs to W . (Note that deg v(t) < d.)

(ii) If ω1(t), ω1(t) ∈ W , then ω1(t) + ω1(t) ∈ W and ω1(t)ω1(t) ∈ W .(iii) Let ω(t) ∈ W and p, q ∈ N, p > 0. Then the polynomials pω(t) + q,

ω(pt + q), ω(t)− ω(t− 1),(

t + q

q

), and

(t + q

q

)−

(t + q − p

q

)belong to W .


(iv) If m, c1, . . . , ck ∈ N+ (k ≥ 1), then the numerical polynomial ω(t) =(t + m

m

)+

k∑i=1

ci

(t + m − i

m

)belongs to W .

(v) Let E ⊆ Nm (m ≥ 1) and let ωE(t1, . . . , tp) be a dimension polynomial ofE associated with some partition of the set Nm into p disjoint subsets (p ≥ 1).Then ωE(t + h1, . . . , t + hp) ∈ W for any non-negative integers h1, . . . , hp.(vi) The set W is well-ordered with respect to the order ≺.

In the rest of this section we consider subsets of Zm (m ≥ 1). The results weare going to present are of the same type as the statements of Theorems 1.5.2and 1.5.7 for subsets of Nm. The proofs can be found in [110, Section 2.5].

In what follows, Z+ and Z− will denote the set of all positive and the set ofall non-positive integers, respectively, and the set Zm will be considered as theunion

Zm =⋃

1≤j≤2m

Z(m)j (1.5.5)

where Z(m)1 , . . . ,Z(m)

2m are all distinct Cartesian products of m factors each ofwhich is either Z− or N. We assume that Z(m)

1 = Nm and call a set Z(m)j

(1 ≤ j ≤ 2m) an ortant of Zm. Furthermore, we suppose that a partition(σ1, . . . , σp) (p ≥ 1) of the set Nm = 1, . . . , m is fixed (that is, we fix arepresentation Nm in the form (1.5.1)), and mi = Card σi for i = 1, . . . , p(clearly,

∑pi=1 mi = m).

We shall consider Zm as a partially ordered set with respect to the order defined as follows: (x1, . . . , xm) (y1, . . . , ym) if and only if (x1, . . . , xm) and(y1, . . . , ym) belong to the same ortant of Zm and |xi| ≤ |yi| for i = 1, . . . , m.For any set B ⊆ Zm, VB will denote the set of all m-tuples in Zm which ex-ceed no element of B with respect to the order . (In other words, an elementv = (v1, . . . , vm) belongs to VB if and only if b v for any b ∈ B). Further-more, if r1, . . . , rp ∈ Nm, then B(r1, . . . , rp) will denote the set of all elements(x1, . . . , xm) ∈ B such that

∑j∈σi

|xj | ≤ ri for i = 1, . . . , p.

Definition 1.5.12 A subset V of Nm is said to be an initial subset of Nm ifthe inclusion v ∈ V implies that v′ ∈ V for any element v′ ∈ Nm such thatv′ ≤P v. Similarly a subset V of Zm is said to be an initial subset of Zm if theinclusion v ∈ V implies that v′ ∈ V for any element v′ ∈ Zm such that v′ v.

Let us consider the mapping γ : Zm → N2m such that γ(z1, . . . , zm) =(maxz1, 0, . . . ,maxzm, 0,max−z1, 0, . . . ,max−zm, 0), and let us fix thepartition (σ′

1, . . . , σ′p) of the set N2m = 1, . . . , 2m such that σ′

i = σi

⋃j +m|j ∈ σi (1 ≤ i ≤ p). The following proposition gives some properties of themapping γ and initial subsets of Nm and Zm.

Proposition 1.5.13 (i) Let B ⊆ Zm and r1, . . . , rp ∈ N. Then Card B(r1, . . . , rp) = Card γ(B)(r1, . . . , rp) and γ(VB) = Vγ(B)

⋂γ(Zm).


(ii) A set V ⊆ Nm is an initial subset of Nm if and only if V = VA for somefinite set A ⊆ Nm. Similarly, a set V ⊆ Zm is an initial subset of Zm if andonly if V = VC for some finite set C ⊆ Zm.(iii) If V is an initial subset of Zm, then γ(V) is an initial subset of N2m.

The following result can be considered as a fundamental theorem on dimen-sion polynomials of subsets of Zm.

Theorem 1.5.14 Let B be a subset of Zm (m ≥ 1) and let (σ1, . . . , σp) (p ≥ 1)be a fixed partition of the set Nm. Furthermore, let mi = Card σi (i = 1, . . . , p).Then there exists a numerical polynomial φB(t1, . . . , tp) in p variables t1, . . . , tpwith the following properties.

(i) φB(r1, . . . , rp)=Card VB(r1, . . . , rp) for all sufficiently large (r1, . . . , rp) ∈Np.

(ii) The total degree of the polynomial φB does not exceed m and degtiφB ≤ mi

for i = 1, . . . , p.(iii) Let γ(B) = A ⊆ N2m and let aj (1 ≤ j ≤ m) denote the 2m-tuple in N2m

whose jth and (j + m)th coordinates are 1 and all other coordinates are equalto 0. Then φB(t1, . . . , tp) = ωA′(t1, . . . , tp) where A′ =

⋃mi=1aj

⋃A and ωA′ is

the dimension polynomial of the set A associated with the partition (σ′1, . . . , σ

′p)

of N2m such that σ′i = σi

⋃j + m|j ∈ σi (1 ≤ i ≤ p).

(iv) Let Bj = B⋂

Z(m)j and let Aj be a subset of Nm obtained by replacing

every element b ∈ Bj with an element b′ whose coordinates are the absolutevalues of the corresponding coordinates of b (j = 1, . . . , 2m). Then

φB(t1, . . . , tp) =2m∑j=1

ωAj(t1, . . . , tp)

+p∏

j=1

[mj∑i=0

(−1)mj−i2i

(mj

i

)(tj + i

i

)]

−2m

p∏j=1

(tj + mj

mj

)(1.5.6)

where ωAj(t1, . . . , tp) (1 ≤ j ≤ 2m) is the dimension polynomial of the set Aj

associated with the partition (σ1, . . . , σp) of Nm (see Definition 1.5.3).(v) If B = ∅, then deg φB = m and

φB(t1, . . . , tp) =p∏

j=1

[mj∑i=0

(−1)mj−i2i

(mj

i

)(tj + i

i

)]

=p∏

j=1

[mj∑i=0

2i

(mj

i

)(tji

)]

=p∏

j=1

[mj∑i=0

(mj

i

)(tj + i

mj

)](1.5.7)

(vi) φB(t1, . . . , tp) = 0 if and only if (0, . . . , 0) ∈ B.


Definition 1.5.15 The polynomial φB(t1, . . . , tp), whose existence is estab-lished by Theorem 1.5.14, is called the Z-dimension polynomial of the setB ⊆ Zm associated with the partition (σ1, . . . , σp).

Exercise 1.5.16 With the notation of Theorem 1.5.14, show that

φB(t1, . . . , tp) =2m∑j=1

ωAj(t1, . . . , tp) +

[m1−1∑i=0

(−1)m1−i2i

(m1

i

)(t1 + i

i

)]

×p∏

λ=2

[mλ∑i=0

(−1)mλ−i2i

(mλ

i

)(tλ + i

i

)]

+p−2∑ν=1

2m1+···+mν

ν∏l=1

(tl + ml

ml

)

×[

mν+1−1∑i=0

(−1)mν+1−i2i

(mν+1

i

)(tν+1 + i

i

)]

×p∏

λ=ν+2

[mλ∑i=0

(−1)mλ−i2i

(mλ

i

)(tλ + i

i

)]

+ 2m−mp

p−1∏ν=1

(tν + mν

mν

)

×[

mp−1∑i=0

(−1)mp−i2i

(mp

i

)(tp + i

i

)](1.5.8)

[Hint: Prove, first, thatp∏

i=1

(xi + yi) = x1 . . . xp + y1

p∏λ=2

(xλ + yλ) +

p−2∑ν=1

x1 . . . xνyν+1

p∏λ=ν+2

(xλ +yλ)+x1 . . . xp−1yp for any real numbers x1, . . . , xp,

y1, . . . , yp (use induction on p).]

In the case p = 1, Theorem 1.5.14 implies the following result.

Theorem 1.5.17 Let B be a subset of Zm (m ≥ 1). Then there exists anumerical polynomial φB(t) in one variable t with the following properties.

(i) φB(r) = CardVB(r) for all sufficiently large r ∈ N;(ii) deg φB ≤ m.(iii) There exist subsets A1, . . . , A2m of Nm such that

φB(t) =2m∑j=1

ωAj(t) +

m−1∑i=0

(−1)m−i2i

(m

i

)(t + i

i

)(1.5.9)

where ωAj(t) (1 ≤ j ≤ 2m) is the Kolchin dimension polynomial of the set Aj

(see Definition 1.5.5);


(iv) If B = ∅, then deg φB = m and

φ∅(t)=m∑

i=0

(−1)m−i2i

(m

i

)(t + i

i

)=

m∑i=0

2i

(m

i

)(t

i

)=

m∑i=0

(m

i

)(t + i

m

);

(1.5.10)(v) φB(t) = 0 if and only if (0, . . . , 0) ∈ B.

Definition 1.5.18 With the notation of Theorem 1.5.17, The polynomialφB(t), is said to be a standard Z-dimension polynomial of the set B ⊆ Zn.

Proposition 1.5.19 (see [110, Proposition 2.5.17]). The set of all standard Z-dimension polynomials of subsets of Zm (for all m = 1, 2, . . . ) coincides withthe set of all Kolchin polynomials W .

If B ⊆ Nm and a partition (σ1, . . . , σp) of Nm is fixed, then one canassociate with the set B two numerical polynomials, the dimension polyno-mial ωB(t1, . . . , tp) and Z-dimensional polynomial φB(t1, . . . , tp). It is easyto see that these two polynomials are not necessarily equal. For example,ω∅ = φ∅ as it follows from Theorem 1.5.2(iii) and Theorem 1.5.14(v). An-other illustration is given by the dimension and Z-dimension polynomialsof a set B ⊆ Nm whose elements do not contain zero coordinates. Treat-ing B as a subset of Zm we obtain that VB = VB

⋃(Zm \ Nm) whence

CardVB(r1, . . . , rp) = CardVB(r1, . . . , rp) + Card (Zm \ Nm)(r1, . . . , rp) =

CardVB(r1, . . . , rp) +p∏

i=1

⎡⎣ mi∑j=0

(−1)mi−j2j

(mi

j

)(ri + j

j

)⎤⎦−p∏

i=1

(ri + mi

mi

)for

all sufficiently large (r1, . . . , rp) ∈ Np. Therefore, in this case

φB(t1, . . . , tp) = ωB(t1, . . . , tp) +p∏

i=1

⎡⎣ mi∑j=0

(−1)mi−j2j

(mi

j

)(ti + j

j

)⎤⎦−

p∏i=1

(ti + mi

mi

). (1.5.11)

Example 1.5.20 Let us consider the set B = (1, 2), (3, 1) as a subset of Z2

and fix a partition (σ1, σ2) of the set N2 = 1, 2 such that σ1 = 1 andσ2 = 2. The dimension polynomial of the set B (treated as a subset of N2))was found in Example 1.5.9: ωB(t1, t2) = t1 + t2 + 3. Now we can use formula(1.5.11) to find the Z-dimension polynomial φB(t1, t2):φB(t1, t2) = ωB(t1, t2) +

[∑1j=0(−1)1−j2j

(1j

)(t1+j

j

)] [∑1j=0(−1)1−j2j

(1j

)(t2+j

j

)]−(

t1+11

)(t2+1

1

)= t1 + t2 +3+ [−1+2(t1 +1)][−1+2(t2 +1)]− (t1 +1)(t2 +1) =

3t1t2 + 2t1 + 2t2 + 3.

Let B be a subset of Zm (m ∈ N, m ≥ 1) and for any l ∈ N, 0 ≤ l ≤ m,let ∆(l,m) denote the set of all l-element subsets of the set Nm = 1, . . . , m.


Furthermore, for any δ ∈ ∆(l,m), let Bδ denote the set of all elements of B whosecoordinates with indices from the set δ are equal to zero, and let Bδ denote thesubset of Zm−l obtained by omitting in every element of Bδ coordinates withindices in δ (these coordinates are equal to zero). Finally, let Bδj = Bδ

⋂Z(m−l)

j

(1 ≤ j ≤ 2m−l) and let Bδj be the set of all elements b ∈ Nm−l such that b isobtained from some element b ∈ Bδj by replacing the coordinates of b with theirabsolute values. The following analog of Theorem 1.5.7 gives one more formulafor computation of the Z-dimension polynomial of a set B ⊆ Zm associated withthe partition (σ1, . . . , σp) of Nm. The proof of this result can be found in [110,Section 2.5].

Theorem 1.5.21 With the above notation and with the fixed partition (1.5.1)of the set Nm,

φB(t1, . . . , tp) =m∑

l=0

(−1)l2m∑j=1

∑δ∈∆(l,m)

ωBδj(t1, . . . , tp) (1.5.12)

where ωBδj(t1, . . . , tp) is the dimension polynomial of the set Bδj ⊆ Nm−l (see

Definition 1.5.3) that corresponds to the partition of Nm−l into disjoint subsetsσ′

i = σi \ δ, 1 ≤ i ≤ p. (If some σ′i is empty, then the polynomial ωBδj

does notdepend on the corresponding variable ti.)

Exercise 1.5.22 Use the last theorem to show that the Z-dimension polyno-mial in two variables associated with the set (1, 2), (3, 1) ⊆ Z2 in Exam-ple 1.5.20 is equal to 3t1t2 + 2t1 + 2t2 + 3. (We consider the same partition(σ1, σ2) of the set N2 = 1, 2: σ1 = 1 and σ2 = 2.)

Let B ⊆ Zm (m ∈ N, m ≥ 1) and let B⋂Zm

j = ∅ for j = 1, . . . , 2m.If b = (b1, . . . , bm) ∈ Zm and bi1 , . . . , bik

are all zero coordinates of b, wherek ≥ 1, 1 ≤ i1 < · · · < ik, let b′ denote the element of Zm obtained from b byreplacing every its zero coordinate with 1. (For example, if b = (−3, 0, 2, 0) ∈ Z4,then b′ = (−3, 1, 2, 1).) Let B′ be a subset of Zm obtained by adjoining to Ball elements of the form b′ where b is an element of B with at least one zerocoordinate. It is easy to see that the Z-dimension polynomials φB(t1, . . . , tp)and φB′(t1, . . . , tp) are equal (both polynomials correspond to partition (1.5.1)of the set Nm) and B′ ⋂Z(m)

j = ∅ for j = 1, . . . , 2m.Let B′

j = (|b1|, . . . , |bm|) | (b1, . . . , bm) ∈ B′ ⋂Zmj ⊆ Nm (1 ≤ j ≤ 2m).

Then B′j = ∅ hence deg ωB′

j(t1, . . . , tp) < m for j = 1, . . . , 2m (see Theo-

rem 1.5.2(iii)). Furthermore, the relation (1.5.6) implies that deg φB = deg φB′ <

m. In particular, if B⋂Z(m)

j = ∅, then the standard Z-dimension polynomial

of the set B can be represented in the form φB(t) =m−1∑i=0

ai

(t + i

i

)where

a0, . . . , ai−1 ∈ Z. The following theorem gives the leading coefficient am−1 ofsuch a polynomial in the case m = 2.


Theorem 1.5.23 Let B be a finite subset of Z2 whose elements are pairwiseincomparable with respect to the order and let r be the number of elementsof B which have at least one zero coordinate. Suppose that each of the setsBj = B⋂

Z(2)j (1 ≤ j ≤ 4) is nonempty and Bj = bj1, . . . , bjn(j) where

bjν = (bjν1, bjν2) (j = 1, . . . , 4; n(j) ≥ 1; 1 ≤ ν ≤ n(j)). Furthermore, letbj1 = min

1≤ν≤n(j)|bjν1|, bj2 = min

1≤ν≤n(j)|bjν2|, and let b = (bj1, bj2). Then

φB(t) = a1t + a0

where

a1 =4∑

j=1

(bj1 + bj2) + r − 4.

The results on dimension polynomials of subsets of Nm and Zm can be easilygeneralized to subsets of Nm×Zn where m and n are positive integers. As before,let ≤P and denote, respectively, the product order on Nm and the order onZn introduced in this section (and based on the corresponding partition of Zn).Then one can consider Nm × Zn as a partially ordered set with respect to theorder such that (a1, . . . , am, b1, . . . , bn) (a′

1, . . . , a′m, b′1, . . . , b

′n) if and only

if (a1, . . . , am) ≤P (a′1, . . . , a

′m) and (b1, . . . , bn) (b′1, . . . , b

′n). Furthermore, for

any A ⊆ Nm × Zn, let WA denote the set of all (m + n)-tuples from Nm × Zn

that exceed no element of A with respect to the order .Let (σ1, . . . , σp) and (σ′

1, . . . , σ′q) be fixed partition of the sets Nm and Nn,

respectively (p, q ∈ N, p ≥ 1, q ≥ 1). Then for any (r1, . . . , rp+q) ∈ Np+q and forany B ⊆ Nm×Zn, B(r1, . . . , rp+q) will denote the set (a1, . . . , am, b1, . . . , bn) ∈B | ∑i∈σ1

ai ≤ r1, . . . ,∑

i∈σpai ≤ rp,

∑i∈σ′

1bi ≤ rp+1, . . . ,

∑i∈σ′

qbi ≤ rp+q.

A set W ⊆ Nm × Zn is called an initial subset of Nm × Zn if the inclusionw ∈ W implies that w′ ∈ W for any (m + n)-tuple w′ ∈ Nm × Zn such thatw′ w.

Let us consider a mapping ρ : Nm × Zn → Nm+2n such that ρ((a1, . . . , am,b1, . . . , bn)) = (a1, . . . , am,maxb1, 0, . . . , maxbn, 0, max−b1, 0, . . . ,max−bn, 0) for any element (a1, . . . , am, b1, . . . , bn) ∈ Nm × Zn. The follow-ing result is an analog of Proposition 1.5.13.

Proposition 1.5.24 (i) Let A ⊆ Nm × Zn and r1, . . . , rp+q ∈ N. ThenCard A(r1, . . . , rp+q) = Card ρ(A)(r1, . . . , rp+q) and ρ(WA) = Wρ(A)

⋂ρ(Nm×

Zn).(ii) A set W ⊆ Nm×Zn is an initial subset of Nm×Zn if and only if W = WA

for some finite set A ⊆ Nm × Zn.(iii) If W is an initial subset of Nm × Zn, then ρ(W ) is an initial subset of

Nm+2n.

The following statement generalizes theorems 1.5.2 and 1.5.14.

Theorem 1.5.25 Let A be a subset of Nm × Zn (m ≥ 1, n ≥ 1) and let(σ1, . . . , σp) and (σ′

1, . . . , σ′q) (p ≥ 1, q ≥ 1) be fixed partitions of the sets


Nm and Nn, respectively. Furthermore, let mi = Card σi (i = 1, . . . , p) andnj = Card σ′

j (j = 1, . . . , q). Then there exists a numerical polynomialψA(t1, . . . , tp+q) in p + q variables t1, . . . , tp+q with the following properties.

(i) ψA(r1, . . . , rp+q) = Card WA(r1, . . . , rp+q) for all sufficiently large (r1, . . . ,rp+q) ∈ Np+q .

(ii) The total degree of the polynomial ψA does not exceed m+n, degtiψA ≤ mi

for i = 1, . . . , p, and degtjψA ≤ nj for j = 1, . . . , q .

(iii) Let B = ρ(A)⋃e1, . . . , en where ek (1 ≤ k ≤ n) denotes the (m + 2n)-

tuple in Nm+2n whose (m+k)th and (m+k+n)th coordinates are equal to 1 andall other coordinates are equal to 0. Then ψA(t1, . . . , tp+q) = ωB(t1, . . . , tp+q)where ωB is the dimension polynomial of the set B associated with the partition(σ1, . . . , σp+q) of Nm+2n such that σi = σi for i = 1, . . . , p and σp+j = σ′

j

⋃k+n|k ∈ σ′

j for j = 1, . . . , n.(iv) If A = ∅, then deg ψA = m + n and

ψA(t1, . . . , tp+q) =p∏

i=1

(ti + mi

mi

) q∏j=1

[nj∑

k=0

(−1)nj−k2k

(nj

k

)(tj + k

k

)]

=p∏

i=1

(ti + mi

mi

) q∏j=1

[nj∑

k=0

2k

(nj

k

)(tjk

)]

=p∏

i=1

(ti + mi

mi

) q∏j=1

[nj∑

k=0

(nj

k

)(tj + k

nj

)](1.5.13)

(v) ψA(t1, . . . , tp) = 0 if and only if (0, . . . , 0) ∈ A.

PROOF. By Proposition 1.5.24(iii), ρ(WA) is an initial subset of Nm+2n,and by Proposition 1.5.24(ii), there exists a set A′ ⊆ Nm+2n such thatρ((WA) = VA′ . Therefore (see Proposition 1.5.24(i)), Card WA(r1, . . . , rp+q) =CardVA′(r1, . . . , rp+q) for all r1, . . . , rp+q ∈ N. Applying Theorem 1.5.2 we ob-tain that there exists a numerical polynomial ψA(t1, . . . , tp+q) in p+ q variablest1, . . . , tp+q that satisfies conditions (i) and (ii) of our theorem.

It is easy to see that a set A′ with the property ρ((WA) = VA′ can be chosenas the set of all minimal (with respect to the product order on Nm+2n) ele-ments of the set Nm+2n \ρ(WA) = Nm+2n \ (Vρ(A)

⋂ρ(Nm × Zn)) = (Nm+2n \

Vρ(A))⋃

(Nm+2n \ ρ(Nm × Zn)). Furthermore, all minimal elements of the setNm+2n \Vρ(A) are contained in ρ(A) and any element a ∈ Nm+2n \ρ(Nm ×Zn)exceeds some ei (1 ≤ i ≤ n) with respect to the product order (since a /∈ρ(Nm ×Zn), there exists an index j (1 ≤ j ≤ n) such that both (m + j)th and(m+n+j)th cordinates of a are positive). Thus, the set B = ρ(A)

⋃e1, . . . , ensatisfies the condition ρ(WA) = VB whence φA(t1, . . . , tp+q) = ωB(t1, . . . , tp+q)(see Theorem 1.5.2).

If A = ∅, then for any r1, . . . , rp+q ∈ N, WA(r1, . . . , rp+q) = (a1, . . . , am,b1, . . . , bn) ∈ Nm × Zn|∑i∈σ1

ai ≤ r1, . . . ,∑

i∈σpai ≤ rp,

∑i∈σ′

1bi ≤ rp+1, . . . ,


∑i∈σ′

qbi ≤ rp+q. By Proposition 1.4.9, CardWA(r1, . . . , rp+q) =

p∏i=1

(ri + mi

mi

)q∏

j=1

[nj∑

k=0

(−1)nj−k2k

(nj

k

)(rj + k

k

)]=

p∏i=1

(ri + mi

mi

) q∏j=1

[nj∑

k=0

2k

(nj

k

)(rj

k

)]=

p∏i=1

(ri + mi

mi

) q∏j=1

[nj∑

k=0

(nj

k

)(rj + k

nj

)]for all sufficiently large (r1, . . . , rp+q) ∈

Np+q that implies formulas (1.5.13).Finally, it remains to note that the inclusion (0, . . . , 0) ∈ A is equivalent to

the equality WA = ∅, i. e., to the equality ψA(t1, . . . , tp+q) = 0.

Definition 1.5.26 The polynomial ψA(t1, . . . , tp+q) whose existence is estab-lished by Theorem 1.5.25, is called the N-Z-dimension polynomial of the setA ⊆ Nm ×Zn associated with the partitions (σ1, . . . , σp) and (σ′

1, . . . , σ′q) of the

sets Nm and Nn, respectively.

1.6 Basic Facts of the Field Theory

In this section we present some fundamental definitions and results on fieldextensions that will be used in the subsequent chapters. The proofs of most ofthe statements can be found in [8], [144], [167], and [171, Vol. II]. (If none ofthese books contains a statement, we give the corresponding reference.)

In what follows we adopt the standard notation and conventions of the clas-sical field theory. The characteristic of a field K will be denoted by Char K. IfK is a subfield of a field L, we say that L is a field extension or an overfield ofthe field K. In this case we also say that we have a field extension L/K. If Fis a subfield of L containing K, we say that F/K is a subextension of L/K orthat F is an intermediate field of L/K. This subextension is said to be properif F = K and F = L.

If K is a field, then by an isomorphism of K into a field M we mean anisomorphism of K onto some subfield of M . If L and M are field extensions ofthe same field K, then a field homomorphism φ : L → M such that φ(a) = a forall a ∈ K is called a K-homomorphism or a homomorphism over K. Of coursesuch a homomorphism is injective, so it is also called a K-isomorphism of L intoM , or an isomorphism of L/K into M/K. Two overfields L and M of a field Kare said to be K-isomorphic, if there is a K-isomorphism of L onto M . In thissituation we also say that the field extensions L/K and M/K are K-isomorphic.

Finitely generated and finite field extensions

Recall that if L/K is a field extension and S ⊆ L, then K(S) denotes thesmallest subfield of L containing K and S, that is, the intersection of all subfieldsof L containing K and S. Obviously, K(S) is the set of all elements of the formf(a1, . . . , am)/g(a1, . . . , am) where f and g are polynomials in m variables withcoefficients in K (m ≥ 1) and a1, . . . , am ∈ S. If L = K(S), we say that L isobtained by adjoining S to K; S is called the set of generators of L over K or

1.6. BASIC FACTS OF THE FIELD THEORY 65

the set of generators of the field extension L/K. If there exists a finite set ofgenerators of L/K, we say that L is a finitely generated field extension of K orthat L/K is finitely generated. The following theorem summarizes some well-know facts about finitely generated field extensions. As usual, if F1, F2, . . . , Fn

are intermediate fields of L/K (that is, Fi are subfields of L containing K),then F1F2 . . . Fn denotes the compositum of the fields F1, F2, . . . , Fn, i. e., thefield K(F1

⋃ · · ·⋃Fn). (Obviously, this is the smallest intermediate field of L/Kcontaining all the fields Fi.)

Theorem 1.6.1 Let L/K and M/L be field extensions (so that L is an inter-mediate field between K and M). Then

(i) If L/K and M/L are finitely generated, so is M/K. More precisely, ifL = K(S) and M = K(Σ) for some finite sets S and Σ, then M = K(S ∪ Σ).

(ii) If M/K is finitely generated, then M/L and L/K are finitely generated.

(iii) Let F be another intermediate field between K and M . Then:(a) If L/K is finitely generated, so is FL/F .

(b) If L/K and F/K are finitely generated, then FL/K is finitely generated.

If L/K is a field extension, then its degree, that is the dimension of L regardedas a vector space over K, will be denoted by L : K. If L : K < ∞, the fieldextension is said to be finite.

Theorem 1.6.2 Let M/K be a field extension and L an intermediate field ofM/K.

(i) If (xi)i∈I is a basis of M over L and (yj)i∈J is a basis of L over K, then(xiyj)i∈I,j∈J is a basis of M over K. Thus, M : K = (M : L)(L : K).(In particular, M : K is finite if and only if both M : L and L : K are finite).

(ii) If F is another intermediate field of M/K and B a basis of L over K,then B spans FL over F . In particular, if L : K < ∞, then FL : F ≤ L : K.(iii) If L and F are intermediate fields of M/K and L : K < ∞, F : K < ∞,

then FL : K ≤ (F : K)(L : K). The equality holds if the integers F : K andL : K are relatively prime.(iv) If S ⊆ M , then L(S) : K(S) ≤ L : K and L(S) : L ≤ K(S) : K.(v) If M = L(Σ) for some finite set Σ, then there exists a finitely generated

field extension K ′ of K such that K ′ ⊆ L and M : L = K ′(Σ) : K ′.

Algebraic extensions

An element a of a field extension L of K is called algebraic over the fieldK if there is a nonzero polynomial f(X) ∈ K[X] such that f(a) = 0; if such apolynomial does not exist, a is said to be transcendental over K. If every elementof a field L is algebraic over its subfield K, we say that the field extension L/Kis algebraic or L is algebraic over K.


Theorem 1.6.3 Let L/K be a field extension and let an element a ∈ L bealgebraic over K. Then

(i) There exists a unique monic irreducible polynomial p(X) ∈ K[X] whichhas a as a root. A polynomial f(X) ∈ K[X] has a as a root if and only if p(X)divides f(X) in K[X]. The polynomial p(X) is called the minimal polynomialof a over K; it is denoted by Irr(a,K).

(ii) The field K(a) is isomorphic to the quotient field of K[X]/(p(X)) andhas a basis 1, a, . . . , an−1 where n = deg p(X). Furthermore, K(a) = K[a] andK(a) : K = deg Irr(a,K).(iii) Let φ : K → M be a field homomorphism and let pφ(X) be a polynomial

obtained by applying φ to every coefficient of p(X). If b is a root of pφ(X) in M ,then there exists a unique field homomorphism φ′ : K(a) → M which extends φand sends a onto b.

The following statement summarizes basic properties of algebraic field exten-sions.

Theorem 1.6.4 Let L/K be a field extension.(i) If L/K is finite, then L is algebraic over K.(ii) If L = K(a1, . . . , an) and elements a1, . . . , an are algebraic over K, then

L/K is a finite field extension (hence L is algebraic over K). In this case L =K[a1, . . . , an].(iii) Let L = K(S) for some (not necessarily finite) set S. If every element of

S is algebraic over K, then L is algebraic over K.(iv) Suppose that the field extension L/K is algebraic, and let a be an element

of some overfield of L. If a is algebraic over L, then a is algebraic over K.(v) Let M be a field extension of L, so that K ⊆ L ⊆ M . If M/K is alge-

braic, then both M/L and L/K are algebraic. If, conversely, M/L and L/K arealgebraic, then M is algebraic over K.(vi) If M is a field extension of L, L/K is algebraic, and F is a subfield of M

containing K, then the field extension LF/F is algebraic.(vii) If F and G are two intermediate fields of L/K such that the field exten-sions F/K and G/K are algebraic, then the compositum FG is an algebraicfield extension of K.(viii) The elements of L which are algebraic over K form a subfield of L. Thissubfield is called the algebraic closure of K in L.(ix) If L/K is algebraic and φ : L → L is an endomorphism of L over K, then

φ is an automorphism of the field L.

Splitting fields and normal extensions. Algebraic and normalclosures

Let K be a field, K[X] a ring of polynomials in one indeterminate X overK, and F a family of polynomials in K[X]. A splitting field of F over K is afield L ⊃ K such that each polynomial f ∈ F splits over L (i.e., f is a product


of linear polynomials in L[X]) and L is the minimal overfield of K with thisproperty. (In other words, if each f ∈ F splits over some intermediate field L1

of L/K, then L1 = L.) A field extension L/K is called normal if L is a splittingfield of some family F ⊆ K[X]. An element a in some overfield of a field K iscalled normal if the field extension K(a)/K is normal.

Theorem 1.6.5 Let K be a field. Then(i) Every family of polynomials F ⊆ K[X] possesses a splitting field.(ii) Let φ : K → K ′ be a field isomorphism and let φ∗ : K[X] → K ′[X]

be the isomorphism of polynomial rings induced by φ (φ∗ :∑m

i=0 aiXi →∑m

i=0 φ(ai)Xi). If L ⊇ K is a splitting field of a family F ⊆ K[X] andL′ ⊇ K ′ is a splitting field of the family φ∗(F) ⊆ K ′[X], then φ can be extendedto an isomorphism L → L′.(iii) Any two splitting fields of a family F ⊆ K[X] are K-isomorphic.(iv) If L is a splitting field of a polynomial f ∈ K[X] of degree n, then L : K

divides n!; in particular, L : K ≤ n!.

Recall that a field K is said to be algebraically closed if it satisfies one of thefollowing equivalent conditions:

(1) every nonconstant polynomial f ∈ K[X] has a root in K;(2) every polynomial f ∈ K[X] splits into linear factors over K;(3) if L/K is an algebraic field extension, then L = K.

An algebraic closure of a field K is an algebraically closed overfield L ⊇ Ksuch that any algebraically closed intermediate field of L/K coincides with L.

Theorem 1.6.6 Let K/K be a field extension. Then the following conditionsare equivalent.

(i) K is an algebraic closure of K.(ii) K/K is algebraic, and K is algebraically closed.(iii) K/K is algebraic, and every polynomial f ∈ K[X] splits over K.(iv) K is the splitting field over K of the family F = K[X].(v) K/K is algebraic, and for every chain K ⊆ L ⊆ M of algebraic field

extensions and every field homomorphism φ : L → K, there exists a field homo-morphism φ : M → K extending φ.(vi) K/K is algebraic, and for every algebraic extension L/K, there exists a

field homomorphism L → K that leaves the field K fixed (such a homomorphismis said to be a K-embedding of L into K).

Theorem 1.6.7 (i) Every field has an algebraic closure, and any two algebraicclosures of a field K are K-isomorphic.

(ii) If K is an algebraic closure of a field K, then every K-endomorphism ofK is a K-automorphism.

Normal field extensions can be characterized as follows.


Theorem 1.6.8 Let L/K be an algebraic field extension and L an algebraicclosure of L (and hence of K). Then the following conditions are equivalent.

(i) L/K is normal.(ii) If an irreducible polynomial f ∈ K[X] has a root in L, it splits over L.(iii) Every K-embedding L → L maps L to L.(iv) Every K-automorphism L → L maps L to L.(v) If K ⊆ F ⊆ L ⊆ M is a sequence of field extensions and φ : F → M is

a K-homomorphism, then φ(F ) ⊆ L, and there exists a K-automorphism ψ ofthe field L whose restriction on F is φ.

If L : K < ∞, then each of the conditions (i) - (v) is equivalent to the fact thatL is a splitting field of some polynomial f ∈ K[X].

The following two theorems deal with the problem of embedding of an alge-braic extensions of a field K into a normal extension of K. If L and M are twooverfields of K, then the set of all K-isomorphisms (also called K-embeddings)of L into M is denoted by EmbK(L,M) (the number of elements of the last setis denoted by |EmbK(L,M)|).Theorem 1.6.9 Let N/K be a normal field extension, a ∈ N , and f(X) =Irr(a,K). Then the following conditions (i) - (v) on an element b ∈ N areequivalent.

(i) b is a root of f(X).(ii) Irr(b,K) = f(X).(iii) There is a K-isomorphism φ : K(a) → K(b) with φ(a) = b.(iv) There is a unique K-isomorphism φ : K(a) → K(b) with φ(a) = b.(v) There is a K-automorphism ψ : N → N with ψ(a) = b.

Furthermore, if m denotes the number of distinct roots of f(X), then |EmbK

(K(a), N)| = m ≤ deg f = K(a) : K.

Theorem 1.6.10 Let K ⊆ L ⊆ M ⊆ N be a sequence of field extensions suchthat N is normal over K. Furthermore, for every φ ∈ EmbK(L,N), let φ∗

denote an automorphism of N extending φ (such an extension exists by state-ment (v) of Theorem 1.6.8). Then the mapping EmbK(L,N)×EmbL(M,N) →EmbK(M,N), that sends a pair (φ, ψ) to φ∗ ψ, is a bijection. In particular,|EmbK(M,N)| = |EmbK(L,N)| · |EmbL(M,N)|.Theorem 1.6.11 (i) If L/K is a normal field extension and F is an interme-diate field of L/K, then L/F is also normal.

(ii) The class of normal extensions is closed under lifting: If K ⊆ L ⊆ M is asequence of field extensions, L/K is normal, and F is an intermediate field ofM/K, then the compositum FL is normal over F .(iii) If Lii∈I is a family of fields, each normal over a field K, and each

contained in a single larger field, then the compositum of the fields Li (i ∈ I)and

⋂i∈I Li are normal over K.


Let L/K be an algebraic field extension. A field L′ ⊇ L is called a normalclosure of L/K (or a normal closure of L over K) if L′/K is normal and L′ is aminimal field extension of L with this property (that is, if M is any intermediatefield of L′/L which is normal over K, then M = L′).

Exercise 1.6.12 Let K be a field and L = K(Σ) where every element a ∈ Σis algebraic over K. Let L be an algebraic closure of L (and hence also of K).Furthermore, for every a ∈ Σ, let Za denote the set of all roots of the polynomialIrr(a,K) in L. Prove that K(

⋃a∈Σ Za) is a normal closure of L/K.

The following theorem summarizes basic properties of the normal closure.

Theorem 1.6.13 Let L/K be an algebraic field extension. Then(i) A field N ⊇ L is a normal closure of L/K if and only if N is a splitting

field of the family Irr(a,K) | a ∈ L over K.(ii) There is a normal closure of L/K. Any two normal closures of L/K are

K-isomorphic.(iii) Let M be a field extension of L normal over K, let L = K(S) for some

set S ⊆ L, and let T denote the set of all elements of M that are zeros ofthe minimal polynomials over K of the elements of S. Then K(T ) is a normalclosure of L/K.(iv) Let N be a normal closure of L/K. Then N : K < ∞ if and only if

L : K < ∞.(v) If φ is a K-homomorphism of L into its algebraic closure L, then φ(L) is

contained in the normal closure of L/K that lies in L.(vi) Let L : K < ∞ and let φ1, . . . , φn be different K-embeddings of L into

an algebraic closure K of K. Then the compositum N = (φ1L) . . . (φnL) is anormal closure of L/K.(vii) If L : K < ∞ and N is a normal closure of L/K, then there exist subfields

L1, . . . , Lr of N such that each Li is K-isomorphic to L and N = L1 . . . Ln.(viii) Let M be an overfield of L such that the field extension M/K is normal.Then there exists a unique subfield of M which is a normal closure of L overK; this subfield is the intersection of all intermediate fields between L and Mthat are normal over K.

Separable and purely inseparable field extensions. Perfect fields

Let K be a field and K[X] a ring of polynomials in one indeterminate Xover K. A polynomial f(X) ∈ K[X] is called separable if it has no multipleroots in its splitting field. (By Theorem 1.6.5, if f(X) has distinct roots in onesplitting field, it has distinct roots in any its splitting field.)

Proposition 1.6.14 Let f ∈ K[X] be an irreducible polynomial over a field K.Then the following conditions are equivalent.

(i) f divides f ′ (as usual, f ′ denotes the derivative of f).(ii) f ′ = 0.


(iii) Char K is a prime number p, and f(X) = g(Xp) for some polynomialg(X) ∈ K[X].(iv) The polynomial f is not separable.

Consequently, a polynomial g ∈ K[X] is separable if and only if gcd(g, g′) = 1in K[X].

Let L be a field extension of K. An element a ∈ L is said to be separable overK if a is algebraic over K and Irr(a,K) is separable. Equivalently, we couldsay that an element a ∈ L is separable over K if it is algebraic over K and it isa simple root of Irr(a,K).

If K is a field of prime characteristic p, f(X) an irreducible polynomial inK[X], and e is a maximum nonnegative integer such that f(X) = g(Xpe

) forsome polynomial g(Y ) ∈ k[Y ], then the number e is called the exponent ofinseparability or the degree of inseparability of f(X), and degf(X)

pe is called theseparable degree or reduced degree of f(X). Clearly, f(X) is separable if and onlyif its exponent of inseparability is 0.

An algebraic field extension L/K is called separable or separably algebraic(we also say that L is separable over K or L is separably algebraic over K) ifevery element of L is separable over K. Clearly, if Char K = 0, then every irre-ducible polynomial in K[X] is separable and every algebraic field extension L/Kis separable. A normal and separable field extension L of a field K is said to be aGalois extension of K (in this case we also say that L/K is a Galois field exten-sion or that L is Galois over K). It is easy to see that a field extension L/K isGalois if and only if L is a splitting field of a set of separable polynomials over K.

Theorem 1.6.15 Let L/K be an algebraic field extension. Then(i) If a ∈ L is separable over K, then so are all elements f(a) where f ∈

K[X]. Consequently, the field extension K(a)/K is separable.(ii) Let K be a field of prime characteristic p and let an element a be algebraic

over K. Then the field extension K(a)/K is separable if and only if K(a) =K(ap).(iii) Suppose that Char K = p = 0. Then for any element a ∈ L, there exists

n ∈ N such that apn

is separable over K.(iv) If L = K(Σ), then L/K is separable if and only if every element of Σ is

separable over K.(v) If L/K is separable and N is a normal closure of L/K, then N/K is

separable.(vi) Let F be an intermediate field of L/K. Then the extension L/K is sepa-

rable if and only if L/F and F/K are separable field extensions.(vii) Let F1 and F2 be intermediate fields of L/K. If F1/K is separable, thenso is F1F2/F2. If F1/K and F2/K are separable, then so is F1F2/K.(viii) Let F1 and F2 be intermediate fields of L/K such that F1/K and F2/K arenormal separable extensions of finite degree. Then F1F2 : K = (F1 : K)(F2 :K)/[F1 ∩ F2 : K].


(ix) Let N be a normal field extension of K containing L. Then |EmbK(L,N)| ≤L : K, and the equality holds if and only if L/K is separable.

(x) If L/K is a finite Galois extension and M is any field extension of K,then LM/M is a Galois extension and LM : M = L : L ∩ M .

Proposition 1.6.16 Let K be a field of a prime characteristic p and let L bea field extension of K. If L/K is separably algebraic, then L = KLp (as usual,Lp denotes the field of all elements xp where x ∈ L). Conversely, if L = KLp

and the field extension L/K is finite, then L is separably algebraic over K.

Let L be an overfield of a field K. A primitive element of L over K is anelement a ∈ L such that L = K(a). If such an element exists, then L/K is saidto be a simple field extension. The following result is known as the PrimitiveElement Theorem.

Theorem 1.6.17 (i) A finite field extension L/K is simple if and only if thereexist only a finite number of intermediate fields of L/K.

(ii) Let L/K be a field extension such that L = K(a1, . . . , an, b) where ele-ments a1, . . . , an are separable over K and b is algebraic over K. Then L/K isa simple extension. In particular, if K has characteristic 0 or if K is a finitefield, then any finite extension of K is simple.(iii) Every finite separable field extension L/K is simple. If, in addition, the

field K is infinite, then there exist infinitely many primitive elements of L over K.(iv) If L = K(a1, . . . , an) is a finite separable field extension of an infinite

field K, then a primitive element b of L over K can be chosen in the formb =

∑ni=1 λiai with λi ∈ K (1 ≤ i ≤ n).

A field is called perfect if every its algebraic extension is separable (or equiva-lently, if every irreducible polynomial with coefficients in this field is separable).The following theorem contains basic properties of perfect fields.

Theorem 1.6.18 Let K be a field.(i) If Char K = 0, then K is perfect.(ii) If Char K = p = 0, then K is perfect if and only if every element of K

has a pth root in K, i.e., if and only if the Frobenius homomorphism σ : K → Kgiven by σ(x) = xp is surjective (and hence an automorphism of K).(iii) Every finite field is perfect.(iv) Every algebraically closed field is perfect.(v) If K is perfect and L an algebraic field extension of K, then L is perfect.(vi) K is perfect if and only if any its algebraic closure is separable over K.

Let K be a field of characteristic p = 0 and let K be an algebraic closure ofK. For every n ≥ 1, the set K1/pn

= a ∈ K | apn ∈ K is obviously a subfieldof K containing K. Furthermore, it is easy to see that K ⊆ K1/p ⊆ K1/p2 ⊆ . . .and

⋃∞n=1 K1/pn

is an intermediate field of K/K. This field is called the perfect


closure of K in K; it is denoted by P(K). It is immediate from the definitionthat P(K) is the smallest perfect subfield of K containing K. Furthermore, twoperfect closures of K are K-isomorphic, and if L is any perfect overfield of K,then there is a unique K-isomorphism of P(K) into L.

Let K be a field and let a be an element in some overfield of K. We say thata is purely inseparable over the field K if a is algebraic over K and Irr(a,K)has the form (x − a)n for some n ≥ 1 (that is, the polynomial Irr(a,K) hasonly a single root in its splitting field). A field extension L/K is called purelyinseparable if it is algebraic and every element of L is purely inseparable over K.

The following theorem and two propositions summarize basic properties ofpure inseparability.

Theorem 1.6.19 Let L/K be an algebraic field extension. Then(i) An element a ∈ L is both separable and purely inseparable over K if and

only if a ∈ K.(ii) Suppose that Char K = p = 0. An element a ∈ L is purely inseparable

over K if and only if there exists n ∈ N such that apn ∈ K. If nonnegativeintegers n with this property exist and e is the smallest such a number, thenIrr(a,K) = Xpe − apn

, so that K(a) : K = pe.(iii) Let Σ ⊆ L and let every element of the set Σ be purely inseparable over

K. Then the field extension K(Σ)/K is purely inseparable.(iv) Let F be an intermediate field of L/K. If a ∈ L is purely inseparable over

K, then it is also purely inseparable over F .(v) Let F be an intermediate field of L/K. Then L/K is purely inseparable if

and only if L/F and F/K are purely inseparable.(vi) Let F1 and F2 be intermediate fields of L/K. If F1/K is purely insepara-

ble, then so is F1F2/F2. If F1/K and F2/K are purely inseparable, then so isF1F2/K.

Proposition 1.6.20 Let L be an algebraic field extension of a field K of char-acteristic p = 0. Let K be an algebraic closure of K, K1/p = a ∈ K | ap ∈ K,and K1/p∞

= a ∈ K | apn ∈ K for some n ∈ N. Then the following conditionsare equivalent.

(i) L/K is purely inseparable.(ii) For every a ∈ L, Irr(a,K) = Xpm − b for some m ∈ N and b ∈ K.(iii) No element of L \ K is separable over K.(iv) Each a ∈ L has a power apn

in K.(v) There is a K-homomorphism of L into K1/p∞

.(vi) There is only one K-homomorphism of L into K.

Proposition 1.6.21 Let K be a field of characteristic p = 0. Then(i) Every purely inseparable field extension of K is normal over K.(ii) Let L be an overfield of K. If L is purely inseparable over K and M is

a field extension of L, then the natural embedding L → M (x → x for every


x ∈ L) is the only K-homomorphism from L to M . Conversely, if there exists anoverfield M of L such that M/L is normal and the natural embedding L → Mis the only K-homomorphism from L to M , then L/K is purely inseparable.(iii) If L is a finite purely inseparable field extension of K, then there exists

e ∈ N such that L : K = pe and Lpe ⊆ K.

Let L/K be an algebraic field extension. Then the sets S(L/K) = x ∈ L |xis separable over K and P (L/K) = x ∈ L |x is purely inseparable over Kare called the separable closure of K in L (or a separable part of L over K) andpurely inseparable closure of K in L (or a purely inseparable part of L over K),respectively. It is easy to see that S(L/K) and P (L/K) are intermediate fieldsof L/K and S(L/K) ∩ P (L/K) = K (see Theorem 1.6.19(i)). By a separable(purely inseparable) closure of a field K we mean the separable (respectively,purely inseparable) closure of K in its algebraic closure K. If the separableclosure of K coincides with K, we say that the field K is separably algebraicallyclosed.

Proposition 1.6.22 Let L/K be an algebraic field extension and let S =S(L/K), P = P (L/K).

(i) The extension S/K is separable, and the extensions P/K and L/S arepurely inseparable.

(ii) L/P is separable if and only if L = SP .(iii) If L/K is normal, then so is S/K.(iv) Let L/K be normal, G the group of K-automorphisms of L, and F the

fixed field of G in L (that is, F = a ∈ L | g(a) = a for every g ∈ G). ThenF = P , L/P is separable, and L = SP .

If L/K is an algebraic field extension, then S(L/K) : K and L : S(L/K)are called, respectively, the separable degree (or the separable factor of the degree)and inseparable degree (or degree of inseparability) of L over K; they are denotedby [L : K]s and [L : K]i, respectively.

It is easy to see that [L : K]i = 1 if and only if L/K is separable (whichis always the case if Char K = 0). Furthermore, Theorem 1.6.2(i) implies thatL : K = [L : K]s[L : K]i.

Theorem 1.6.23 Let L/K be an algebraic field extension, S = S(L/K), andN a normal field extension of K containing L. Then

(i) The mapping EmbK(L,N) → EmbK(S,N) (α → α |S) is a bijection.(ii) [L : K]s = |EmbK(L,N)|.(iii) Let L denote an algebraic closure of the field L. Then any embedding

L → L is uniquely determined by its restriction on S.(iv) If a ∈ L, then [K(a) : K]s is equal to the number of roots of the polynomial

Irr(a,K).(v) If F is any intermediate field of L/K, then [L : K]s = [L : F ]s[F : K]s

and [L : K]i = [L : F ]i[F : K]i.


(vi) Let K be a field of a prime characteristic p and a ∈ L. Then[K(a) : K]i = pd where d is the greatest nonnegative integer such that Irr(a,K)can be written as a polynomial in Xpd

.(vii) If Char K = p = 0 and L/K is finite, then [L : K]i is a power of p.(viii) If Σ ⊆ L and F , G are intermediate fields of L/K such that F ⊆ G, then[G(Σ) : F (Σ)]s ≤ [G : F ]s and [G(Σ) : G]s ≤ [F (Σ) : F ]s.

Let L/K be a field extension and let elements a1, . . . , ar ∈ L be algebraic overK. We denote the r-tuple (a1, . . . , ar) by a and the field extension K(a1, . . . , ar)by K(a). Let K be the normal closure of K(a) over K and let EmbK(K(a), K) =φ1, . . . , φd be the set of all distinct K-isomorphisms of K(a) into K (φ1 is theidentity isomorphism of K(a) onto itself).

Suppose, first, that the elements a1, . . . , ar are separable over K (e. g.,Char K = 0). Then d = K(a) : K (see Theorem 1.6.15(ix) ). Let a(i) =(φi(a1), . . . , φi(ar)) (1 ≤ i ≤ d). Then the set a(i) | 1 ≤ i ≤ d is called thecomplete set of conjugates of a over K.

Let L be an overfield of K such that L and K(a) are contained in a commonfield. Then we can assume that L and K are also contained in a common overfield(see Corollary 1.6.41 below). Clearly, conjugates of a over L are also conjugatesof a over K, hence a complete set of conjugates of a over L contains at mostd elements. Thus, a(1), . . . , a(d) fall into disjoint sets which are complete sets ofconjugates with respect to L. Let there be m such sets, Σ1, . . . ,Σm, and let b(i),1 ≤ i ≤ m, denote one of the a(j) in Σi. Setting L(b(i)) = L(b(i)

1 , . . . , b(i)r ), we

see the L(b(i)) : L is the number of conjugates of a in Σi whencem∑

i=1

L(b(i)) : L = K(a) : K (1.6.1)

If elements a1, . . . , ar are not separable over K and the conjugates a(i) ofthe r-tuple a are defined as before, then the number of sets d, counted withoutmultiplicities is [K(a) : k]s (see Theorem 1.6.23(iii)). If we assign to each image

of a the multiplicity[K(a) : K][K(a) : K]s

, then the number of conjugates of a counted

with multiplicities is [K(a) : K]. Introducing a field L as before, we see that thenumber of conjugates of a over L counted with their multiplicities is less than orequal to the number of conjugates of a over K counted with their multiplicities(it follows from Theorem 1.6.2(iv)). Let M and N denote the separable parts ofK(a) over K and L(a) over L, respectively. Then the multiplicity of a over K isK(a) : M , while the multiplicity of a over L is L(a) : N . By Theorem 1.6.2(iv),we have K(a) : M = K(a) : K(M) ≥ L(a) : L(M) ≥ L(a) : N , so that themultiplicity of a over L is less than or equal to the multiplicity of a over K.

Let b(1), . . . , b(m) be defined as before. Then the arguments used in the caseof separable elements ai and the last inequality of multiplicities imply that

m∑i=1

[L(bi) : L]s = [K(a) : K]s (1.6.2)


andm∑

i=1

[L(bi) : L] ≤ K(a) : K . (1.6.3)

Linearly disjoint extensions

Let L be a field extension of a filed K, and let F1 and F2 be two intermediatefields (or domains) of L/K. We say that F1 and F2 are linearly disjoint overK if the following condition is satisfied: If ai | i ∈ I and bj | j ∈ J are,respectively, families of elements in F1 and F2 which are linearly independentover K, then the family aibj | (i, j) ∈ I × J is linearly independent over K.

Theorem 1.6.24 Let L be a field extension of a field K, and let F1 and F2 betwo intermediate fields of L/K. Then the following conditions are equivalent.

(i) F1 and F2 are linearly disjoint over K.(ii) If ai | i ∈ I is a basis of F1 over K (i. e., a basis of F1 as a vector

K-space) and bj | j ∈ J is a basis of F2 over K, then aibj | (i, j) ∈∈ I × Jis a basis of F1F2 over K.(iii) If a family ai | i ∈ I ⊆ F1 is linearly independent over K, then it is

linearly independent over F2.(iv) There is a basis of F1 over K which is linearly independent over F2.(v) Let ψ : F1

⊗K F2 → F1F2 be a homomorphism of K-algebras such that

x⊗

y → xy for any generator x⊗

y of F1

⊗K F2 (x ∈ F1, y ∈ F2). Then

ψ is injective. (Note that the image of ψ is the K-algebra F1[F2] = F2[F1] ofall elements of the form a1b1 + · · · + ambm where ai ∈ F1 and bi ∈ F2. Inparticular, if at least one of the field extensions F1/K, F2/K is algebraic, thenF1F2 = F1[F2] = F2[F1] and so the map ψ is surjective.)

If F1/K and F2/K are finite, then each of conditions (i) - (v) is equivalentto the equality F1F2 : K = (F1 : K)(F2 : K).

If the field extensions F1/K and F2/K are normal and separable or if one ofthem is a finite Galois extension, then each of conditions (i) - (v) is equivalentto the equality F1 ∩ F2 = K.

The following proposition gives some more properties of linearly disjoint filedextensions.

Proposition 1.6.25 Let L/K be a field extension and let F1 and F2 be twointermediate fields of L/K that are linearly disjoint over K. Then

(i) Every basis of the vector K-space F1 is a basis of the vector F2-space F1F2.In particular, F1 : K = F1F2 : F2.

(ii) F1

⋂F2 = K.

(iii) Let M/K be another filed extension of K and let N be an overfield of M .Then L and N are linearly disjoint over K if and only if L and M are linearlydisjoint over K and LM and N are linearly disjoint over M .


Algebraic dependence. Transcendence bases

Let L/K be a field extension and S ⊆ L. We say that an element a ∈ L isalgebraically dependent on S over K and write a ≺K S (or a ≺ S if it is clearwhat field K is considered) if a is algebraic over K(S). If a is not algebraicallydependent on S over K, that is, if a is transcendental over K(S), we say that ais algebraically independent of S over K and write a ⊀K S (or a ⊀ S).

A set S ⊆ L is said to be algebraically dependent over K if there exists s ∈ Ssuch that s ≺K S \ s, that is, s is algebraic over K(S \ s). If s ⊀K S \ sfor all s ∈ S, that is, if s is transcendental over K(S \ s) for every s ∈ S, thenS is said to be algebraically independent over K. (In particular, an empty set isalgebraically independent over K.)

Theorem 1.6.26 Algebraic dependence is a dependence relation in the senseof Definition 1.1.4.

Corollary 1.6.27 Let L/K be a field extension and let S ⊆ T ⊆ L. Then(i) If S is algebraically dependent over K, then so is T .(ii) If T is algebraically independent over K, then so is S.(iii) If S is algebraically independent over K and a ∈ L is transcendental over

K(S), then S ∪ a is algebraically independent over K.

Theorem 1.6.28 Let L/K be a field extension.(i) A subset S of L is algebraically dependent over K if and only if there

exist distinct elements s1, . . . , sn ∈ S (n ∈ N, n ≥ 1) and a nonzero polynomialf(X1, . . . , Xn) in n indeterminates over K such that f(s1, . . . , sn) = 0.

(ii) Let B = b1, . . . , bm be a subset of L. Then B is algebraically independentover K if and only if b1 is transcendental over K and bi is transcendental overK(b1, . . . , bi−1) for i = 2, . . . , m.(iii) Let a1, . . . , ar, b1, . . . , bs be elements in L. Suppose that r < s and each

bj (1 ≤ j ≤ s) is algebraic over K(a1, . . . , ar). Then the set b1, . . . , bs isalgebraically dependent over K.(iv) Let S ⊆ L and let F be an intermediate field of L/K such that the exten-

sion F/K is algebraic. If the set S is algebraically independent over K, then Sis also algebraically independent over F .

(v) If S ⊆ L and F is an intermediate field of L/K such that F/K is algebraic,then F (S) : K(S) = F : K if and only if the set S is algebraically independentover F . Also, F (S) : F = K(S) : K if only if the set S is algebraically indepen-dent over F . (Recall that one always has the inequalities L(S) : K(S) ≤ L : Kand L(S) : L ≤ K(S) : K, see Theorem 1.6.2(iv).)(vi) Statement (v) remains valid if one replaces the degree by the separable

factor of the degree.(vii) Let K be algebraically closed in L and let S be a set of elements of someoverfield of L which is algebraically independent over L. Then K(S) is alge-braically closed in L(S) and the fields K(S) and L are linearly disjoint over K.


Let L/K be a field extension. A set B ⊆ L is called a transcendence basisof L over K if B is algebraically independent over K and L ≺K B, that is,L is algebraic over K(B). The following two theorems describe fundamentalproperties of transcendence bases.

Theorem 1.6.29 Let L/K be a field extension and let B be a subset of L. Thenthe following statements are equivalent.

(i) B is a transcendence basis of L over K.(ii) B is a maximal algebraically independent subset of L over K.(iii) B is minimal with respect to the property that the extension L/K(B) is

algebraic.

Theorem 1.6.30 Let L/K be a field extension. Then(i) Any two transcendence bases of L/K have the same cardinality. This car-

dinality is called the transcendence degree of L over K (or the transcendencedegree of the extension L/K); it is denoted by trdegKL.

(ii) Let S and T be two subsets of L such that S is algebraically independentover K, L is algebraic over K(T ), and S ⊆ T . Then there exists a transcendencebasis B of L over K such that S ⊆ B ⊆ T .(iii) If L/K is finitely generated and B is a transcendence basis of L over K,

then B is a finite set and the field extension L/K(B) is finite. Furthermore,if Σ ⊆ L and L = K(Σ), then there exists a finite subset S of Σ such thatB ∩ S = ∅ and L = K(B

⋃S).

(iv) Let M be a field extension of L. If B and C are, respectively, transcendencebases of L over K and of M over L, then B∩C = ∅ and B∪C is a transcendencebasis of M over K. Therefore, trdegKM = trdegKL + trdegKL.

If L/K is a field extension and Σ ⊆ L, then by a transcendence basis of theset Σ over K we mean a maximal algebraically independent over K subset of Σ.Theorem 1.6.30(ii) implies that every set Σ ⊆ L contains its transcendence basiswhich is also a transcendence basis of K(Σ) over K.

The following two propositions give some additional properties of transcen-dence degree.

Proposition 1.6.31 Let L/K be a field extension, let F and G be intermediatefields of L/K and S ⊆ L \ F . Then

(i) trdegF FG ≤ trdegKG.(ii) trdegKFG ≤ trdegKF + trdegKG.(iii) trdegK(S)F (S) ≤ trdegKF with equality if S is algebraically independent

over F or algebraic over K.

Proposition 1.6.32 (i) Let K and K1 be fields, and let L and L1 be, respec-tively, algebraically closed field extensions of K and K1 such that trdegKL =trdegK1L1. Then every isomorphism from K to K1 can be extended to an iso-morphism from L to L1.


(ii) If K is a field, then every two algebraically closed field extensions of Khaving the same transcendence degree over K are K-isomorphic.(iii) Let L be an algebraically closed extension of a field K. Then every auto-

morphism of K can be extended to an automorphism of L.(iv) Let L/K be a field extension of finite transcendence degree, and let M

be an algebraically closed field extension of L. Then every K-homomorphismfrom L to M can be extended to a K-automorphism of M . In particular, everyK-endomorphism of L is a K-automorphism.

(v) Let K be a field and let A be an integral domain which is a K-algebra suchthat trdegKA < ∞ (Recall that the transcendence degree of an integral domainA over K is defined as the transcendence degree of the quotient field of thisdomain over K.) Let P be a nonzero prime ideal of A. Then trdegK(A/P ) <trdegKA.

A field extension L/K is called purely transcendental if L = K(B) for sometranscendence basis B of L over K. In this case, if B = bi | i ∈ I, then L isnaturally K-isomorphic to the field of rational functions K((Xi)i∈I) in the setof indeterminates Xi | i ∈ I (with the same index set I). Clearly, if Σ is atranscendence basis of a field extension M/N , then the extensions N(Σ)/N andM/N(Σ) are purely transcendental and algebraic, respectively.

Proposition 1.6.33 Let L/K be a field extension.(i) Suppose that L = K(t) where the element t is transcendental over K. Let

s = f(t)/g(t) ∈ K(t) where the polynomials f(t) and g(t) are relatively primeand at least one of them is nonconstant. (As we have seen, K(t) can be treatedas a field of fractions of the indeterminate t over K, so every element of K(t)can be written as a ratio of two polynomials in t.) Then s is transcendental overK, t is algebraic over K(s), and K(t) : K(s) = maxdeg f(t), deg g(t)).

(ii) If an element t ∈ L is transcendental over K and F is an intermediatefield of the extension K(t)/K, F = K, then K(t) is algebraic over F .(iii) If L/K is purely transcendental, then every element a ∈ L \ K is tran-

scendental over K.(iv) Let F1 and F2 be intermediate fields of L/K which are algebraic and purely

transcendental over K, respectively. Then F1 and F2 are linearly disjoint over K.

Theorem 1.6.34 (Luroth’s Theorem). Let K be a field and t an elementof some overfield of K. If t is transcendental over K and F is an intermediatefield of the extension K(t)/K, F = K, then F = K(s) for some s ∈ K(t).

A transcendence basis B of a field extension L/K is said to be a separatingtranscendence basis of L/K (or a separating transcendence basis of L over K) ifL is separably algebraic over K(B). If a field L has a separating transcendencebasis over its subfield K, then L is said to be separably generated over K. It iseasy to see that if L/K is an algebraic field extension, then L is separable overK if and only if L/K is separably generated.

The following statement uses the notation of Proposition 1.6.20.


Proposition 1.6.35 Let L/K be a field extension and Char K = p = 0. Thenthe following statements are equivalent.

(i) Every finitely generated subextension of L/K is separably generated.(ii) The fields K and K1/p∞

are linearly disjoint over K.(iii) The fields K and K1/p are linearly disjoint over K.

A field extension L/K is called separable if either Char K = 0 or Char K =p = 0 and the conditions of Proposition 1.5.35 are satisfied. (As we have seen,this definition is compatible with the use of the word “separable” in the case ofalgebraic extensions.)

Proposition 1.6.36 (i) If a field extension L/K is separably generated, thenL/K is separable.

(ii) If L = K(Σ) is a finitely generated separable extension of a field K(Card Σ < ∞), then Σ contains a separating transcendence basis of L/K.(iii) Any finitely generated extension of a perfect field is separably generated.(iv) Let L/K be a field extension and F an intermediate field of L/K. Then(a) If L/K is separable, then F/K is separable.(b) If F/K and L/F are separable, then L/K is separable.(c) If L/K is separable and F/K is algebraic, then L/F is separable.

(v) Let F1 and F2 be two intermediate fields of a field extension L/K suchthat F1 and F2 are linearly disjoint over K. Then F1 is separable over K if andonly if F1F2 is separable over F2.

Let N/K be a field extension and let L and M be intermediate fields ofN/K. We say that L and M are algebraically disjoint (or free) over K if thefollowing condition is satisfied: If S and T are, respectively, subsets of L and Mthat are algebraically independent over K, then S ∩ T = ∅ and the set S ∪ T isalgebraically independent over K.

Theorem 1.6.37 Let N/K be a field extension and let L and M be intermedi-ate fields of N/K. Then the following conditions are equivalent.

(i) L and M are algebraically disjoint over K.(ii) Every finite set of elements of L algebraically independent over K remains

such over M .(iii Every finite set of elements of M algebraically independent over K re-

mains such over L.(iv) There exists a transcendence basis of L over K which is algebraically in-

dependent over M .(v) There exist, respectively, transcendence bases S and T of L and M over

K such that the set S ∪ T is algebraically independent over K and S ∩ T = ∅.(vi) Let L′ and M ′ denote, respectively, the algebraic closures of L and M

in N . Then L′ and M ′ are algebraically disjoint over K.


(vii) There exist, respectively, transcendence bases S and T of L and M overK such that K(S) and K(T ) are linearly disjoint over K.

If two intermediate fields L and M of a field extension N/K are algebraicallydisjoint, we also say that L is free from M over K. This terminology comes fromcondition (ii) of the last theorem (the equivalent condition (iii) shows that L isfree from M over K if and only if M is free from L over K).

Theorem 1.6.38 Let N/K be a field extension and let L and M be intermedi-ate fields of N/K which are algebraically disjoint over K. Then

(i) If L1 and M1 are intermediate fields of N/K such that L1 ⊆ L and M1 ⊆M , then L1 and M1 are algebraically disjoint over K.

(ii) The field L ∩ M is algebraic over K.(iii) If S and T are, respectively, transcendence bases of L and M over K, then

S ∪ T is a transcendence bases of the compositum LM over K.(iv) trdegKLM = trdegKL + trdegKM .(v) Every subset of L which is algebraically independent over K is algebraically

independent over M .(vi) Every transcendence basis of L over K is a transcendence basis of LM

over M .(vii) trdegMLM = trdegKL.(viii) If S and T are, respectively, subsets of L and M that are algebraicallyindependent over K, then K(S) and K(T ) are linearly disjoint over K.(ix) If L/K is separable, then LM/M is separable.(x) If L/K and M/K are separable, then LM/K is separable.(xi) Suppose that K is algebraically closed in N . If M is separable over K or

L is separable over K, then M is algebraically closed in LM .

Theorem 1.6.39 Let L and M be intermediate fields of a field extension N/K.(i) If trdegKL < ∞, then L and M are algebraically disjoint over K if and

only if trdegMLM = trdegKL.(ii) If trdegKL < ∞ and trdegKM < ∞, then L and M are algebraically

disjoint over K if and only if trdegKLM = trdegKL + trdegKM .(iii) Let F be an intermediate field of M/K. Then L and M are algebraically

disjoint over K if and only if L and F are algebraically disjoint over K and LFand M are algebraically disjoint over F .(iv) If either L or M is algebraic over K, then L and M are algebraically

disjoint over K.(v) If L and M are linearly disjoint over K, then they are algebraically disjoint

over K.(vi) If L and M are algebraically disjoint over K and either L or M is purely

transcendental over K, then L and M are linearly disjoint over K.


Theorem 1.6.40 Let L and M be field extensions of a field K. Then(i) There exists a field extension G of K such that G contains K-isomorphic

copies of both L and M .(ii) There exists a field extension H of K that contains K-isomorphic copies

L′ and M ′ of L and M , respectively, such that H = L′M ′ and the fields L′ andM ′ are algebraically disjoint over K. (The field H is called the free join of Land M over K.) Let λ : L → L′ and µ : M → M ′ be the K-isomorphisms ofL and M onto L′ and M ′, respectively, and let H ′ be another free join of Land M with the K-isomorphisms λ′ : L → L′′ and µ′ : M → M ′′ of L and M ,respectively, onto their copies in H ′ = L′′M ′′. Then the free joins H and H ′ areequivalent in the sense that there exists a K-isomorphism φ of H onto H ′ suchthat φ coincides with λ′λ−1 on L′ = λ(L) and with µ′µ−1 on M ′ = µ(M).(iii) If L and M are linearly disjoint over K then the compositum LM is a

free join of L and M over K.

Corollary 1.6.41 Let L/K be a field extension and let φ be an isomorphismof K into a field M . Then φ can be extended to an isomorphism φ′ of L into anoverfield of M such that trdegMMφ′(L) = trdegKL.

If two integral domains R and R′ contain the same field K as a subring, thenby a free join of R and R′ over K we mean an integral domain Ω containing Kas a subring, together with two K-isomorphisms τ : R → Ω and τ ′ : R′ → Ωsuch that

(i) Ω = (τR)(τ ′R′) and(ii) the rings τR and τ ′R′ are free over K, that is, whenever X = x1, . . . , xr

and X ′ = x′1, . . . , x

′s are finite subsets of τR and τ ′R′, respectively, such that

the elements of each set are algebraically independent over K, then the elementsof the set X ∪ X ′ are algebraically independent over K.

Clearly, in this case the quotient field of Ω is a free join of the quotient fieldsof R and R′ (in the sense of Theorem 1.6.40(ii)).

Proposition 1.6.42 Let K and L be fields, let R = K[a1, . . . , am] be an integraldomain generated by elements a1, . . . , am over K, and let M be an overfield ofK. Furthermore, let τ1 : R → L and τ2 : M → L be K-homomorphisms that donot map K to 0. Then there exists a free join N of M and R over K such that

(i) N = M ′[a′1, . . . , a

′m] where M ′ is a field and a′

1, . . . , a′m are elements in

some overfield of M ′.(ii) There exist two K-isomorphisms φ1 : K[a′

1, . . . , a′m] → R and φ2 : M ′ →

M and a homomorphism ψ : N → L such that ψ|K[a′1,...,a′

m] = τ1 φ1 andψ|M ′ = τ2 φ2.

The last type of disjointness we would like to mention is the quasi-lineardisjointness of field extensions. Let L and M be field extensions of a field Kcontained in a common overfield N . We say that L and M are quasi-linearlydisjoint over K if the perfect closures P(L) and P(M) of the fields L and M ,respectively, are linearly disjoint over the perfect closure P(K) of K.


With the notation of the last definition, if K is a perfect field (in particular,if Char K = 0), then the quasi-linear disjointness coincides with linear disjoint-ness). If Char K = p > 0, then it is easy to see that L and M are quasi-linearlydisjoint over K if and only if they satisfy the following condition.

Whenever x1, . . . , xn are elements of L such that for any integer e ≥ 0 thepe-th powers of the xi are linearly independent over K, then x1, . . . , xn arelinearly independent over M .

Theorem 1.6.43 Let M be a field extension of a field K and let L1 and L2

be two intermediate fields of M/K which are quasi-linearly disjoint over K andwhose compositum is M . Then

(i) L1 ∩ L2 is a purely inseparable field extension of K.(ii) Let φ be an isomorphism of K onto a field K ′ and let ψ1 and ψ2 be

extensions of φ to isomorphisms of L1 and L2, respectively, into an overfieldN of K ′ such that ψ1(L1) and ψ2(L2) are quasi-linearly disjoint over K ′. Thenthere exists a unique extension ψ of φ to an isomorphism of M into N whosecontraction to Li is ψi (i = 1, 2).

Theorem 1.6.44 (i) Let k,K,L,E be fields having a common overfield suchthat k ⊆ K, k ⊆ E ⊆ L. Then K and L are quasi-linearly disjoint (linearlydisjoint or algebraically disjoint) over k if and only if K and E are quasi-linearlydisjoint (respectively, linearly disjoint or algebraically disjoint) over k, and KEand L are quasi-linearly disjoint (respectively, linearly disjoint or algebraicallydisjoint) over E.

(ii) Let k,K,L,E,E′ be fields having a common overfield such that k ⊆ K,k ⊆ E ⊆ L, and k ⊆ E′ ⊆ L. Then the following conditions are equivalent.

(a) K and E are quasi-linearly disjoint (linearly disjoint or algebraically dis-joint) over k and KE and L are quasi-linearly disjoint (respectively, linearlydisjoint or algebraically disjoint) over E.

(b) K and E′ are quasi-linearly disjoint (linearly disjoint or algebraically dis-joint) over k and KE′ and L are quasi-linearly disjoint (respectively, linearlydisjoint or algebraically disjoint) over E′.

Regular and primary field extensions

A field extension L/K is called regular if K is algebraically closed in L (thatis, every element of L algebraic over K lies in K) and L/K is separable. Theproofs of statements 1.6.40 - 1.6.43 can be found in [119, Chapter III].

Proposition 1.6.45 Let K be a field, K its algebraic closure, and L an over-field of K. Then the extension L/K is regular if and only if the fields L and Kare linearly disjoint over K.

Theorem 1.6.46 (i) If F is an intermediate field of a regular field extensionL/K, then the extension F/K is regular.


(ii) If F is an intermediate field of a field extension L/K such that the exten-sions F/K and L/F are regular, then L/K is also regular.(iii) Every extension of an algebraically closed field is regular.(iv) Let L and M be extensions of a field K contained in a common overfield.

If L/K is regular and L and M are algebraically disjoint over K, then L andM are linearly disjoint over K.

(v) Let L and M be intermediate fields of a field extension N/K.(a) If L/K is regular and L and M are algebraically disjoint over K, then the

extension LM/L is regular.(b) If L/K and M/K are regular and L and M are algebraically disjoint over

K, then the extension LM/K is regular.(c) Let L and M be linearly disjoint over K. Then the extension L/K is

regular if and only if LM/M is regular.

Let K be a field. An overfield L of K is said to be a primary field extensionof K (we also say that the extension L/K is primary) if the algebraic closure ofK in L is purely inseparable over K.

The following statement gives an alternative description of primary fieldextensions.

Proposition 1.6.47 Let L/K be a field extension and let S(K) denote theseparable closure of K (that is, the separable closure of K in its algebraic closureK). Then the extension L/K is primary if and only if L and S(K) are linearlydisjoint over K.

Theorem 1.6.48 Let F be an intermediate field of a field extension L/K.(i) If L/K is primary, then so is F/K.(ii) If F/K and L/F are primary extensions, then so is L/K.(iii) If K = S(K) (that is, K coincides with its separable closure), then every

field extension of K is primary.(iv) Let F/K be primary and let G be another intermediate field of L/K such

that G/K is separable and F and G are algebraically disjoint over K. Then F andG are linearly disjoint over K. Furthermore, FG is a primary extension of G.

(v) Let G be an intermediate field of L/K, and let F/K and G/K be primary.If F and G are algebraically disjoint over K, then FG is a primary extensionof K.(vi) Let G be an intermediate field of L/K such that G and F are linearly

disjoint over K. Then F is primary over K if and only if FG is primary over G.

Proposition 1.6.49 Let K and K ′ be field extensions of the same field k andlet M be a free join of K and K ′ over k. Then

(i) If the extension K/k is primary, then K/k and K ′/k are quasi-linearlydisjoint.

(ii) K and K ′ are quasi-linearly disjoint over k and M/K ′ is a primary (regu-lar) extension if and only if the extension K/k is primary (respectively, regular).


Corollary 1.6.50 Let L,L′,M , and M ′ be fields such that L ⊆ L′ ⊆ M ′ andL ⊆ M ⊆ M ′ are sequences of field extensions. Furthermore, suppose that theextensions L′/L and M/L are quasi-linearly disjoint, and the extension M ′/Mis primary. Then the extension L′/L is also primary.

GALOIS CORRESPONDENCE

Let L be a field extension of a field K. It is easy to check that the set ofall K-automorphisms of the field L is a group with respect to the compositionof automorphisms. This group is denoted by Gal(L/K) and called the Galoisgroup of the field extension L/K. If F is any intermediate field between K andL, then the set of all automorphisms α ∈ Gal(L/K) that leave elements of Ffixed is a subgroup of Gal(L/K) denoted by F ′. On the other hand, if H is anysubgroup of Gal(L/K), then the set of elements a ∈ L such that α(a) = a forevery α ∈ H is an intermediate field between K and L; this field is denoted byH ′. The following theorem is called the fundamental theorem of Galois theory.

Theorem 1.6.51 Let L be a finite Galois field extension of a field K. Then(i) There is a one-to-one inclusion reversing correspondence between interme-

diate fields of L/K and subgroups of Gal(L/K) given by F → F ′ = Gal(L/F )(K ⊆ F ⊆ L) and H → H ′ (H ⊆ Gal(L/K)).

(ii) If H is a subgroup of Gal(L/K), then L : H ′ = |H| (here and below |S|denotes the number of elements of a finite set S) and F : K = Gal(L/K) : H.(iii) A subgroup H of Gal(L/K) is normal if and only if H ′ is a Galois

field extension of K. In this case Gal(H ′/K) is isomorphic to the factor groupGal(L/K)/H.

If L/K is an infinite Galois field extension, then not all subgroups ofGal(L/K) have the form Gal(L/F ) for some intermediate field F between Kand L. However, one can prove an analog of Theorem 1.6.51 for infinite Galoisfield extensions using the following concept of Krull topology.

Let G = Gal(L/K) be the Galois group of a Galois field extension L/K ofarbitrary (finite or infinite) degree. Let I denote the family of all intermediatefields F between K and L such that F : K < ∞ and the field extension F/Kis Galois. Furthermore, let N denote the set of all subgroups H of G such thatH = Gal(L/F ) for some intermediate field F ∈ I.

Proposition 1.6.52 (i) With the above notation, if a1, . . . , an ∈ L, then thereis a field F ∈ I such that ai ∈ F for i = 1, . . . , n.

(ii) Let N ∈ N and let N = Gal(L/F ) for some F ∈ I. Then F = N ′ (we usethe notation introduced before Theorem 1.6.51) and N is a normal subgroup ofG. Furthermore, G/N ∼= Gal(F/K), so |G/N | = |Gal(F/K)| = F : K < ∞.(iii)

⋂N∈N N = e (e denotes the identity of the group G) and

⋂N∈N αN =

α for any α ∈ G.(iv) If N1, N2 ∈ N, then N1 ∩ N2 ∈ N.


The results of the last lemma allows one to introduce a topology on G asfollows: a subset X of G is open if X = ∅ or if X is a union of a family of sets ofthe form αiNi with αi ∈ G, Ni ∈ N. This topology is called the Krull topologyon the group G. It is easy to see that the set αN |α ∈ G, N ∈ N is a basisof the Krull topology. Furthermore, if N ∈ N and α ∈ G, then |G : N | < ∞, soG \ αN is a union of finitely many cosets of N . It follows that αN is a clopen(that is, both closed and open) subset of G, so that the Krull topology on Ghas a basis of clopen sets.

The following statement gives some more properties of this topology.

Proposition 1.6.53 (i) With the above notation, the group G = Gal(L/K) isHausdorff, compact, and totally disconnected in the Krull topology.

(ii) Let H be a subgroup of G and let N = Gal(L/H ′). Then N is the closureof H in the Krull topology on G.

The following statement is called the Fundamental Theorem of InfiniteGalois Theory.

Theorem 1.6.54 Let L be a Galois field extension of a field K, and let G =Gal(L/K).

(i) With the Krull topology on G, the maps F → F ′ = Gal(L/F ) and H → H ′

give an inclusion reversing correspondence between the intermediate fields F ofL/K and the closed subgroups H of G.

(ii) If an intermediate field F of L/K and a subgroup H ⊆ G correspond toeach other via the correspondence described in (i), then |G : H| < ∞ if an onlyif H is open. When this occurs, |G : H| = F : K.(iii) IfH is a subgroup ofG, thenH is normal if and only ifH ′ is aGalois extension

of K. When this occurs, there is a group isomorphism Gal(F/K) ∼= G/H. Withthe quotient topology on G/H this isomorphism is also a homeomorphism.

SPECIALIZATIONS

Definition 1.6.55 Let K be a field and let a = (ai)i∈I be an indexing whosecoordinates lie in some overfield of K (such an indexing is said to be an indexingover K). Then a K-homomorphism φ of the ring K[a] = K[ai | i ∈ I] intoan overfield of K is called a specialization of the indexing a over K. The imageb = φ(a) (which is the indexing (φ(ai) )i∈I with the same index set I) is alsocalled a specialization of a. (In this case we often say that a specializes to b.)

A specialization is called generic if it is an isomorphism.

If the index set I of an indexing a over K is finite, I = 1, . . . , s for somepositive integer s, then we treat a as an s-tuple (a1, . . . , as).

The proofs of the following two propositions and Theorem 1.6.54 can befound in [119, Chapter II, Section 3].

Proposition 1.6.56 Let a = (a1, . . . , as) be an indexing over a field K and letP be a defining ideal of a in the polynomial ring K[X1, . . . , Xs] (that is, P is


the kernel of the natural epimorphism K[X1, . . . , Xs] → K[a1, . . . , as] sending apolynomial f(X1, . . . , Xs) to f(a1, . . . , as)). Furthermore, let b = (b1, . . . , bs) beanother s-tuple over K with the defining ideal P ′ in K[X1, . . . , Xs]. Then

(i) b is a specialization of a if and only if P ⊆ P ′.(ii) b is a generic specialization of a if and only if P = P ′.

Proposition 1.6.57 Let V be an irreducible variety over a field K (elements ofV are s-tuples with coordinates in the universal field over K), let P be a definingideal of V in the polynomial ring K[X1, . . . , Xs], and let a = (a1, . . . , as) be ageneric zero of P . Then an s-tuple b = (b1, . . . , bs) over K belongs to V if andonly if b is a specialization of a.

Theorem 1.6.58 Let K be a field, a = (ai)i∈I an indexing in an overfield of K,and b = (bi)i∈I a specialization of a over K. Then trdegKK(b) ≤ trdegKK(a)with equality only if the specialization is generic. In particular, if every ai (i ∈ I)is algebraic over K, then every specialization of a over K is generic.

Two s-tuples a and b over a field K are said to be equivalent if there is a K-isomorphism of K(a) onto K(b). The following two statements are consequencesof the last theorem.

Corollary 1.6.59 Two s-tuples over a field K are equivalent if and only if theyare specializations of each other.

Corollary 1.6.60 Let K be a field, V a variety over K[X1, . . . , Xs], and a ans-tuple over K. Then a is a generic zero of an irreducible component of V ifand only if there is no s-tuple b ∈ V such that trdegKK(b) > trdegKK(a) andb specializes to a over K.

If b = (bj)j∈J is an indexing over the field K(a) (we use the notation ofDefinition 1.6.55), then by extension of the specialization φ of a over K to bwe mean an extension of φ to a homomorphism of K[a, b] into an overfield ofK; the image of this extended homomorphism is also called an extension of theoriginal specialization.

We say that almost every specialization of a over K has property P if thereexists a nonzero element c ∈ K[a] such that every specialization φ of a over K,which does not specialize c to 0, has property P. It is easy to see that if almostevery specialization of a over K has property P, then the generic specializationof a has this property.

Theorem 1.6.61 [41, Introduction, Theorem VII] Let K be a field and let a =(ai)i∈I be an indexing in an overfield L of K. Furthermore, let b = (b1, . . . , bs)be an s-tuple with coordinates in L and let c be a nonzero element of K[a, b] =K[ai | i ∈ I ∪ b1, . . . , bs]. Then almost every specialization φ of a over Kextends to a specialization φ′ of a, b (treated as an indexing with the index setI ∪ 1, . . . , s) such that φ′(c) = 0.


Definition 1.6.62 Let a and b be two indexings with coordinates in an overfieldof a field K, and let φ be a specialization of a over K. An extension of φ to aspecialization φ′ of a, b over K is called nondegenerate if it satisfies the followingcondition: a subindexing of φ′(b) is algebraically independent over K(φ(a)) if andonly if the corresponding subindexing of b is algebraically independent over K(a).

The proofs of the following three propositions can be found in [41, Chapter 7,Sections 10 - 12, 23].

Proposition 1.6.63 Let K be a field, a = (ai)i∈I an indexing in an overfieldof K, and L the algebraic closure of K(a). Furthermore, let f be a polynomialin the polynomial ring K(a)[X1, . . . , Xn] which is irreducible in L[X1, . . . , Xn].Then almost every specialization φ of a over K extends to the coefficients of f(treated as a finite indexing over K), and the polynomial fφ obtained by applyingφ to every coefficient of f is irreducible in K(φ(a) )[X1, . . . , Xn].

Proposition 1.6.64 Let a be an indexing and b a finite indexing in an overfieldof a field K. Then almost every specialization φ of a over K has a nondegenerateextension to a specialization φ′ of a, b over K. Furthermore, if the field extensionK(a, b)/K(a) is primary, then the nondegenerate extension φ′ is unique.

Proposition 1.6.65 Let K be a field and let a and b be two indexings withcoordinates in some overfield of K such that K(a) and K(b) are quasi-linearlydisjoint over K. Furthermore, let K(φ(a), ψ(b)) be an overfield of K generatedby specializations φ and ψ of a and b over K, respectively. Then the indexing(φ(a), ψ(b)) is a specialization of (a, b) over K.

The following proposition is proved in [118].

Proposition 1.6.66 Let K[x1, . . . , xm, y1, . . . , yn] be the ring of polynomialsin m + n variables over a field K, let (a, b) be an m + n-tuple over K, andlet (a, b) be a specialization of (a, b) over K (we write this as (a, b) −→

K(a, b)).

If trdegKK(a) = trdegKK(a) − t, t ∈ N, then there exists an n-tuple c =(c1, . . . , cn) over K such that

(a, b) −→K

(a, c) −→K

(a, b)

and trdegKK(a, c) ≥ trdegKK(a, b) − t.

If R is an integral domain, then a homomorphism φ of R into a field K iscalled a specialization of the domain R. If R and K have a common subringR0 and φ(a) = a for every a ∈ R0, then φ is said to be a specialization of Rover R0.

Proposition 1.6.67 [105, Chapter 0, Proposition 9]. Let R0 and R be subringsof a field with R0 ⊆ R.

(i) If R is integral over R0, then every specialization of R0 into an alge-braically closed field L can be extended to a specialization of R into L.


(ii) Let x ∈ R. If a specialization φ0 : R0 → L into an algebraically closed fieldL cannot be extended to a specialization R0[x] → L, then φ0 can be extended toa unique specialization φ : R0[x−1] → L such that φ(x−1) = 0.(iii) If R is finitely generated (respectively, finitely generated and separable)

over R0 and u ∈ R, u = 0, then there exists a nonzero element u0 ∈ R0 with thefollowing property: Every specialization φ0 : R0 → L into an algebraically closed(respectively, a separably closed) field L such that φ0(u0) = 0 can be extended toa specialization φ : R → L such that φ(u) = 0 (respectively, such that φ(u) = 0and φ(R) is separable over φ(R0)).

PLACES OF FIELDS OF ALGEBRAIC FUNCTIONS

Let K be a field. By a field of algebraic functions of one variable over K wemean a field L containing K as a subfield and satisfying the following condition:there is an element x ∈ L which is transcendental over K, and L is a finite fieldextension of K(x).

If K is a subfield of a field L, then by a V -ring of L over K we mean asubring A of L such that K ⊆ A L and for any element x ∈ L \ A, one hasx−1 ∈ A. (If A is V -ring of L over K and L is the quotient field of A, then Ais a valuation ring in the sense of Definition 1.2.48.) It is easy to check that theset of all non-units of a V -ring A form an ideal p in A, so A is a local ring.

Let L be a field of algebraic functions of one variable over a field K. By aplace in L we mean a subset p of L which is the ideal of non-units of some V -ringA of L over K. This V -ring is uniquely determined; in fact, one can easily checkthat A = a ∈ L | ap ⊆ p. The ring A is called the ring of the place p, and thefield A/p is called the residue field of the place.

Exercise 1.6.68 With the above notation and conventions, show that the ringA is integrally closed in L.

The result of the last exercise implies that if K′ is the algebraic closure ofK in L, then K ′ ⊆ A. Thus, the notion of a place in L is the same whether weconsider L as a field of algebraic functions over K or over K ′. Furthermore, sinceK ′ ∩ p = (0), the natural homomorphism of A onto the residue field F = A/pmaps K ′ isomorphically onto a subfield of F , so one can treat F as an overfield ofK ′. The proof of the following results about places of fields of algebraic functionscan be found in [25, Chapter 1].

Proposition 1.6.69 Let L be a field of algebraic functions of one variable overa field K.

(i) If p is a place in L, then the residue field of p is a finite algebraic extensionof K.

(ii) If p is a place in L, and A is the ring of p, then there is an element t ∈ A(called a uniformizing variable at p) such that p = tA and

⋂∞k=1 tkA = (0).

(iii) Let p be a place in L, and let A be the ring of p. Then there exists adiscrete valuation vp of the field L (see Definition 1.2.50) such that A is thevaluation ring of vp.

1.7. DERIVATIONS AND MODULES OF DIFFERENTIALS 89

(iv) Let R be a subring of L containing K and let I be a proper nonzero idealof R. Then there exists a place q of L whose ring B contains R and I ⊆ q

⋂R.

Proposition 1.6.70 Let L be a field of algebraic functions of one variable overa field K, and let x1, . . . , xr be elements of L which are not all in K. Let I be theset of all polynomials f(X1, . . . , Xr) in r variables X1, . . . , Xr over K such thatf(x1, . . . , xr) = 0, and let ξ1, . . . , ξr be elements of K such that f(ξ1, . . . , ξr) = 0for all f ∈ I. Then there exists a place p of L such that xi−ξi ∈ p for i = 1, . . . , r.

Let L be a field of algebraic functions of one variable over a field K, let p bea place in L, and let vp be the corresponding discrete valuation. We say that asequence (xn) of elements of L converges at p to x ∈ L if limn→∞ vp(x−xn) = ∞.Of course, in this case, (xn) is a Cauchy sequence at p, that is, limn→∞ vp(xn+1−xn) = ∞. It is easy to see that the set of all Cauchy sequences at p form a ring Swith respect to natural (coordinatewise) operations. Furthermore, the set of allsequences converging to 0 at p form an ideal J of S. The ring S/J is denoted byLp and called the p-adic completion of the field L. It can be shown (the detailscan be found, for example, in [25, Chapter III]) that Lp is a field containing Las its subfield (if we identify an element x ∈ L with the sequence (xn) wherexn = x for all n). Furthermore, the function vp can be naturally extended to adiscrete valuation Lp → Z∪∞ (denoted by the same symbol vp) with respectto which Lp is complete, that is, every Cauchy sequence at p converges at p.

Proposition 1.6.71 Let L be a field of algebraic functions of one variable overa field K, p a place in L, and A the ring of p. Suppose that the residue fieldA/p is separable over K. Then every element a in the p-adic completion Lp ofthe field L has a representation a =

∑∞k=r aktk where r ∈ Z, t is a uniformizing

variable, and ak ∈ L for every k. (The representation is understood in the sensethat the sequence of partial sums of the series converges to a at p.) Furthermore,if a ∈ A, then r ≥ 0; and if a ∈ p, then r ≥ 1.

1.7 Derivations and Modules of Differentials

Throughout this section, by a ring we shall always mean a commutative ring.Let A be a ring and M an A-module. A mapping D : A → M is called a

derivation from A to M if D(a+b) = D(a)+D(b) and D(ab) = aD(b)+bD(a) forany a, b ∈ A. The set of all derivations from A to M is denoted by Der(A,M).It is easy to see that if a, b ∈ A and D1 and D2 are derivations from A toM , then the mapping aD1 + bD2 is also a derivation from A to M . Therefore,Der(A,M) can be naturally considered as an A-module; it is called the moduleof derivations from A to M .

If A is an algebra over a ring K, then a derivation D from A to an A-module M is said to be K-linear if D is a homomorphism of K-modules. Theset of all K-linear derivations from A to M is denoted by DerK(A,M), it can benaturally treated as an A-submodule of Der(A,M). The A-module DerK(A,A)of all K-linear derivations from A to A will be also denoted by DerKA.


Example 1.7.1 Let R = K[X1, . . . , Xn] be the ring of polynomials in variablesX1, . . . , Xn over a field K. Then every partial derivative ∂/∂Xi (1 ≤ i ≤ n) isa derivation from R to R. Moreover, it is easy to see that if Si denotes thepolynomial ring in the set of variables X1, . . . , Xn \ Xi, then ∂/∂Xi is anSi-linear derivation from R to R.

Exercises 1.7.2 Let R be a subring of a ring S and let D ∈ Der(R,S).1. Prove that D(an) = nan−1D(a) for any a ∈ R and any integer n ≥ 1.2. Show that Ker D is a subring of R (in particular, Ker D contains the

prime subring of R).3. Show that if R is a field, then Ker D is a subfield of R.4. Let A be a subring of R. Show that A ⊆ Ker D if and only if D is A-linear.5. Prove that if D′ is another derivation from R to S, then [D,D′] = DD′−

D′D ∈ Der(R,S). ([D,D′] is called the Lie bracket of D and D′.)6. Show that Dn(xy) =

∑ni=0

(ni

)Di(x)Dn−i(y) for any x, y ∈ R.

7. Prove that if S is a field and F is the quotient field of R, then D canbe extended in a unique way to a derivation D′ ∈ Der(F, S). Moreover, the

extension is given by the formula D′(x

y) =

yD(x) − xD(y)y2

.

8. Let S = R[X1, . . . , Xn] be the ring of polynomials in variables X1, . . . , Xn

over R. For any f ∈ S, let fD be obtained from the polynomial f by applyingD to all coefficients of f . Prove that the mapping f → fD is a derivation of Sextending D.

9. Let R be a field and let T = R(X1, . . . , Xn) be the field of rationalfunctions in variables X1, . . . , Xn over R. Prove that if K is a field extension ofT , then partial derivations ∂/∂Xi (1 ≤ i ≤ n) form a basis of the vector T -spaceDerR(T, K).

10. Let R be a field of characteristic p > 0 and let S and K be field extensionsof R. Prove that every R-derivation from S to K is an Sp(R)-derivation, so thatDerR(S,K) = DerSp(R)(S,K).

The following propositions describe the conditions when one can extend aderivation from a field K to a derivation from some field extension of K. Theproofs of the statements can be found, for example, in [8, Section 4.5].

Proposition 1.7.3 Let K be a field and L a finitely generated field extensionof K, L = K(η1, . . . , ηn). Let K[X1, . . . , Xn] be the ring of polynomials invariables X1, . . . , Xn over K, M a field extension of L, and D ∈ Der(K,M).Furthermore, let I denote the ideal f ∈ K[X1, . . . , Xn] | f(η1, . . . , ηn) = 0of the ring K[X1, . . . , Xn], and let S be a system of generators of the idealI. Finally, let ζ1, . . . , ζn be any elements in M . Then D can be extended toa derivation D′ : L → M with D′(ηi) = ζi (i = 1, . . . , n) if and only if

fD(η1, . . . , ηn) +n∑

i=1

ζi∂f

∂Xi= 0 for all f ∈ S. (As in Exercise 1.7.2.8, fD

denotes the polynomial obtained from f by applying the derivation D toall coefficients of f .) If such an extension D′ exists, then it is unique and


one has D′g(η1, . . . , ηn) = gD(η1, . . . , ηn) +n∑

i=1

ζi∂g

∂Xi(η1, . . . , ηn) for any

g ∈ K[X1, . . . , Xn].

Proposition 1.7.4 Let K be subfield of a field L, D ∈ Der(K,L), and S ⊆ L.If the set S is algebraically independent over K, then for every map φ : S → L,there exists a unique derivation D′ : K(S) → L such that D′ extends D andD′(s) = φ(s) for all s ∈ S.

Proposition 1.7.5 Let L be a separable algebraic field extension of a field Kand let M be a field extension of L. Then every derivation D ∈ Der(K,M) canbe extended, in a unique way, to a derivation D′ : L → M .

Corollary 1.7.6 Let K be a field, L a field extension of K, and M a fieldextension of L. If S = s1, . . . , sn is a finite subset of L such that L is aseparable algebraic field extension of K(S), then dimMDerK(L,M) ≤ n.

Proposition 1.7.7 Let L/K be a finitely generated separable field extension.Then trdegKL = dimLDerKL. If x1, . . . , xn is a separating transcendencebasis for L/K and F = K(x1, . . . , xn), then there is a basis D1, . . . , Dn ofthe vector L-space DerKL such that the restriction of Di on F is ∂/∂xi (i =1, . . . , n).

Proposition 1.7.8 Let K be a field of a prime characteristic p, let L be a fieldextension of K such that Lp ⊆ K, and let M be a field extension of L. Thena derivation D : K → M can be extended to a derivation from L to M if andonly if D is an Lp-derivation.

Proposition 1.7.9 Let K be a field, L a field extension of K, and M a field ex-tension of L. Then the extension L/K is separable if and only if every derivationfrom K to M can be extended to a derivation from L to M .

Exercises 1.7.10 Let K be a field of a prime characteristic p and let L be afield extension of K.

1. Prove that if M is a field extension of L, then KLp = a ∈ L |D(a) = 0for every D ∈ DerK(L,M).

2. Show that Kp = a ∈ K |D(a) = 0 for every D ∈ DerK(K,L).3. Prove that the field K is perfect if and only if the only derivation from

K to L is the zero derivation.

If A is an algebra over a commutative ring K, then the module of differentials(or module of Kahler differentials) of A over K, written ΩA|K , is the A-modulegenerated by the set of symbols da | a ∈ A subject to the relations

d(ab) = adb + bda (Leibniz rule) andd(αa + βb) = αd(a) + βd(b) (K-linearity)

for any a, b ∈ A, α, β ∈ K.


Equivalently, one can say that, ΩA|K = F/N where F is a free A-modulegenerated by the set da | a ∈ A and N is an A-submodule of F generated byall elements d(a + b) − da − db, d(ab) − adb − bda, and dα for a, b ∈ A, α ∈ K.

The mapping d : A → ΩA|K defined by d : a → da (a ∈ A) is a K-linearderivation called the universal K-linear derivation. The image da of an elementa ∈ A under this derivation is said to be the differential of a. (In order toindicate the ring K and K-algebra A, we shall sometimes write dA|K for d.)

The following universal property of the derivation d is a direct consequenceof the definition.

Proposition 1.7.11 With the above notation, let D : A → M be a K-linearderivation from A to an A-module M . Then there is a unique A-module homo-morphism f : ΩA|K → M such that fd = D.

Corollary 1.7.12 Let A be an algebra over a ring K and let M be an A-module.Then DerK(A,M) ∼= HomA(ΩA|K ,M).

The next proposition gives some properties of modules of differentials asso-ciated with field extensions. The proofs of its statements can be found in [144,Chapter V, Section 23].

Proposition 1.7.13 Let L be a field extension of a field K. Then(i) dimLΩL|K = dimLDerKL.(ii) If L is finitely generated over K, L = K(η1, . . . , ηm), then the elements

dη1, . . . , dηm generate ΩL|K as a vector L-space (hence dimLΩL|K < ∞).(iii) If L/K is a separable algebraic extension, then ΩL|K = 0.(iv) If x1, . . . , xn is a separating transcendence basis for the extension L/K,

then dx1, . . . , dxn is a basis of the vector L-space ΩL|K . Conversely, if thefield extension L/K is separably generated and y1, . . . yn are elements of L suchthat dy1, . . . , dyn is a basis of ΩL|K over L, then y1, . . . yn is a separatingtranscendence basis for L/K.

(v) If Char K = 0, then a family of elements ηii∈I in L is algebraicallyindependent over K if and only if the family dηii∈I in ΩL|K is linearly inde-pendent over L.

The following construction and theorem give an alternative description ofmodules of differentials.

Let K be a commutative ring, A a K-algebra, and µ : A⊗

K A → A thenatural homomorphism of K-algebras (µ : x

⊗y → xy for all x, y ∈ A). If

I = Ker µ, then I/I2 can be naturally treated as an A-module (with respect tothe canonical structure of an A

⊗K A/I-module and the natural isomorphism

of K-algebras A⊗

K A/I ∼= A). It is easy to see that a⊗

1 − 1⊗

a ∈ I forevery a ∈ A and the mapping e : A → I/I2 defined by a → (a

⊗1−1

⊗a)+I2

(a ∈ A) is a derivation from A to I/I2. The following result shows that the pair(I/I2, e) is isomorphic to the pair (ΩA|K , d) in the natural sense. (As before,ΩA|K denotes the module of differentials of A over K and d : A → ΩA|K is theuniversal K-linear derivation.)


Proposition 1.7.14 With the above notation, there exists an isomorphism ofA-modules φ : ΩA|K → I/I2 such that φd = e.

Example 1.7.15 Let A be an algebra over a commutative ring K generated (asa K-algebra) by a family xii∈I . Then the family dxii∈I generate ΩA|K as anA-module, and if the family xii∈I is algebraically independent over K, then

ΩA|K is a free A-module with the basis dxii∈I . Indeed, letm∑

k=1

akdxik= 0

for some ik ∈ I, ak ∈ A (1 ≤ k ≤ m). For every j ∈ I, let∂

∂xjdenote the

corresponding partial derivation of A to itself. By Corollary 1.7.12, for everyk = 1, . . . , m, there exist homomorphisms of A-modules φik

: ΩA|K → A suchthat

φik(dxj) =

∂xj

∂xik

=

1, if j = ik,

0, if j = ik

for any j ∈ I. Since ar = φir(

m∑k=1

akdxik) = φir

(0) = 0 for r = 1, . . . , m, we

obtain that ΩA|K is a free A-module with the basis dxii∈I .

Exercises 1.7.16 Let A A′

K K ′

be a commutative diagram of ring homomorphisms.1. Prove that there exists a natural homomorphism of A-modules ΩA|K →

ΩA′|K′ that induces a natural homomorphism of A-modules ΩA|K⊗

K A′ →ΩA′|K′

2. Prove that if A′ = A⊗

K K ′, then the homomorphism ΩA|K⊗

K A′ →ΩA′|K′ considered in the preceding exercise is an isomorphism.

3. Show that if S is a multiplicative set in A and A′ = S−1A, then ΩA′|K ∼=ΩA|K

⊗A A′ ∼= S−1ΩA|K .

Theorem 1.7.17 Let K, A, and B be commutative rings, and let f : K → Aand g : A → B be ring homomorphisms (due to which B can be consider as bothK- and A-algebra). Then

(i) There exists an exact sequence of B-modules

ΩA|K⊗A

Bα−→ ΩB|K

β−→ ΩB|A → 0 (1.7.1)

where α : (dA|Ka)⊗

b → bdB|Kg(a) and β : bdB|Kb′ → bdB|Ab′ for all a ∈ Aand b, b′ ∈ B.


(ii) α has a left inverse (that is, α is injective and Imα is a direct summandof the B-module ΩB|K) if and only if every K-linear derivation from A to aB-module M can be extended to a derivation from B to M .

Exercise 1.7.18 minu1pt Let K be a ring, K ′ and A two K-algebras, andA′ = K ′ ⊗

K A. Show that if S is a multiplicative set in A, then ΩA′|K′ ∼=ΩA|K

⊗K K ′ ∼= ΩA|K

⊗A A′ and ΩS−1A|K ∼= ΩA|K

⊗A S−1A.

Theorem 1.7.19 ([138, Theorem 25.2]) Let K be a ring, A a K-algebra, Ian ideal of A, B = A/I, and C = A/I2. Let φ : I → ΩA|K

⊗A B be a ho-

momorphism of A-modules such that φ(x) = dA|Kx⊗

1 for all x ∈ I, and letα : ΩA|K

⊗A B → ΩB|K be the homomorphism of B-modules in the sequence

(1.7.1). Furthermore, let ψ : I/I2 → ΩA|K⊗

A B denote a homomorphism ofB-modules induced by φ (obviously φ(I2) = 0). Then

(i) The sequence of B-modules

I/I2 ψ−→ ΩA|K⊗A

Bα−→ ΩB|K → 0 (1.7.2)

is exact.(ii) ΩA|K

⊗A B ∼= ΩC|K

⊗C B.

(iii) The homomorphism ψ has a left inverse if and only if the extension 0 →I/I2 → C → B → 0 of the K-algebra B by I/I2 is trivial over K.

We conclude this section with some facts about skew derivations and skewpolynomial rings.

Let A be a ring (not necessarily commutative) and let α be an injectiveendomorphism of A. An additive mapping δ : A → A is called an α-derivation(or a skew derivation of A associated with α) if δ(ab) = α(a)δ(a)+ δ(a)b for anya, b ∈ A.

If δ is an α-derivation of a ring A, then one can define the corresponding ringof skew polynomials A[X;α, δ] in one indeterminate X as follows: the additivegroup of A[X;α, δ] is the additive group of the ring of polynomials in X withleft-hand coefficients from A, and the multiplication in A[X;α, δ] is defined bythe rule Xa = α(a)X + δ(a) (a ∈ A) and the distributive laws. (A is naturallytreated as a subring of A[X;α, δ].) Elements of the ring A[X;α, δ] are calledskew polynomials in X with coefficients in A. As in the case of a polynomialring, every skew polynomial f ∈ A[X;α, δ] has a unique representation in theform f = anXn + · · · + a1X + a0 with ai ∈ A (i = 0, . . . , n); n is said to be thedegree of f and denoted by deg f . Obviously, deg (fg) ≤ deg f + deg g for anyf, g ∈ A[X;α, δ].

The concept of a ring of skew polynomials in one indeterminate can be nat-urally generalized to the case of several indeterminates. Let σ1 = α1, . . . , αnand σ2 = β1, . . . , βm be two sets of injective endomorphisms of a ring A, andlet ∆ = δ1, . . . , δm be the set of skew derivations of the ring A into itself asso-ciated with the endomorphisms β1, . . . , βm, respectively. Furthermore, assume


that every two elements of the set σ1 ∪ σ2 ∪∆ commute (as mappings of A intoitself).

Let Ω be a free commutative semigroup with free generators X1, . . . , X2m+n,so that every element ω ∈ Ω has a unique representation as ω = Xk1

1 . . . Xk2m+n

2m+n

(k1, . . . , k2m+n ∈ N). Then the set S of all formal finite sums∑

ω∈Ω aωω(aω ∈ A for all ω ∈ Ω, and only finitely many coefficients aω are not equalto zero) can be naturally considered as a left A-module. Moreover, the setS becomes a ring if for any a ∈ A, one defines the multiplication in S bythe rules Xia = βi(a)Xi + δi(a) (i = 1, . . . , m), Xja = βj−m(a)Xj (j =m + 1, . . . , 2m), Xka = αk−2m(a)Xk (k = 2m + 1, . . . , 2m + n), and the dis-tributive laws. (A is naturally treated as a subring of S.) The ring S is calledthe ring of skew polynomials associated with (σ1, σ2,∆). This ring is denoted byA[X1, . . . , X2m+n; δ1, . . . , δm;β1, . . . , βm;α1, . . . , αn] (sometimes we write δi(βi)instead of δi, 1 ≤ i ≤ m); its elements are called skew polynomials in the inde-terminates X1, . . . , X2m+n.

As in the usual polynomial ring in several variables, a power product ω =Xk1

1 . . . Xk2m+n

2m+n ∈ Ω is called a monomial and the number deg ω =∑2m+n

i=1 ki iscalled the degree of ω. The degree deg f of a skew polynomial f =

∑ω∈Ω aωω ∈

S is defined as maxdeg ω |ω ∈ Ω, aω = 0. It is easy to see that deg (fg) ≤deg f + deg g for any two skew polynomials f and g.

The following statement generalizes the classical Hilbert basis theorem forpolynomial rings (Theorem 1.2.11(vi)). The proof of this result in the case ofskew polynomial rings in one indeterminate can be found in [26, Section 0.8].We leave to the reader the generalization of this proof to the case of severalindeterminates.

Theorem 1.7.20 With the above notation, suppose that the ring A is left Noe-therian and α1, . . . , αn, β1, . . . , βm are automorphisms of A. Then the ring ofskew polynomials S = A[X1, . . . , X2m+n; δ1, . . . , δm;β1, . . . , βm;α1, . . . , αn] isleft Noetherian.

Recall that an integral domain A is called a left (respectively, right) Oredomain if for any nonzero elements x, y ∈ A we have Ax∩Ay = (0) (respectively,xA ∩ yA = (0)).

Proposition 1.7.21 ([26, Proposition 8.4]) Let A be a left Ore domain, α aninjective endomorphism and δ an α-derivation of A. Then the ring of skewpolynomials A[X;α, δ] is a left Ore domain.

Exercises 1.7.22 Let A be a ring, α an injective endomorphism, and δ anα-derivation of A.

1. Prove that if A is an integral domain and α is injective, then the ring ofskew polynomials A[X;α, δ] is an integral domain.

2. Show that if A is an integral domain and there exists a nonzero elementx ∈ A such that xα(A)∩α(A) = (0), then the ring A[X;α, δ] is not a right Oredomain.


1.8 Grobner Bases

In what follows we give some basic concepts and results of the theory of Grobneror standard bases. The detail presentation of this theory (with the proofs of thestatements of this section) can be found in [1], [9], [54], and [57].

Let K be a field and R = K[x1, . . . , xn] a ring of polynomials in n variablesx1, . . . , xn over K. As usual, by a monomial in R we mean a power productt = xk1

1 . . . xknn with k1, . . . , kn ∈ N; the integer deg t =

∑ni=1 ki is called the

degree of t.In what follows the commutative semigroup of all monomials will be denoted

by T . If t and t′ are two monomials in T , we say that t′ divides t (or that t isa multiple of t′) and write t′|t if there exists t′′ ∈ T such that t = t′t′′. (In this

case the monomial t′′ is written ast

t′.)

A total order < on the set T is called monomial (or admissible) if it satisfiesthe following two conditions:

(i) 1 < t for any t ∈ T, t = 1.(ii) If t1, t2 ∈ T and t1 < t2, then tt1 < tt2 for every t ∈ T .

(As usual, the inequality t1 < t2 (t1, t2 ∈ T ) can be written as t2 > t1 andt1 ≤ t2 (or t2 ≥ t1) means that either t1 < t2 or t1 = t2.)

The following examples show some particular monomial orders.

Examples 1.8.1 1. The lexicographic order <lex on the set of monomials T isdefined as follows: if t = xk1

1 . . . xknn , t′ = xl1

1 . . . xlnn , then t <lex t′ if and only

if (k1, . . . , kn) is less than (l1, . . . , ln) with respect to the lexicographic order onNn.

2. The degree lexicographic order <Deglex on T is defined by the followingcondition: if t and t′ are as in the previous example, then t <Deglex t′ if andonly if (deg t, k1, . . . , kn) is less than (deg t′, l1, . . . , ln) with respect to the lexi-cographic order on Nn+1.

3. The degree reverse lexicographic order <Degrevlex on the set of monomialsT : if t and t′ are as above, then t <Degrevlex t′ if and only if deg t < deg t′

or if deg t = deg t′ and the last indeterminate with different exponents in tand t′ has higher exponent in t. In other words, t <Degrevlex t′ if and only if(deg t, ln, . . . , l1) is less than (deg t′, kn, . . . , k1) with respect to the lexicographicorder on Nn+1.

We leave the proof of the following statement to the reader as an exercise.

Proposition 1.8.2 The set T is well-ordered with respect to any monomialorder on T .

Exercises 1.8.3 1. Let < be any monomial order on T and let A be an n× n-matrix with nonnegative integer entries, which is invertible over Q. For anyt = xk1

1 . . . xknn , t′ = xl1

1 . . . xlnn ∈ T , let k and l denote the column-vectors

(k1, . . . , kn)∗ and (l1, . . . , ln)∗, respectively (∗ denotes the transposition). Let

1.8. GROBNER BASES 97

Ak = (k′1, . . . , k

′n)∗ and Al = (l′1, . . . , l

′n)∗. Prove that the order <A on the set T

such that t <A t′ if and only if xk′1

1 . . . xk′

nn < x

l′11 . . . x

l′nn is also a monomial order

on T . (We say that <A is a matrix order defined by the given monomial order< and the matrix A ∈ GL(n,Q).)

Obviously, every polynomial f ∈ K[x1, . . . , xn] has a unique (up to the orderof the terms in the sum) representation as

f = a1t1 + · · · + artr (1.8.1)

where ti ∈ T , 0 = ai ∈ K (i = 1, . . . , r), and ti = tj whenever i = j. Such arepresentation of f will be called standard; the monomials ti, elements ai ∈ K,and the products aiti (1 ≤ i ≤ r) will be referred to as monomials, coefficients,and terms of f , respectfully (ai is said to be a coefficient of ti; it is denoted bycf (ti)).

If < is a total (in particular, monomial) order on T , we define the leadingmonomial of f , written lm<(f) (or lm(f) if the order is fixed), to be the greatestmonomial of f with respect to <. The coefficient of lm<(f) is called the leadingcoefficient of f ; it is denoted by lc<(f) (or lc(f)). The term lm<(f) is said tobe the leading term of f ; it is denoted by lt<(f) (or lt(f)). We say that a term

at divides bt′ (t, t′ ∈ T , 0 = a, b ∈ K) if t|t′. Then we writebt′

atfor a−1b

t′

t.

In what follows, we assume that a monomial order < on the set T is fixed.It is easy to see that for any two polynomials f, g ∈ K[x1, . . . , xn], one has

lm(fg) = lm(f)lm(g) and lm(f + g) ≤ maxlm(f), lm(g)with equality if and only if the leading terms of f and g do not cancel in thesum.

An ideal I in K[x1, . . . , xn] is called a monomial ideal if it is generatedby a set of monomials. Applying the Hilbert Basis Theorem one obtains thatevery monomial ideal in K[x1, . . . , xn] can be generated by a finite number ofmonomials (this statement is known as Dickson’s Lemma).

In what follows, if J is any ideal of K[x1, . . . , xn], then lt(J) will denote themonomial ideal generated by the set lt(f) | f ∈ J.Definition 1.8.4 Let K[x1, . . . , xn] be a ring of polynomials in n variablesx1, . . . , xn over a field K, and let a monomial order < on the set of all monomi-als of K[x1, . . . , xn] be fixed. Let J be an ideal of K[x1, . . . , xn]. A finite subsetG = g1, . . . , gr of J is called a Grobner basis (or a standard basis) of theideal J if (lt(g1), . . . , lt(gr)) = lt(J).

Equivalently, G = g1, . . . , gr is a Grobner basis of J if for any f ∈ J ,lm(f) is a multiple of some lm(gi) (1 ≤ i ≤ r). A finite set G ⊆ K[x1, . . . , xn]is called a Grobner basis if it is a Grobner basis of the ideal (G).

Proposition 1.8.5 With the above notation (and a fixed monomial order on theset of terms of K[x1, . . . , xn]), every nonzero ideal of the ring K[x1, . . . , xn] hasa Grobner basis. Furthermore, any Grobner basis of an ideal J ⊆ K[x1, . . . , xn]generates J .


Now we are going to introduce the concept of reduction of polynomials thatleads to another characterization of Grobner bases. In what follows, K is a field,K[X] = K[x1, . . . , xn] is a ring of polynomials in n variables x1, . . . , xn over K,and T is the set of all monomials of K[X].

Definition 1.8.6 Given f, g, h ∈ K[X] with g = 0, we say that f reducesto h modulo g in one step, written f

g−→ h, if and only if lm(g) divides anonzero monomial t of f , so that t = t′lm(g) for some t′ ∈ T , and h =f − cf (t)(lc(g))−1g.

Definition 1.8.7 Let f, h,f1, . . . , fm be polynomials in K[X], fi = 0 for i =1, . . . , m, and F = f1, . . . , fm. We say that f reduces to h modulo F , denotedf

F−→ h, if and only if there exists a sequence of indices i1, . . . , ip ∈ 1, . . . , mand a sequence of polynomials h1, . . . , hp ∈ K[X] such that

ffi1−−→ h1 . . .

fip−1−−−→ hp−1

fip−−→ h.

A polynomial h ∈ K[X] is said to be reduced with respect to a set of polyno-mials F = f1, . . . , fm ⊆ K[X] if h = 0 or no monomial of h is divisible by anylm(fi), 1 ≤ i ≤ m. In other words, h cannot be reduced modulo F . If f

F−→ hand h is reduced with respect to F , then h is said to be a remainder for f withrespect to F .

The reduction process based on Definition 1.8.6 leads to the following algo-rithm that produces a remainder for a polynomial f ∈ K[X] with respect toa set of polynomials F = f1, . . . , fm ⊆ K[X]. This algorithm produces quo-tients q1, . . . , qm ∈ K[X] and a remainder h ∈ K[X] such that f = q1f1 + · · ·+qmfm + h.

Algorithm 1.8.8 (f, f1, . . . , fm; q1, . . . , qm, h)Input: f ∈ K[X], F = f1, . . . , fm ⊆ k[X] with fi = 0 (i = 1, . . . , m)Output: q1, . . . , qm, h ∈ K[X] such that

f = q1f1 + · · · + qmfm + h, h is reduced with respect to F , andmaxlm(q1)lm(f1), . . . , lm(qm)lm(fm), lm(h) = lm(f)

Beginq1 := 0, . . . , qm := 0, g := f, h := 0While g = 0 DoIf there exists i, 1 ≤ i ≤ m, such that lm(fi) divides lm(g), Thenchoose the smallest i such that lm(fi) divides lm(g)

qi := qi + lc(g)lc(fi)−1 lm(g)lm(fi)

g := g − lc(g)lc(fi)−1 lm(g)lm(fi)

fi

Elseh := h + lt(g)g := g − lt(g)


Theorem 1.8.9 Let I be an ideal in a ring of polynomials K[X] = K[x1, . . . , xn]over a field K, and let G = g1, . . . , gr be a subset of I. Then the followingstatements are equivalent.

(i) G is a Grobner basis of I.

(ii) A polynomial f belongs to I if and only if fG−→ 0.

(iii) A polynomial f belongs to I if and only if f can be represented as f =∑ri=1 higi with lm(f) = maxlm(hi)lm(gi) | 1 ≤ i ≤ r.

Exercises 1.8.10 In what follows K[X] denotes the ring of polynomialsK[x1, . . . , xn] in the set of variables X = x1, . . . , xn over a field K.

1. Let G be a finite set of nonzero polynomials in K[X]. The reductionrelation G−→ is said to be confluent if for all f, f1, f2 ∈ K[X] such that f

G−→ f1

and fG−→ f2, there exists h ∈ K[X] such that f1

G−→ h and f2G−→ h. Prove that

G is a Grobner basis if and only if G−→ is confluent.2. Let I be an ideal of K[X] and G a Grobner basis of I. Let TG denote the

set of all monomials t ∈ T such that lm(g) does not divide t for every g ∈ G.Prove that t + I | t ∈ TG is a basis of the vector K-space K[X]/I.

3. Let G be a Grobner basis of an ideal I in K[X], let L be a field extension ofK, and let J be the ideal of the polynomial ring L[X] = L[x1, . . . , xn] generatedby I. Show that G is also a Grobner basis of J .

Let K[X] = K[x1, . . . , xn] be the polynomial ring in variables x1, . . . , xn

over a field K. Let f and g be two nonzero polynomials in K[X] and let Ldenote the least common multiple of lm(f) and lm(g). (Recall that the leastcommon multiple of two monomials t1 = xi1

1 . . . xinn and t2 = xj1

1 . . . xjnn , denoted

by lcm(t1, t2), is the monomial xmaxi1,j11 . . . x

maxin,jnn .) Then the polynomial

S(f, g) =L

lt(f)f − L

lt(g)g (1.8.2)

is called the S-polynomial of f and g.

The following theorem gives the theoretical foundation for computingGrobner bases.

Theorem 1.8.11 (Buchberger criterion) With the above notation, let G =g1, . . . , gr be set of nonzero polynomials in K[X]. Then G is a Grobner basisof the ideal I = (g1, . . . , gr) if and only if S(gi, gj)

G−→ 0 for all gi, gj ∈ G, i = j.

The corresponding algorithm of computation of Grobner bases is also dueto B. Buchberger.

Algorithm 1.8.12 (s, f1, . . . , fs; r, g1, . . . , gr)Input: F = f1, . . . , fs ⊆ K[x1, . . . , xn] with fi = 0 (1 ≤ i ≤ s)Output: G = g1, . . . , gr, a Grobner basis of the ideal (f1, . . . , fs)


BeginG := F , G = fi, fj | fi, fj ∈ G, fi = fj (1 ≤ i, j ≤ s)WhileG = ∅ DoChoose any f, f ′ ∈ G

G := G \ f, f ′S(f, f ′) G−→ h, where h is reduced with respect to GIf h = 0 ThenG := G

⋃(f, h) | f ∈ GG := G

⋃h

Let us show that this algorithm terminates and produces a Grobner basis ofthe ideal I = (f1, . . . , fs).

Suppose that the algorithm does not terminate. Then our process leads to astrictly increasing infinite sequence of subsets of K[x1, . . . , xn]

F = G1 G2 . . .

where Gi = Gi−1∪hi for some nonzero element hi ∈ I such that hi is reducedwith respect to Gi−1. It follows that lt(G1) lt(G2) . . . which contradictsthe Hilbert Basis Theorem (Theorem 1.2.11(vi)). Thus, the algorithm producesa finite set G = g1, . . . , gr in a finite number of steps. Since F ⊆ G ⊆ I,(G) = I. Furthermore, S(g, g′) G−→ 0 by construction. Applying Theorem 1.8.11we obtain that G is a Grobner basis of the ideal I.

The concept of a Grobner basis for an ideal of K[X] can be naturally ex-tended to a similar concept for a submodule of a finitely generated free K[X]-module. Let E be such a module with free generators e1, . . . , es. Then elementsof the form tei, where t is a monomial in K[X] and 1 ≤ i ≤ s, are called mono-mials. Elements of E of the form am, where a ∈ K and m is a monomial, arecalled terms. If m1 = t1ei and m2 = t2ej are two monomials in E, we say thatm1 divides m2 and write m1|m2 if there exists a monomial t in K[X] such thatm2 = tm1 (that is, i = j and t1 divides t2 in K[X]). In this case we writet =

m2

m1. If a, b ∈ K, b = 0, we say that the term am1 divides bm2 if m1|m2. In

this case we writebm2

am1for a−1b

m2

m1. The least common multiple lcm(m1,m2) of

the monomials m1 and m2 is defined as lcm(t1, t2)ei if i = j; if i = j, we definelcm(m1,m2) = 0.

A monomial order on E is a total order < on the set of all monomials of Esuch that for any two monomials m1,m2 ∈ E and for any monomial t ∈ K[X],t = 1, the inequality m1 < m2 implies tm1 < tm2.

Example 1.8.13 Let < be a monomial order on K[x]. Then one can obtaina monomial order on E (denoted by the same symbol <) as follows: for anymonomials m1 = t1ei, m2 = t2ej ∈ E, m1 < m2 if and only if either t1 < t2 ort1 = t2 and i < j.


In what follows we fix a monomial order < on E. Obviously, every nonzeroelement f ∈ E has a unique representation as

f = a1t1ei1 + · · · + aptpeip(1.8.3)

where t1, . . . , tp are monomials in K[X] such that t1ei1 > · · · > tpeip, a1, . . . , ap

are nonzero elements of K, and 1 ≤ i1, . . . , ip ≤ s. The monomial t1ei1 is calledthe leading monomial of f ; it is denoted by lm(f). The coefficient a1 is said tobe the leading coefficient of f and a1t1ei1 is said to be the leading term of f ;they are denoted by lc(f) and lt(f), respectively.

Definition 1.8.14 With the above notation, for any elements f, g, h ∈ E,

g = 0, we say that f reduces to h modulo g in one step, written fg−→ h, if lt(g)

divides a term u that appears in f and h = f − u

lt(g)g.

Definition 1.8.15 Let f, h ∈ E and F = f1, . . . , fk ⊆ E, fi = 0 for i =1, . . . , k. We say that f reduces to h modulo F , denoted f

F−→ h, if and onlyif there exists a sequence of indices i1, . . . , iq ∈ 1, . . . , k and a sequence of

elements h1, . . . , hq ∈ E such that ffi1−−→ h1 . . .

fiq−1−−−→ hp−1

fiq−−→ h.

An element f ∈ E is said to be reduced with respect to a set F = f1, . . . , fk ⊆E if f = 0 or no monomial of f is divisible by any lm(fi), 1 ≤ i ≤ k (so that f

cannot be reduced modulo F ). If fF−→ h and h is reduced with respect to F , then

h is said to be a remainder for f with respect to F . The reduction process basedon Definition 1.8.14 leads to an algorithm that produces a remainder for anelement f ∈ E with respect to a set F . This algorithm is similar to Algorithm1.8.8, it produces quotients q1, . . . , qk ∈ E and a remainder r ∈ E such thatf = q1f1 + · · · + qkfk + r.

Definition 1.8.16 With the above notation, let M be a K[X]-submodule of E.A set of nonzero elements G = g1, . . . , gr ⊆ M is called a Grobner basisof M if for any f ∈ M , there exists gi ∈ G such that lm(gi) divides lm(f). Aset G ⊆ E is said to be a Grobner basis if it is a Grobner basis of the K[X]-submodule (G) it generates.

Theorem 1.8.17 Let M be a K[X]-submodule of the free K[X]-module E andG = g1, . . . , gr ⊆ M , gi = 0 for i = 1, . . . , r. Then the following statementsare equivalent.

(i) G is a Grobner basis of any M .

(ii) f ∈ M if and only if fG−→ 0.

(iii) For any f ∈ M , there exist h1, . . . , hr ∈ K[X] such that f =∑r

i=1 higi

and lm(f) = maxlm(higi) | 1 ≤ i ≤ r.(iv) For any f ∈ E, if f

G−→ r1, fG−→ r2, and r1, r2 are reduced with respect to

G, then r1 = r2.


The Buchberger criterion for Grobner bases of submodules of E can beformulated similarly to Theorem 1.8.11. In this case, by the S-polynomial (alsocalled an S-element) of two nonzero elements f, g ∈ E we mean the element

S(f, g) =L

lt(f)f − L

lt(g)g

where L = lcm(lm(f), lm(g)).

Theorem 1.8.18 (Buchberger criterion for modules) Let G = g1, . . . , gr bea set of nonzero elements in E. Then G is a Grobner basis of a submoduleM = (G) of E if and only if S(gi, gj)

G−→ 0 for all gi, gj ∈ G, i = j.

Based on this theorem one can formulate an algorithm of computation ofGrobner bases of submodules of E. We leave this formulation, as well as theproof of the termination of this algorithm, to the reader as an exercise.

Chapter 2

Basic Conceptsof Difference Algebra

2.1 Difference and Inversive Difference Rings

In what follows we keep the basic notation and conventions of Chapter 1. Inparticular, by a ring we always mean an associative ring with unity, every ringhomomorphism is unitary (maps unity onto unity), every subring of a ring con-tains the unity of the ring. Unless otherwise indicated, by the module over aring A we mean a left A-module. Every module over a ring is unitary and everyalgebra over a commutative ring is also unitary.

Definition 2.1.1 A difference ring is a commutative ring R together with afinite set σ = α1, . . . , αn of pairwise commuting injective endomorphisms ofthe ring R into itself. The set σ is called a basic set of the difference ring R,and the endomorphisms α1, . . . , αn are called translations.

Thus, a difference ring R with a basic set σ = α1, . . . , αn is a commutativering possessing n additional unitary operations αi : a → αi(a) (a ∈ R, 1 ≤ i ≤ n)such that αi(a) = 0 if and only if a = 0, αi(a + b) = αi(a) + αi(b), αi(ab) =αi(a)αi(b), αi(1) = 1, and αi(αj(a)) = αj(αi(a)) for any a ∈ R, 1 ≤ i, j ≤ n.(Formally speaking, a difference ring is an (n+1)-tuple (R,α1, . . . , αn) where Ris a ring, and α1, . . . , αn are mutually commuting injective endomorphisms of R.Then R is said to be an underlying ring of our difference ring. However, unless thenotation is inconvenient or ambiguous, we always write R for (R,α1, . . . , αn).)

If α1, . . . , αn are automorphisms of R, we say that R is an inversive differencering with the basic set σ.

In what follows, a difference ring R with a basic set σ = α1, . . . , αn will bealso called a σ-ring. If α1, . . . , αn are automorphisms of R , then σ∗ will denotethe set α1, . . . , αn, α−1

1 , . . . , α−1n . An inversive difference ring with a basic set

σ will be also called a σ∗-ring.

103

104 CHAPTER 2. BASIC CONCEPTS OF DIFFERENCE ALGEBRA

If the basic set σ of a σ-ring R consists of a single element, R is said to bean ordinary difference ring. If Cardσ > 1, we say that R is a partial differencering with the basic set σ or a partial σ-ring.

If a difference ring R with a basic set σ is a field, it is called a difference (orσ-) field. If R is inversive, it is called an inversive difference field or a σ∗-field.(As in the case of difference rings, from the formal point of view, a differencefield is a pair consisting of a field K and a set of its mutually commuting injective(that is, nonzero) endomorphisms; then the field K is said to be the underlyingfield of our difference field.)

Let R be a difference ring with a basic set σ = α1, . . . , αn and R0 asubring of the ring R such that α(R0) ⊆ R0 for any α ∈ σ. Then R0 is called adifference (or σ-) subring of R and R is said to be a difference (or σ-) overringof R0 or a difference (or σ-) ring extension of R0. In this case we use the samesymbols αi (1 ≤ i ≤ n) for the endomorphisms of the basic set of R and theirrestrictions on R0. If the σ-ring R is inversive and R0 a σ-subring of R suchthat α−1(R0) ⊆ R0 for any α ∈ σ, then R0 is said to be a σ∗-subring of R, andR is called a σ∗-overring of R0 or a σ∗-ring extension of R0. If R is a difference(σ-) field and R0 a subfield of R such that α(a) ∈ R0 for any a ∈ R0, α ∈ σ,then R0 is said to be a difference (or σ-) subfield of R; R, in turn, is called adifference (or σ-) field extension or a difference (or σ-) overfield of R0. In thiscase we also say that we have a σ-field extension R/R0. If R is inversive and itssubfield R0 is a σ∗-subring of R, then R0 is said to be an inversive difference (orσ∗-) subfield of R while R is called an inversive difference (or σ∗-) field extensionor an inversive difference (or σ∗-) overfield of R0. (We also say that we have aσ∗-field extension R/R0.) If R0 ⊆ R1 ⊆ R is a chain of σ- (σ∗-) field extensions,we say that R1/R0 is a difference or σ- (respectively, inversive difference or σ∗-)field subextension of R/R0 or an intermediate difference (σ-) or, respectively,inversive difference (σ∗-) field of R/R0.

If R is a difference ring with a basic set σ and J is an ideal of the ring Rsuch that α(J) ⊆ J for any α ∈ σ, then J is called a difference (or σ-) ideal ofR. If a prime (maximal) ideal P of a σ-ring R is closed with respect to σ (thatis α(P ) ⊆ P for any α ∈ σ), it is called a prime difference ideal or a primeσ-ideal (respectively, a maximal difference ideal or a maximal σ-ideal) of R.

A difference (σ-) ideal J of a σ-ring R is said to be reflexive if for anytranslation α, the inclusion α(a) ∈ J (a ∈ R) implies a ∈ J . Clearly, if R isan inversive σ-ring, then a difference ideal J of R is reflexive if and only if it isclosed under all inverse automorphisms α−1, where α ∈ σ. In this case we alsosay that J is a σ∗-ideal of the σ∗-ring R. A prime (maximal) reflexive σ-idealof a σ-ring R is also called a prime (respectively, maximal) σ∗-ideal of R.

Examples 2.1.2 1. Any commutative ring can be considered as both differ-ence and inversive difference ring with a basic set σ consisting of one or severalidentity automorphisms.

2. Let z0 be a complex number and let U be a region of the complex planesuch that z + z0 ∈ U whenever z ∈ U (e. g., U = z ∈ C|(Re z)(Re z0) ≥ 0).Furthermore, let MU denote the field of all functions of one complex variable

2.1. DIFFERENCE AND INVERSIVE DIFFERENCE RINGS 105

meromorphic in U . Then MU can be treated as an ordinary difference fieldwhose basic set consists of one translation α such that α(f(z)) = f(z + z0) forany function f(z) ∈ MU . This difference field is denoted by MU (z0). It is clearthat MU (z0) is an inversive difference field if and only if z − z0 ∈ U for anyz ∈ U . In this case α−1 : f(z) → f(z − z0) for any f(z) ∈ MU .

3. Let z0 be a nonzero complex number and let V be a region of the complexplane such that zz0 ∈ U whenever z ∈ V (e.g., |z0| ≤ 1 and V = z ∈ C | |z| ≤ rfor some positive real number r). Then the field of all functions of one complexvariable meromorphic in the region V can be considered as an ordinary differencefield with one translation β such that β(f(z)) = f(z0z) for any function f(z) ∈MV . This difference field is denoted by M∗

V (z0). It is easy to see that if zz0

∈ Vfor any z ∈ V , then M∗

V (z0) can be treated as an inversive difference field(β−1 : f(z) → f( z

z0) for any f(z) ∈ MV ).

4. Let A be a ring of functions of n real variables continuous on then-dimensional real space Rn. Let us fix some real numbers h1, . . . , hn andconsider a set of mutually commuting injective endomorphisms α1, . . . , αn ofthe ring A such that (αif)(x1, . . . , xn) = f(x1, . . . , xi−1, xi + hi, xi+1, . . . , xn)(i = 1, . . . , n). Then A can be treated as a difference ring with the basic setσ = α1, . . . , αn. This ring is denoted by A0(h1, . . . , hn).

Similarly, one can introduce the difference structure on the ring Cp(Rn) ofall functions of n real variables that are continuous on Rn together with alltheir partial derivatives up to the order p (p ∈ N or p = +∞). It is easy tosee that Cp(Rn) can be considered as a difference ring with the basic set σ =α1, . . . , αn described above. This difference ring is denoted by Ap(h1, . . . , hn).Clearly, Ap(h1, . . . , hn) is a σ-subring of a σ-ring Aq(h1, . . . , hn) whenever p > q.

Difference rings Ap(h1, . . . , hn) often arise in connection with equationsin finite differences when the ith partial finite difference ∆if(x1, . . . , xn) =f(x1, . . . , xi−1, xi + hi, xi+1, . . . , xn)− f(x1, . . . , xn) of a function f(x1, . . . , xn)∈ Cp(Rn) is written as ∆if = (αi − 1)f (1 ≤ i ≤ n).

5. Let K be an ordinary difference field with one translation α and letR = K[x] be a polynomial ring in one indeterminate x over K. Then for anypolynomial f(x) ∈ R \K, the endomorphism α can be extended to an injectiveendomorphism α of the ring R such that α(x) = f(x). Thus, any polynomialf(x) ∈ R \ K induces a structure of an ordinary difference ring on R such thatK becomes a difference subring of R.

6. Let R = K[x1, x2, . . . ] be a polynomial ring in a denumerable set of in-determinates X = x1, x2, . . . over a field K. Then any injective mapping βof the set X into itself induces an injective endomorphism β of the ring R suchthat β(a) = a for any a ∈ K and β(xi) = β(xi) for i = 1, 2, . . . . More gen-eral, if K is a difference field with a basic set σ = α1, . . . , αn and β1, . . . , βn

are n mutually commuting injective mappings of the set X into itself, thenthe ring R = K[x1, x2, . . . ] can be considered as a difference ring with a basicσ = α1, . . . , αn where αi(a) = αi(a) for all a ∈ R and αi(xj) = βi(xj) forj = 1, 2, . . . (1 ≤ i ≤ n). In this case K becomes a σ-subring of the σ-ring R.


Let R be a difference (in particular, an inversive difference) ring with a basicset σ. An element c ∈ R is said to be a constant if α(a) = a for any α ∈ σ. It iseasy to see that the set of all constants of the ring R is a σ-subring of R (or aσ∗-subring of R, if the difference ring R is inversive). This subring is called thering of constants of R, it is denoted by CR.

Exercises 2.1.3 1. Find the rings of constants of the difference rings inExamples 2.1.2.

2. Let K be a field and K[x] a polynomial ring in one indeterminate x overK. Show that not every automorphism of the ring K[x] can be extended to anautomorphism of the ring of formal power series K[[x]].

3. Let K be a field and let R = K(x1, . . . , xs) be the field of rational fractionsin s indeterminates x1, . . . , xs over K.

Prove that if rational fractions g1, . . . , gs ∈ R are algebraically independentover K, then there exists a unique injective endomorphism α of the field R suchthat α(xi) = gi for i = 1, . . . , s and α(a) = a for any a ∈ K. Furthermore, showthat if g1, . . . , gs are homogeneous linear rational fractions in x1, . . . , xs, then αis an automorphism of the field R.

4. Let an integral domain R be an ordinary σ∗-ring with a basic set σ = α.Prove that the polynomial ring R[x] in one indeterminate x is a σ∗-overring ofR if and only if the extension of the automorphism α to the ring R[x] is givenby α(x) = ax + b where a, b ∈ R and a is a unit of R.

5. Let K((x)) denote the field of all formal (convergent) Laurent series in onevariable x over a field K. Show that K((x)) can be considered as an ordinarydifference field with a translation α defined by α(x) = ax where a is an arbitraryelement of the field K.

6. Show that the nilradical and Jacobson radical of any inversive differencering are inversive difference ideals.

7. Let R = K[x] be a polynomial ring in one indeterminate x over a fieldK of zero characteristic. Then R can be considered as a difference field with atranslation α such that (αf)(x) = f(x + 1) for any polynomial f(x) ∈ R. Showthat this difference ring does not contain proper difference ideals.

8. Let S = K[x] be a polynomial ring in one indeterminate x over a field Kof zero characteristic considered as an ordinary difference ring with a basic setσ = β such that (βf)(x) = f(ax) for some nonzero element a ∈ K. Provethat (x) is the only prime difference ideal of the σ-ring S.

If R is a difference ring with a basic set σ = α1, . . . , αn, then Tσ (orT , if it is clear what basic set is considered) will denote the free commutativesemigroup with identity generated by α1, . . . , αn. Elements of Tσ will be writtenin the multiplicative form αk1

1 . . . αknn (k1, . . . , kn ∈ N) and naturally treated as

endomorphisms of the ring R. If the σ-ring R is inversive, then Γσ (or Γ, if we abasic set in our considerations is fixed) will denote the free commutative groupgenerated by σ. It is clear that elements of Γσ (written in the multiplicativeform αi1

1 . . . αinn where i1, . . . , in ∈ Z) act on the ring R as automorphisms and

Tσ is a subsemigroup of Γσ.


For any a ∈ R and for any τ ∈ Tσ, the element τ(a) is said to be a transformof a. If the σ-ring R is inversive, then an element γ(a) a ∈ R, γ ∈ Γσ) is alsocalled a transform of a.

Let R be a difference ring with a basic set σ and S ⊆ R. Then the intersectionof all σ-ideals of R containing S is denoted by [S]. Clearly, [S] is the smallestσ-ideal of R containing S; as an ideal, it is generated by the set TσS = τ(a)|τ ∈Tσ, a ∈ S. If J = [S], we say that the σ-ideal J is generated by the set S whichis called a set of difference (or σ-) generators of J . If S is finite, S = a1, . . . , ak,we write J = [a1, . . . , ak] and say that J is a finitely generated difference (or σ-)ideal of the σ-ring R. (In this case elements a1, . . . , ak are said to be difference(or σ-) generators of J .)

It is easy to see that if J is a σ-ideal of a difference (σ-) ring R, thenJ∗ = a ∈ R | τ(a) ∈ J for some τ ∈ Tσ is a reflexive σ-ideal of R, which iscontained in any reflexive σ-ideal of R containing J . The ideal J∗ is called areflexive closure of J . Clearly, if S ⊆ R, then the smallest inversive σ-ideal ofR containing S is the reflexive closure [S]∗ of [S]. If the σ-ring R is inversive,then [S]∗ is generated by the set ΓσS = γ(a)|γ ∈ Γσ, a ∈ S. If S is finite,S = a1, . . . , ak, we write [a1, . . . , ak]∗ for I = [S]∗ and say that I is a finitelygenerated σ∗-ideal of R. (In this case, elements a1, . . . , ak are said to be σ∗-generators of I.)

A difference (σ-) ring R is called simple if the only σ-ideals of R are (0) andR. It is easy to see that the ring of constants of a simple difference ring is afield.

Let R be a difference ring with a basic set σ, R0 a σ-subring of R andB ⊆ R. The intersection of all σ-subrings of R containing R0 and B is calledthe σ-subring of R generated by the set B over R0, it is denoted by R0B.(As a ring, R0B coincides with the ring R0[τ(b)|b ∈ B, τ ∈ Tσ] obtained byadjoining the set τ(b)|b ∈ B, τ ∈ Tσ to the ring R0.) The set B is said to bethe set of difference (or σ-) generators of the σ-ring R0B over R0. If this set isfinite, B = b1, . . . , bk, we say that R′ = R0B is a finitely generated difference(or σ-) ring extension (or overring) of R0 and write R′ = R0b1, . . . , bk. If Ris a σ-field, R0 a σ-subfield of R, and B ⊆ R, then the intersection of allσ-subfields of R containing R0 and B is denoted by R0〈B〉 (or R0〈b1, . . . , bk〉 ifB = b1, . . . , bk is a finite set). This is the smallest σ-subfield of R containingR0 and B; it coincides with the field R0(τ(b)|b ∈ B, τ ∈ Tσ). The set B iscalled the set of difference (or σ-) generators of the σ-field R0〈B〉 over R0. IfCardB < ∞, we say that R′ = R0〈B〉 is a finitely generated difference (or σ-)field extension or a finitely generated difference (or σ-) overfield of R0. We alsosay that R′/R0 is a finitely generated difference (σ-) field extension.

Let R be an inversive difference ring with a basic set σ, R0 a σ∗-subringof R and B ⊆ R. Then the intersection of all σ∗-subrings of R containing R0

and B is the smallest σ∗-subring of R containing R0 and B. This ring coincideswith the ring R0[γ(b)|b ∈ B, γ ∈ Γσ]; it is denoted by R0B∗. The set Bis said to be a set of inversive difference (or σ∗-) generators of R0B∗ overR0. If B = b1, . . . , bk is a finite set, we say that S = R0B∗ is a finitely


generated inversive difference (or σ∗-) ring extension (or overring) of R andwrite S = R0b1, . . . , bk∗.

Finally, if R is a σ∗-field, R0 a σ∗-subfield of R and B ⊆ R, then theintersection of all σ∗-subfields of R containing R0 and B is denoted by R0〈B〉∗.This is the smallest σ∗-subfield of R containing R0 and B; it coincides withthe field R0(γ(b)|b ∈ B, γ ∈ Γσ). The set B is called the set of inversivedifference (or σ∗-) generators of the σ∗-field extension R0〈B〉∗ over R0. If Bis finite, B = b1, . . . , bk, we write R0〈b1, . . . , bk〉∗ for R0〈B〉∗ and say thatR′ = R0〈B〉∗ is a finitely generated inversive difference (or σ∗-) field extension(or overfield) of R0. We also say that the σ∗-field extension R′/R0 is finitelygenerated.

In what follows we shall often consider two or more difference ringsR1, . . . , Rp with the same basic set σ = α1, . . . , αn. Formally speaking,it means that for every i = 1, . . . , p, there is some fixed mapping νi from the setσ into the set of all injective endomorphisms of the ring Ri such that any twoendomorphisms νi(αj) and νi(αk) of Ri commute (1 ≤ i ≤ p, 1 ≤ j, k ≤ n). Weshall identify elements αj (1 ≤ j ≤ n) with their images νi(αj) and say thatelements of the set σ act as mutually commuting injective endomorphisms ofthe ring Ri (i = 1, . . . , p).

Definition 2.1.4 Let R1 and R2 be two difference rings with the same basicset σ = α1, . . . , αn. A ring homomorphism φ : R1 → R2 is called a difference(or σ-) homomorphism if φ(α(a)) = α(φ(a)) for any α ∈ σ, a ∈ R1.

Clearly, if φ : R1 → R2 is a σ-homomorphism of inversive difference rings,then φ(α−1(a)) = α−1(φ(a)) for any α ∈ σ, a ∈ R1.

If a σ-homomorphism is an isomorphism (endomorphism, automorphism,etc), it is called a difference (or σ-) isomorphism (respectively, difference (or σ-)endomorphism, difference (or σ-) automorphism, etc.) In accordance with thestandard terminology, a difference isomorphism of a difference (σ-) ring R ontoa difference subring of a σ-ring S is called a difference (or σ-) isomorphism ofR into S.

If R1 and R2 are two σ-overrings of the same difference ring R0 with abasic set σ, then a σ-homomorphism φ : R1 → R2 is said to be a difference(or σ-) homomorphism over R0 or a σ-R0-homomorphism, if φ(a) = a for anya ∈ R0. In this case we also say that φ is a difference R0-homomorphism or aσ-R0-homomorphism from R1 to R2.

Example 2.1.5 Let A be the set of all sequences a = (a1, a2, . . . ) of elementsof an algebraically closed field C. Consider an equivalence relation on A suchthat a = (a1, a2, . . . ) is equivalent to b = (b1, b2, . . . ) if and only if an = bn forall sufficiently large n ∈ N (that is, there exists n0 ∈ N such that an = bn forall n > n0). Clearly, the corresponding set S of equivalence classes is a ring withrespect to coordinatewise addition and multiplication of class representatives.This ring can be treated as an ordinary difference ring with respect to themapping α sending an equivalence class with a representative (a1, a2, a3, . . . ) to


the equivalence class with the representative (a2, a3, . . . ). It is easy to see thatthis mapping is well-defined and it is an automorphism of the ring S. The fieldC can be naturally identified with the ring of constants of the difference ring S.

Let C(z) be the field of rational functions in one complex variable z. ThenC(z) can be considered as an ordinary difference field with respect to the au-tomorphism β such that β(z) = z + 1 and β(a) = a for any a ∈ C. It is easyto see that if C = C in the above construction of the ring S, then the mappingφ : C(z) → S that sends a function f(z) to the equivalence class of the element(f(0), f(1), . . . ) is an injective difference ring homomorphism.

Definition 2.1.6 Let R be a difference ring with a basic set σ = α1, . . . , αn.A σ-overring U of R is called an inversive closure of R, if elements of the setσ act as pairwise commuting automorphisms of the ring U (they are denoted bythe same symbols α1, . . . , αn) and for any a ∈ U , there exists an automorphismτ ∈ Tσ of the ring U such that τ(a) ∈ R.

Proposition 2.1.7 (i) Every difference ring has an inversive closure.(ii) If U1 and U2 are two inversive closures of a σ-ring R, then there exists a

difference R-isomorphism of U1 onto U2.(iii) Let R be a difference ring with a basic set σ and U an inversive difference

ring containing R as a σ-subring. Then U contains an inversive closure of R.(iv) If a difference ring R is an integral domain (a field), then its inversive

closure is also an integral domain (respectively, a field).(v) Let φ be a difference homomorphism of a difference ring R1 with a basic set

σ onto a difference (σ-) ring R2. If R∗1 and R∗

2 are inversive closures of R1 andR2, respectively, then there is a unique extension of φ to a σ-homomorphism φ∗

of R∗1 onto R∗

2. Furthermore, if R∗1 and R∗

2 are contained in a common σ∗-ringand ψ∗ is a σ-homomorphism of R∗

1 onto R∗2 is nontrivial (that is, ψ∗(a) = a for

at least one element a ∈ R∗1), then the restriction of ψ∗ to R1 is also nontrivial.

PROOF. (i) First of all, let us construct an inversive closure of an ordinarydifference ring R with a basic set σ = α. Let R′ = α(R) and let R∗ be aring that is isomorphic to R and such that R

⋂R∗ = ∅. If β : R → R∗ is the

corresponding ring isomorphism, we set (R′)∗ = β(R′), so that βα(R) = (R′)∗.Now, replacing the elements of (R′)∗ by the corresponding elements of R, wetransfer R∗ into an overring R1 of the ring R that will be also denoted byRα. If ρ : R∗ → Rα denotes this replacement, then the mapping (ρβ)−1 isan isomorphism of Rα onto its subring R that extends α. We shall denotethis isomorphism by the same letter α and treat Rα as a σ-overring of theσ-ring R.

Proceeding by induction we can construct an increasing chain of σ-rings R0 =R ⊆ R1 ⊆ R2 ⊆ . . . such that Rn = Rα

n−1 for n = 1, 2, . . . . Let us set R =⋃n∈N Rn and define an injective homomorphism R → R that extends α (it is

denoted by the same letter) as follows. If a ∈ R, then a belongs to some σ-ringRn (n ∈ N). The appropriate element α(a) ∈ Rn is defined as an image of a


under the mapping α : R → R. Since for any k = 0, 1, . . . , Rk+1 is a σ-overringof Rk, the image α(a) does not depend on the choice of the ring Rn containinga, so that our extension of α is well defined. It is easy to see that the σ-ring Ris an inversive closure of R.

Now suppose that R is a difference ring with a basic set σ = α1, . . . , αnwhere n > 1. Then R can be treated as a difference ring with the basic setσ1 = α1 and we can use the above procedure to construct an inversive closureR1 of this σ1-ring. The ring R1 can be considered as a σ-overring of the σ-ring R where the actions of the elements α2, . . . , αn are defined as follows: ifa ∈ R1 and r is the smallest nonnegative integer such that αr

1(a) ∈ R, thenαi(a) = α−r

1 αiαr1(a) for i = 2, . . . , n. Note that if a ∈ R1 and αs

1(a) ∈ R forsome s > r, then α−s

1 αiαs1(a) = α−s

1 αiαs−r1 αr

1(a) = α−r1 αiα

r1(a). (Using this fact

one can easily check that α2, . . . , αn are mutually commuting endomorphismsof R1.)

Considering R1 as a difference ring with the basic set σ2 = α2 we can repeatthe previous procedure to construct an inversive closure R2 of this σ2-ring andextend α1, . . . , αn to injective endomorphisms of the ring R2. Continuing thisprocess we arrive (after n steps) at an inversive closure U of the σ-ring R. (Itis clear that U is an inversive σ-overring of R and for any a ∈ U , there existsτ ∈ Tσ such that τ(a) ∈ R.)

(ii) Let U1 and U2 be two inversive closures of a σ-ring R. Let us define amapping φ : U1 → U2 as follows: if a ∈ U1 and τ is an element of the semigroupTσ such that b = τ(a) ∈ R, then φ(a) = τ−1(b). (In the last equality we meanthat τ−1 is the inverse of the element τ treated as an automorphism of the ringU2.) It is easy to see that the mapping φ is well-defined (the value φ(a) doesnot depend on the choice of an element τ ∈ Tσ such that τ(a) ∈ Tσ) and φ is aσ∗-isomorphism such that φ(a) = a for any a ∈ R.(iii) Let U be an inversive σ-ring containing R as its σ-subring, and let S =

a ∈ U |τ(a) ∈ R for some element τ ∈ Tσ. It is easy to see that S is a σ-subringof U and an inversive closure of the σ-ring R.(iv) If U is an inversive closure of a σ-ring R, then for any nonzero element

a ∈ U there exists an element τ ∈ Tσ such that τ(a) ∈ R. Clearly, if τ(a)is not a zero divisor in R, a is not a zero divisor in U . Also, if the elementτ(a) is invertible, then a is invertible too, a−1 = τ−1(τ(a)−1). Thus, if R is anintegral domain (a field), then its inversive closure U is also an integral domain(respectively, a field).

(v) Theextensionoftheσ-homomorphismφtoaσ-homomorphismφ∗ : R∗1 → R∗

2

is defined in a natural way: if a ∈ R∗1, then there exists τ ∈ Tσ such that τ(a) ∈ R1.

Then φ∗(a) = τ−1(φ(τ(a))). It is easy to check that this definition does notdepend on the choice of en element τ ∈ Tσ such that τ(a) ∈ R1 and the restrictionof φ∗ on R1 coincides with φ. To prove the second part of statement (v), let ψdenote the restriction of ψ∗ to R1 and suppose that ψ(a) = a for any a ∈ R1.Then for any u ∈ R∗

1, there is τ ∈ Tσ such that τ(u) ∈ R1. Then we would haveψ∗(u) = ψ∗(τ−1(τ(u))) = τ−1ψ∗(τ(u)) = τ−1ψ(τ(u)) = τ−1(τ(u)) = u thatcontradicts the non-triviality of ψ∗.


If H is an inversive difference field with a basic set σ and G a σ-subfieldof H, then the set a ∈ H|τ(a) ∈ G for some τ ∈ Tσ is a σ∗-subfield of Hdenoted by G∗

H (or G∗ if one considers subfields of a fixed σ∗-field H). This fieldis said to be the inversive closure of G in H. Clearly, G∗

H is the intersection ofall σ∗-subfields of H containing G.

Proposition 2.1.8 Let H be a σ∗-field and let ∗ be the operation that assigns toeach σ-subfield F ⊆ H its inversive closure in H. Let F and G be two σ-subfieldsof H and 〈F,G〉 denote the σ-field F 〈G〉 = G〈F 〉 (the “σ-compositum” of F andG). Then

(i) F ∗∗ = F ∗.(ii) 〈F,G〉∗ = 〈F ∗, G∗〉.(iii) Every σ-isomorphism of F onto G has a unique extension to a σ-

isomorphism of F ∗ onto G∗.(iv) If K is a σ-subfield of F and the σ-subfield G of H is such that F and G

are algebraically disjoint (linearly disjoint, quasi-linearly disjoint) over K, thenF ∗ and G∗ are algebraically disjoint (linearly disjoint, quasi-linearly disjoint)over K∗.

(v) Let F ⊆ F0 ⊆ G where F0 is the algebraic part (the purely inseparable part,the separable part) of G over F . Then F ∗

0 is the algebraic part (respectively, thepurely inseparable part or the separable part) of G∗ over F ∗.

PROOF. The first statement is obvious. If a ∈ 〈F,G〉∗, then there exists τ ∈Tσ such that τ(a) =

φ(g1, . . . , gk)ψ(g1, . . . , gk)

where g1, . . . , gk ∈ G and φ and ψ are

polynomials in k variables with coefficients in F . Applying τ−1 to the bothsides of the last equality we obtain that a ∈ 〈F ∗, G∗〉. The opposite inclusion〈F ∗, G∗〉 ⊆ 〈F,G〉∗ can be checked in a similar way.

Let ρ : F → G be a σ-isomorphism of F onto G. For every a ∈ F ∗, thereexists τ ∈ Tσ such that τ(a) ∈ F . It is easy to check that the mapping ρ∗ thatsends a to τ−1ρτ(a) is well defined (i. e., it does not depend on the choice ofτ ∈ Tσ such that τ(a) ∈ F ) and it extends ρ to a σ-isomorphism of F ∗ onto G∗.

Suppose that F and G are algebraically disjoint over their common σ-subfieldK. Let x1, . . . , xr and y1, . . . , ys be finite subsets of F ∗ and G∗, respectively, suchthat the elements of each set are algebraically independent over K∗. Then thereexists τ ∈ Tσ such that τ(xi) ∈ F , τ(yj) ∈ G (1 ≤ i ≤ r, 1 ≤ j ≤ s). Clearly,the elements τ(x1), . . . , τ(xr) are algebraically independent over K (otherwisethe set x1, . . . , xr would not be algebraically independent over K∗) and soare elements τ(y1), . . . , τ(ys. Since F and G are algebraically disjoint over K,the set τ(x1), . . . , τ(xr), τ(y1), . . . , τ(ys) is algebraically independent over Kwhence elements x1, . . . , xr, y1, . . . , ys are algebraically independent over K∗.The other parts of statement (iv) can be proved in the same way.

We leave the proof of statement (v) to the reader as an exercise.

Proposition 2.1.9 Let F be a difference field with a basic set σ = α1, . . . , αn,G a σ-overfield of F and G∗ an inversive closure of G.


(i) If G is an algebraic closure (perfect closure, separable algebraic closure)of F , then G∗ is an algebraic closure (respectively, perfect closure or separablealgebraic closure) of F ∗.

(ii) With conditions of (i), if F is inversive, then G is also inversive.(iii) If G/F is a primary (regular) field extension, then G∗/F ∗ is also a pri-

mary (respectively, regular) field extension.

PROOF. Let β = α1 · α2 · · · · · αn, and let Fβ , Gβ and G∗β denote the fields

F , G and G∗, respectively, treated as ordinary difference fields with the basicset σ′ = β. Then one can easily see that G∗

β is the inversive closure of Gβ .Let H denote the algebraic closure of the field G∗. Then the automorphism

β of G∗ can be extended to an automorphism of H, so that H becomes aσ′-overfield of G∗

β . (Indeed, by Theorem 1.6.6 (v) (with K = L = G∗ andM = K = H), there are extensions of β and β−1 to endomorphisms β1 andβ2 of the field H, respectively. Then β1β2 and β2β1 are G∗-endomorphisms ofH each of which is, according to Theorem 1.6.7(ii), an automorphism of H. Itfollows that β1 is an automorphism of H extending β.) Now, for each listed in(i) assumption on G/F , Gβ is respectively the algebraic part, purely inseparablepart, or separable part of H over Fβ . Applying parts (i) and (v) of Proposition2.1.8 we obtain that G∗

β is respectively the algebraic part, purely inseparablepart, or separable part of H over the inversive closure F ∗

β of Fβ in G∗β . Since H

is an algebraic closure of G∗β the field G∗

β is respectively an algebraic closure,perfect closure, or separable algebraic closure of F ∗

β , that is, of F ∗.Statement (ii) follows from the fact that an overfield of a field K can contain

at most one algebraic closure (perfect closure, separable algebraic closure) of K.Let us prove (iii). Suppose that the field extension G/F is primary, and let

F0 denote the purely inseparable part of G over F . Then F0 is also the algebraicpart of G over F . Applying Proposition 2.1.8 (v) we obtain that F ∗

0 is both thepurely inseparable part of G∗ over F ∗ and the algebraic part of G∗ over F ∗.Thus, G∗/F ∗ is primary.

Now suppose G/F is regular. With the notation of the proof of (i), let H1

and H2 denote the algebraic part of H over Fβ and the algebraic part of Hover F ∗

β , respectively. Because H is inversive, statement (v) of Proposition 2.1.8implies that G∗

β and H2 are linearly disjoint over F ∗β . Hence G∗

β/F ∗β is a regular

extension, and thus G∗/F ∗ is a regular field extension.

Let R be a difference ring with a basic set σ. A subset S of the ring R is saidto be a σ-subset (or an invariant subset) of R if α(s) ∈ S for any s ∈ S, α ∈ σ. Ifthe σ-ring R is inversive and S is a σ-subset of R such that α−1(s) ∈ S for anys ∈ S, α ∈ σ, then S is said to be a σ∗-subset of the ring R. By a multiplicativeσ-subset of a σ-ring R we mean a σ-subset S of R such that 1 ∈ S, 0 /∈ S,and st ∈ S whenever s ∈ R and t ∈ R. (Thus, a multiplicative σ-subset of aσ-ring R is a multiplicative subset of R closed with respect to the actions of thetranslations.) Similarly, a multiplicative σ∗-subset of an inversive σ-ring R is amultiplicative σ-subset of R such that α−1(s) ∈ S for any s ∈ S, α ∈ σ.


Proposition 2.1.10 Let S be a multiplicative σ-subset of a σ-ring R and letS−1R be the ring of fractions of R with denominators in S. Then S−1R hasa unique structure of a σ-ring such that the natural injection ν : R → S−1R

( a → a

1) becomes a σ-homomorphism. If the σ-ring R is inversive, and S is a

multiplicative σ∗-subset of R, then S−1R is an inversive σ-overring of R.

PROOF. For any elementa

s∈ S−1R (a ∈ R, s ∈ S) and for any α ∈ σ,

let α(a

s

)=

α(a)α(s)

. Ifa

s=

b

t(a, b ∈ R; s, t ∈ S), then u(at − bs) = 0 for some

u ∈ S whence α(u)(α(a)α(t) − α(b)α(s)) = 0. Since α(u) ∈ S,α(a)α(s)

=α(b)α(t)

, so

the actions of the elements of σ on the ring S−1R are well defined. It is easy tosee that elements of σ act as injective endomorphisms of S−1R and the naturalinjection ν : R → S−1R is a σ-homomorphism. We leave the proof of the lastpart of the statement to the reader as an exercise.

If S is a multiplicative σ-subset of a σ-ring R, then the ring S−1R is said tobe a σ-ring of fractions of R with denominators in S. If the σ-ring R is inversiveand S is a multiplicative σ∗-subset of R, then S−1R is called the σ∗-ring offractions of R with denominators in S.

It is easy to see that if P is a difference prime ideal of a difference ring Rwith a basic set σ, then S = R \ P is a multiplicative σ-subset of R if and onlyif P is reflexive. If this is the case, then RP is a local difference (σ−) ring.

The following statement describes a universal property of a σ-ring of fractions.

Proposition 2.1.11 Let R and R′ be difference rings with the same basic set σ,S a multiplicative σ-subset of R, and φ : R → R′ a ring σ-homomorphism suchthat φ(s) is a unit of R′ for any s ∈ S. Then φ factors uniquely through the em-bedding ν : R → S−1R: there exists a unique σ-homomorphism ψ : S−1R → R′

such that ψ ν = φ.

PROOF. Let us set ψ(a

s

)= φ(a)φ(s)−1 for any element

a

s∈ S−1R

(a ∈ R, s ∈ S). Then the mapping ψ : S−1R → R′ is well defined. Indeed,

ifa

s=

b

t(b ∈ R, t ∈ S), then u(at − bs) = 0 for some u ∈ S whence

φ(u)(φ(a)φ(t) − φ(b)φ(s)) = 0, so that ψ(a

s

)= φ(a)φ(s)−1 = φ(b)φ(t)−1 =

ψ

(b

t

)(recall that φ(u) is a unit of the ring R′). Furthermore, it is easy to see

that ψ is a σ-homomorphism and ψ ν = φ.Conversely, let χ : S−1R → R′ be a σ-homomorphism of rings such that

χ ν = φ. If s ∈ S, then ν(s) is a unit of the ring S−1R and χ(ν(s))−1 =

χ(ν(s)−1). Sincea

s=

(a

1

)(1s

)= ν(a)ν(s)−1 in the ring S−1R (a ∈ R, s ∈ S),

we have χ(a

s

)= χ(ν(a))χ(ν(s))−1 = φ(a)φ(s)−1 = ψ

(a

s

)for any element

a

s∈ S−1R, so ψ is unique.


The last proposition shows that if a difference ring R with a basic set σ isan integral domain, then the quotient field Q(R) of the ring R can be naturallyconsidered as a σ-overring of R. (We identify an element a ∈ R with its canonicalimage a

1 in the field Q(R).) In this case Q(R) is said to be the quotient σ-fieldof R. It is easy to see that if the σ-ring R is inversive, then its quotient σ-fieldQ(R) is also inversive. Furthermore, it follows from Proposition 2.1.11 that ifa σ-field K contains an integral domain R as its σ-subring, then K containsa unique σ-subfield F which is a quotient σ-field of R. Clearly, if the σ-fieldK is inversive, then the inversive closure of F in K is the quotient field of theinversive closure of R in K.

Let R be a difference ring with a basic set σ = α1, . . . , αn and let T bethe free commutative semigroup generated by σ. An element a ∈ R is said tobe periodic with respect to αi or αi-periodic (1 ≤ i ≤ n) if there exists a positiveinteger ki such that αki

i (a) = a. If σ′ ⊆ σ, then an element a ∈ R is calledσ′-periodic if it is periodic with respect to every αi ∈ σ′. A σ-periodic elementis said to be periodic. Clearly, an element a is periodic if and only if there existsk ∈ N+ such that αk

i (a) = a for i = 1, . . . , n.Given α ∈ σ, an element a ∈ R is said to be α-invariant if α(a) = a. If

σ′ ⊆ σ, then an element a ∈ R is said to be σ′-invariant if it is α-invariant forevery α ∈ σ′. Clearly, a σ-invariant element is a constant; it will be also calledan invariant element. (Obviously, if a is such an element, then τ(a) = a forevery τ ∈ T ).

A difference (σ-) ring R is called invariant if all its elements are invariant.It is called periodic if there exists a positive integer k such that αk(a) = a forall a ∈ R, α ∈ σ. Otherwise, the σ-ring R is called aperiodic.

Let R be a difference ring with a basic set σ and σ′ ⊆ σ. Then the set ofall invariant (respectively, σ′-invariant) elements of R is a σ-subring of R calledthe difference (or σ-) subring of invariant (respectively, σ′-invariant) elements.The set of all periodic (respectively, σ′-periodic) elements of R is a σ-subringof R called the difference (or σ-) subring of periodic (respectively, σ′-periodic)elements. Of course, the last ring is not necessarily periodic (σ′-periodic). If Ris a difference field and σ′ ⊆ σ, then its subrings of σ′-invariant and σ′-periodicelements are difference subfields of R.

Theorem 2.1.12 Let M be a difference field with a basic set σ = α1, . . . , αnand let K and L be its σ-subfields of invariant and periodic elements, respec-tively. Then the field L is algebraically closed in M and algebraic over K. (Inother words, L is the algebraic closure of K in M .) This statement remains validif σ′ ⊆ σ and K and L are σ-subfields of σ′-invariant and σ′-periodic elementsof M , respectively.

PROOF. Let a ∈ L and let k be a positive integer such that αk(a) = a forevery α ∈ σ. Then the symmetric functions of elements αi1

1 . . . αinn (a), 0 ≤ iν ≤

k − 1 (ν = 1, . . . , n), take their values in K, hence a is algebraic over K (it is aroot of a polynomial of degree kn with coefficients in K).

Now let u ∈ M be algebraic over L. Then there exists a polynomial f(X) =amXm + · · · + a0 in one variable X with coefficients in L such that f(u) = 0

2.2. RINGS OF DIFFERENCE POLYNOMIALS 115

and am = 0. Then for any α ∈ σ and for any h ∈ N, αh(u) is a zero of thepolynomial fh(X) = αh(am)Xm + · · · + αh(a)0. Since each aj (0 ≤ j ≤ m) isperiodic, there is only finitely many distinct equations of the form fh(X) = 0,h ∈ N, and they have only finitely many zeros. Therefore, there exist p, q ∈ Nsuch that 0 ≤ p < q and αp(u) = αq(u) for every α ∈ σ. Then αq−p(u) = u forevery α ∈ σ, so that u ∈ L. The last part of the theorem can be proved in thesame way.

As in Section 1.1, a set of the form a = a(i)|i ∈ I will be referred to asan indexing (with the index set I). If J ⊆ I, the set a(i)|i ∈ J is said to be asubidexing of a.

The following definition introduces the concept of a specialization which isstudied in Chapter 5.

Definition 2.2.13 Let K be a difference field with a basic set σ and let a =a(i)|i ∈ I be an indexing of elements in a σ-overfield of K. Then a difference(or σ-) specialization of a over K is a σ-homomorphism φ of Ka into aσ-overfield of K that leaves K fixed. The image φa = φa(i)|i ∈ I is also calleda difference (or σ-) specialization of a over K. A σ-specialization φ is calledgeneric if it is a σ-isomorphism. Otherwise it is called proper.

2.2 Rings of Difference and Inversive DifferencePolynomials

Let R be a difference ring with a basic set σ = α1, . . . , αn, Tσ the free com-mutative semigroup generated by σ, and U = uλ|λ ∈ Λ a family of elementsin some σ-overring of R. We say that the family U is transformally (or σ-algebraically) dependent over R, if the set Tσ(U) = τ(uλ)|τ ∈ Tσ, λ ∈ Λ isalgebraically dependent over the ring R (that is, there exist elements v1, . . . , vk ∈Tσ(U) and a non-zero polynomial f(X1, . . . , Xk) with coefficients in R such thatf(v1, . . . , vk) = 0). Otherwise, the family U is said to be transformally (or σ-algebraically) independent over R or a family of difference (or σ-) indeterminatesover R. In the last case, the σ-ring S = R(uλ)λ∈Λσ is called the algebra ofdifference (or σ-) polynomials in the difference (or σ-) indeterminates (uλ)λ∈Λover R. The elements of S are called difference (or σ-) polynomials.

If a family consisting of a single element u is σ-algebraically dependent overR, the element u is said to be transformally algebraic (or σ-algebraic) over theσ-ring R. If the set τ(u)|τ ∈ T is algebraically independent over R, we saythat u is transformally (or σ-) transcendental over the ring R.

Let K be a σ-field, L a σ-overfield of K, and A ⊆ L. We say that the setA is σ-algebraic over K if every element a ∈ A is σ-algebraic over K. If everyelement of L is σ-algebraic over K, we say that L is a transformally algebraicor a σ-algebraic field extension of the σ-field K, or that L/K is a σ-algebraicfield extension. (By an algebraic σ-field extension of a difference (σ-) field K wemean a σ-overfield L of K such that every element of L is algebraic over K inthe usual sense.)


Proposition 2.2.1 Let R be a difference ring with a basic set σ and I anarbitrary set. Then there exists an algebra of σ-polynomials over R in a family ofσ-indeterminates with indices from the set I. If S and S′ are two such algebras,then there exists a σ-isomorphism S → S′ that leaves the ring R fixed. If Ris an integral domain, then any algebra of σ-polynomials over R is an integraldomain.

PROOF. Let T = Tσ and let S be the polynomial R-algebra in the set ofindeterminates yi,τi∈I,τ∈T with indices from the set I × T . For any f ∈ Sand α ∈ σ, let α(f) denote the polynomial in S obtained by replacing everyindeterminate yi,τ that appears in f by yi,ατ and every coefficient a ∈ R byα(a). We obtain an injective endomorphism S → S that extends the originalendomorphism α of R to the ring S (this extension is denoted by the same letterα). Setting yi = yi,1 (where 1 denotes the identity of the semigroup T ) we obtaina σ-algebraically independent over R set yi|i ∈ I such that S = R(yi)i∈I(we identify yi,τ (i ∈ I, τ ∈ T ) with τyi). Thus, S is an algebra of σ-polynomialsover R in a family of σ-indeterminates yi|i ∈ I.

Let S′ be another algebra of σ-polynomials over R in a family of differenceindeterminates zi|i ∈ I (with the same index set I). Then the mapping S → S′

that leaves elements of R fixed and sends every τyi to τzi(i ∈ I, τ ∈ T ) is clearlya σ-isomorphism. The last statement of the proposition is obvious.

Let R be an inversive difference ring with a basic set σ, Γ = Γσ, I a set, and S∗

a polynomial ring in the set of indeterminates yi,γi∈I,γ∈Γ with indices from theset I ×Γ. If we extend the automorphisms β ∈ σ∗ to S∗ setting β(yi,γ) = yi,βγ forany yi,γ and denote yi,1 by yi, then S∗ becomes an inversive difference overringof R generated (as a σ∗-overring) by the family (yi)i∈I. Obviously, this familyis σ∗-algebraically independent over R, that is, the set γ(yi)|γ ∈ Γ, i ∈ I isalgebraically independent over R. (Note that a set is σ∗-algebraically dependent(independent) over an inversive σ-ring if and only if this set is σ-algebraicallydependent (respectively, independent) over this ring.) The ring S∗ = R(yi)i∈I∗is called the algebra of inversive difference (or σ∗-) polynomials over R in the setof inversive difference (or σ∗-) indeterminates (yi)i∈I. The elements of S∗ arecalled inversive difference (or σ∗-) polynomials. It is easy to see that S∗ is aninversive closure of the ring of σ-polynomials R(yi)i∈I over R. Furthermore, ifa family (ui)i∈I from some σ∗-overring of R is σ-algebraically independent overR, then the inversive difference ring R(ui)i∈I∗ is naturally σ-isomorphic to S∗.Any such overring R(ui)i∈I∗ is said to be an algebra of inversive difference (orσ∗-) polynomials over R in the set of σ∗-indeterminates (ui)i∈I. We obtain thefollowing analog of Proposition 2.2.1.

Proposition 2.2.2 Let R be an inversive difference ring with a basic set σ andI an arbitrary set. Then there exists an algebra of σ∗-polynomials over R in afamily of σ∗-indeterminates with indices from the set I. If S and S′ are twosuch algebras, then there exists a σ∗-isomorphism S → S′ that leaves the ringR fixed. If R is an integral domain, then any algebra of σ∗-polynomials over Ris an integral domain.


Let R be a σ-ring, R(yi)i∈I an algebra of difference polynomials in afamily of σ-indeterminates (yi)i∈I, and (ηi)i∈I a set of elements in someσ-overring of R. Since the set τ(yi)|i ∈ I, τ ∈ Tσ is algebraically indepen-dent over R, there exists a unique ring homomorphism φη : R[τ(yi)i∈I,τ∈Tσ

] →R[τ(ηi)i∈I,τ∈Tσ

] that maps every τ(yi) onto τ(ηi) and leaves R fixed. Clearly,φη is a surjective σ-homomorphism of R(yi)i∈I onto R(ηi)i∈I; it is calledthe substitution of (ηi)i∈I for (yi)i∈I . Similarly, if R is an inversive σ-ring,R(yi)i∈I∗ an algebra of σ∗-polynomials over R and (ηi)i∈I a family of el-ements in a σ∗-overring of R, one can define a surjective σ-homomorphismR(yi)i∈I∗ → R(ηi)i∈I∗ that maps every yi onto ηi and leaves the ring Rfixed. This homomorphism is also called the substitution of (ηi)i∈I for (yi)i∈I .(It will be always clear whether we talk about substitutions for difference orinversive difference polynomials.) If g is a σ- or σ∗- polynomial, then its imageunder a substitution of (ηi)i∈I for (yi)i∈I is denoted by g((ηi)i∈I). The kernelof a substitution is an inversive difference ideal of the σ-ring R(yi)i∈I (or theσ∗-ring R(yi)i∈I∗); it is called the defining difference (or σ-) ideal of the fam-ily (ηi)i∈I over R. (In the case of σ∗-polynomials, this ideal is called a definingσ∗-ideal of the family.)

If K is a σ- (or σ∗-) field and (ηi)i∈I is a family of elements in some its σ-(respectively, σ∗-) overfield L, then K(ηi)i∈I (respectively, K(ηi)i∈I∗) is anintegral domain (it is contained in the field L). It follows that the defining σ-idealP of the family (ηi)i∈I over K is a prime inversive difference ideal of the ringK(yi)i∈I (respectively, of the ring of σ∗-polynomials K(yi)i∈I∗). Therefore,the difference field K〈(ηi)i∈I〉 can be treated as the quotient σ-field of the σ-ringK(yi)i∈I/P . (In the case of inversive difference rings, the σ∗-field K〈(ηi)i∈I〉∗can be considered as a quotient σ-field of the σ∗-ring K(yi)i∈I∗/P .)

If R is a difference (σ-) ring and A a σ-polynomial in the ring R(yi)i∈I,then A can be considered as a polynomial in the ring R[τyi | i ∈ I, τ ∈ Tσ](which is a polynomial ring in the set of variables τyi | i ∈ I, τ ∈ Tσ). Thetotal degree of A as such a polynomial is denoted by deg A; it is called thedegree of A. The degree of A with respect to a variable τyi (τ ∈ Tσ, i ∈ I)is denoted by degτyi

A. The degree of a σ∗-polynomial A in a σ∗-polynomialalgebra R(yi)i∈I∗ over an inversive (σ-) ring R, as well as the degree of Awith respect to a variable γyi (γ ∈ Γσ, i ∈ I), is defined and denoted in thesame way.

Let K be a difference field with a basic set σ and s a positive integer. Byan s-tuple over K we mean an s-dimensional vector a = (a1, . . . , as) whosecoordinates belong to some σ-overfield of K. If the σ-field K is inversive, thecoordinates of an s-tuple over K are supposed to lie in some σ∗-overfield of K.If each ai (1 ≤ i ≤ s) is σ-algebraic over the σ-field K, we say that the s-tuplea is σ-algebraic over K.

Definition 2.2.3 Let K be a difference (inversive difference) field with a basicset σ and let R be the algebra of σ- (respectively, σ∗-) polynomials in finitelymany σ- (respectively, σ∗-) indeterminates y1, . . . , ys over K. Furthermore, letΦ = fj |j ∈ J be a set of σ- (respectively, σ∗-) polynomials in R. An s-tuple


η = (η1, . . . , ηs) over K is said to be a solution of the set Φ or a solution ofthe system of algebraic difference equations fj(y1, . . . , ys) = 0 (j ∈ J) if Φ iscontained in the kernel of the substitution of (η1, . . . , ηs) for (y1, . . . , ys). In thiscase we also say that η annuls Φ. (If Φ is a subset of a ring of inversive differencepolynomials, the system is said to be a system of algebraic σ∗-equations.)

As we have seen, if one fixes an s-tuple η = (η1, . . . , ηs) over a σ-field K,then all σ-polynomials of the ring Ky1, . . . , ys, for which η is a solution,form a prime inversive difference ideal. It is called the defining σ-ideal of η.Similarly, if η is an s-tuple over a σ∗-field K, then all σ∗-polynomials g ofthe ring Ky1, . . . , ys∗ such that g(η1, . . . , ηs) = 0 form a prime σ∗-ideal ofKy1, . . . , ys∗ called the defining σ∗-ideal of η over K.

Let Φ be a subset of the algebra of σ-polynomials Ky1, . . . , ys over a σ-field K. An s-tuple η = (η1, . . . , ηs) over K is called a generic zero of Φ if forany σ-polynomial A ∈ Ky1, . . . , ys, the inclusion A ∈ Φ holds if and only ifA(η1, . . . , ηs) = 0. If the σ-field K is inversive, then the notion of a generic zeroof a subset of Ky1, . . . , ys∗ is defined similarly.

Two s-tuples η = (η1, . . . , ηs) and ζ = (ζ1, . . . , ζs) over a σ- (or σ∗-) fieldK are called equivalent over K if there is a σ-homomorphism K〈η1, . . . , ηs〉 →K〈ζ1, . . . , ζs〉 (respectively, K〈η1, . . . , ηs〉∗ → K〈ζ1, . . . , ζs〉∗) that maps each ηi

onto ζi and leaves the field K fixed.

Proposition 2.2.4 Let R be the algebra of σ-polynomials Ky1, . . . , ys or thealgebra of σ∗-polynomials Ky1, . . . , ys∗ over a difference (respectively, inver-sive difference) field K with a basic set σ. Then

(i) A set Φ R has a generic zero if and only if Φ is a prime σ∗-ideal ofR. If (η1, . . . , ηs) is a generic zero of Φ, then K〈η1, . . . , ηs〉 (or K〈η1, . . . , ηs〉∗if we consider the algebra of σ∗-polynomials over a σ∗-field K) is σ-isomorphicto the quotient σ-field of R/Φ.

(ii) Any s-tuple over K is a generic zero of some prime σ∗-ideal of R.(iii) If two s-tuples over K are generic zeros of the same prime σ∗-ideal of R,

then these s-tuples are equivalent.

PROOF. We will prove the proposition for difference (σ-) polynomials. Theproof for the the algebra of inversive difference polynomials is similar.

(i) Let Φ be a prime σ∗-ideal of R = Ky1, . . . , ys and let ηi denote theimage of yi under the natural σ-epimorphism R → R/Φ (1 ≤ i ≤ s). Then thequotient field L of the σ-ring R/Φ can be naturally treated as a σ-overfield ofK, and the s-tuple (η1, . . . , ηs) with coordinates in L is a generic zero of Φ.

Conversely, if a set Φ R has a generic zero ζ = (ζ1, . . . , ζs), then Φ is adefining ideal of ζ. In this case, as we have seen, Φ is a prime reflexive ideal of R.

(ii) Let η = (η1, . . . , ηs) be an s-tuple over K and let φη : Ky1, . . . , ys →Fη1, . . . , ηs be the substitution of η1, . . . , ηs for y1, . . . , ys. Then Ker φη

is a prime σ∗-ideal of R with the generic zero η.(iii) Let ζ = (ζ1, . . . , ζs) be a generic zero of a prime σ∗-ideal P of R (ζ1, . . . , ζs

belong to some σ-overfield of K). Let η1, . . . , ηs denote the canonical images of


the σ-indeterminates y1, . . . , ys, respectively, in the quotient σ-field L of R/P .Then the substitution φζ : Ky1, . . . , ys → Kζ1, . . . , ζs of ζ1, . . . , ζs fory1, . . . , ys induces a σ-isomorphism L = K〈η1, . . . , ηs〉 → K〈ζ1, . . . , ζs〉 thatleaves fixed the field K and sends ηi to ζi (1 ≤ i ≤ s). It follows that everygeneric zero of P is σ-equivalent to the s-tuple η = (η1, . . . , ηs).

The following example, as well as examples 2.2.6 and 2.2.7 below, is due toR. Cohn.

Example 2.2.5 Let us consider the field of complex numbers C as an ordinarydifference field whose basic set σ consists of the identity automorphism α. LetCy be the algebra of σ-polynomials in one σ-indeterminate y over C and let(k)y denote the k-th transform αky (k = 1, 2, . . . ). Furthermore, let M be thefield of functions of one complex variable z meromorphic on the whole complexplane. Then M can be viewed as a σ-overfield of C if one extends α by settingαf(z) = f(z + 1) for any function f ∈ M . It is easy to check that the σ-polynomial A = ((1)y − y)2 − 2((1)y + y) + 1 is irreducible in Cy (when thisring is treated as a polynomial ring in the denumerable set of indeterminatesy, (1)y, (2)y, . . . ). Furthermore, if c(z) is a periodical function from M withperiod 1, then ξ = (z + c(z))2 and η = (c(z)eiπz + 1

2 )2 are solutions of A. (ξ isa solution of the system of the σ-polynomials A and A′ = (2)y − 2 (1)y + y − 2,while η is the solution of the system of A and A′′ = (2)y−y.) Note that the factthat an irreducible σ-polynomial in one σ-indeterminate may have two distinctsets of solutions, each of which depends on an arbitrary periodic function, doesnot have an analog in the theory of differential polynomials.

Let K be a difference field with a basic set σ, R = Ky1, . . . , ys the al-gebra of σ-polynomials in a set of s σ-indeterminates y1, . . . , ys over K, andΦ ⊆ R. Let a = ai,τ |i = 1, . . . , s, τ ∈ Tσ be a family of elements in someσ-overfield of K. The family a (indexed by the set 1, . . . , s × Tσ) is said tobe a formal algebraic solution of the set of σ-polynomials Φ if a is a solutionof Φ when this set is treated as a set of polynomials in the polynomial ringK[yi,τ |i = 1, . . . , s, τ ∈ Tσ]. (This polynomial ring in the denumerable familyof indeterminates yi,τ |i = 1, . . . , s, τ ∈ Tσ coincides with R (yi,τ stands forτ(yi)), but it is not considered as a difference ring, so the solutions of its subsetsare solutions in the sense of the classical algebraic geometry.)

It is easy to see that every solution a = (a1, . . . , as) of a set Φ ⊆ Fy1, . . . , ysproduces its formal algebraic solution a = τ(ai)|i = 1, . . . , s, τ ∈ Tσ. On theother hand, not every formal algebraic solution can be obtained from a solutionin this way.

Example 2.2.6 Let C be the field of complex numbers considered as an ordi-nary difference field whose basic sets consists of the complex conjugation (thatis, σ = α where α(a + bi) = a− bi for any complex number a + bi). Let Cybe the ring of σ-polynomials in one σ-indeterminate y. If A = y2 + 1 ∈ Cy,then the 1-tuples (i) and (−i) are solutions of the σ-polynomial A that pro-duce formal algebraic solutions (i,−i, i,−i, . . . ) and (−i, i,−i, i, . . . ) of A. At


the same time, the sequence (−i, i, i, i, . . . ) is a formal algebraic solution of Awhich is not a solution of this σ-polynomial.

The following example illustrates how formal algebraic solutions of differencepolynomials can be used in the study of ideals of difference polynomials.

Example 2.2.7 LetKybethealgebraofσ-polynomials inoneσ-indeterminatey over an ordinary difference field K with a basic set σ = α. As in Example2.2.5, we shall denote a transform αky by (k)y (k ∈ N) and identify (0)y with y.

Let X denote the set of all sequences (a0, a1, . . . ) whose terms are 1 or −1,and let Σ(X) denote the set of all σ-polynomials that vanish when the termsof the sequence y, (1)y, (2)y, . . . are replaced by the corresponding terms of anysequence from X. (For example, y2((1)y)2 − 1 ∈ Σ(X), but y (1)y /∈ Σ(X)). Weare going to show that Σ(X) = [y2 − 1].

Clearly, [y2 − 1] ⊆ Σ(X). To prove that an arbitrary σ-polynomial A ∈Σ(X) belongs to [y2 − 1], we proceed by induction on the order of A, that is,the maximal integer k ∈ N such that A effectively involves (k)y, but does notinvolve any (j)y with j > k. If the order of A is 0, then A is a regular polynomialin one variable y that has roots 1 and −1. It follows that A is divisible by y2−1,hence A ∈ [y2 − 1]. Now, suppose that the order of A is a positive integer r.Considering A as a polynomial in (r)y, one can write A = B( ((r)y)2 − 1) +Cyr + D where B,C, and D are σ-polynomials of order less than r. Let usreplace y, (1)y, (2)y, . . . first by a sequence (a0, a1, . . . ) ∈ X with ar = 1 andthen by a sequence (b0, b1, . . . ) ∈ X with br = 1. In both cases C and D becomethe same elements c and d of the field K, so the inclusion A ∈ Σ(X) impliesthat c+d = 0 and c−d = 0 whence c = d = 0 A ∈ [y2−1], since C,D ∈ [y2−1]by the induction hypothesis.

As a consequence of the equality [y2 − 1] = Σ(X) we obtain that the σ-ideal [y2 − 1] is reflexive. Indeed, if A ∈ Ky \ [y2 − 1], then there exists asequence (a0, a1, . . . ) ∈ X that does not annul A (when each (i)y (i ∈ N) isreplaced by ai). Then this sequence does not annul τ(A) for any τ ∈ Tσ, henceτ(A) /∈ [y2 − 1].

Another consequence of our description of [y2 − 1] is the fact that this σ-ideal cannot be represented as a product or intersection of two σ-ideals properlycontaining [y2−1]. Moreover, we will show that if I and J are proper σ-ideals ofKy, [y2−1] I and [y2−1] J , then IJ [y2−1]. To prove this, assume thatA1 ∈ I \ [y2 − 1] and A2 ∈ J \ [y2 − 1]. Let A1 be of order r. Since A1 /∈ [y2 − 1],there exists a sequence a = (a0, a1, . . . ) ∈ X that does not annul A when each(i)y (i ∈ N) is replaced by ai. Since αr+1(A2) /∈ [y2 − 1] and y, (1)y, . . . ,(r) ydo not appear in αr+1(A2), one can change the terms ar+1, ar+2, . . . of thesequence a to obtain a sequence b ∈ X that does not annul αr+1(A2). Then bdoes not annul A1α

r+1(A2), hence A1αr+1(A2) ∈ IJ \ [y2 − 1].

If Φ is a subset of an algebra of σ∗-polynomials Ky1, . . . , ys∗ over aninversive difference field K with a basic set σ, then a formal algebraic solu-tion of Φ is defined as a family a∗ = ai,γ |i = 1, . . . , s, γ ∈ Γσ that annulsevery polynomial in Φ when Φ is treated as a subset of the polynomial ring

2.3. DIFFERENCE IDEALS 121

K[γ(yi)|i = 1, . . . , s, γ ∈ Γσ]. As in the case of σ-polynomials, every solutiona = (a1, . . . , as) of a set of σ∗-polynomials generates the formal algebraic solu-tion a∗ = γ(ai)|i = 1, . . . , s, γ ∈ Γσ of this set, but not all formal algebraicsolutions can be obtained in this way.

Definition 2.2.8 Let A be a difference ring with a basic set σ. An A-algebraR is said to be a difference (σ-) algebra over A or a σ-A-algebra if elements ofσ act as mutually commuting injective endomorphisms of R such that α(ax) =α(a)α(x) for any a ∈ A, x ∈ R, α ∈ σ. If the σ-ring A is inversive, then anA-algebra R is said to be an inversive difference A-algebra or a σ∗-A-algebraif R is a σ∗-ring and a σ-A-algebra (in this case α(ax) = α(a)α(x) for anya ∈ A, x ∈ R, α ∈ σ∗).

It is easy to see that the ring of difference polynomials over a differencering A is a difference A-algebra. Also, if A is an inversive difference ring with abasic set σ, then the ring of inversive difference (σ∗-) polynomials over A is aσ∗-A-algebra. Of course, if K is an difference (or inversive difference) field witha basic set σ and L a σ- (respectively, σ∗-) field extension of K, then L can beviewed as a σ-K-algebra (respectively, as a σ∗-K-algebra).

2.3 Difference Ideals

Throughout this section R denotes a difference ring with a basic set σ =α1, . . . , αn and Tσ denotes the free commutative semigroup generated by σ.If R is inversive, then the free commutative group generated by σ is denotedby Γσ. The following definition introduces a class of difference ideals similar toradical ideals in commutative rings.

Definition 2.3.1 A difference ideal I of R is called perfect if for any a ∈ R,τ1, . . . , τr ∈ Tσ, and k1, . . . , kr ∈ N, the inclusion τ1(a)k1 . . . τr(a)kr ∈ J impliesa ∈ J .

It is easy to see that every perfect ideal is reflexive and every reflexive primeideal is perfect. Furthermore, if the σ-ring R is inversive, then a σ-ideal J of Ris perfect if and only if any inclusion γ1(a)k1 . . . γr(a)kr ∈ J (a ∈ R, γ1, . . . , γr ∈Γσ, k1, . . . , kr ∈ N) implies a ∈ J .

If B is a subset of a difference ring R with a basic set σ, then the intersectionof all perfect σ-ideals of R containing B is the smallest perfect ideal containingB. It is denoted by B and called the perfect closure of the set B or the perfectideal generated by B. (It will be always clear whether a1, . . . , ar denotes theset consisting of the elements a1, . . . , ar or the perfect σ-ideal generated by theseelements.)

The ideal B can be obtained from the set B via the following procedurecalled shuffling. For any set M ⊆ R, let M ′ denote the set of all a ∈ R such thatτ1(a)k1 . . . τr(a)kr ∈ M for some τ1, . . . , τr ∈ Tσ and k1, . . . , kr ∈ N (r ≥ 1).Furthermore, let M1 denote the set [M ]′. With this notation, B =

⋃∞i=0 Bi

where B0 = B and Bk+1 = (Bk)1 = [Bk]′ for k = 0, 1, . . . . Indeed, B = B0 ⊆


B, and the inclusion Bk ⊆ B implies [Bk] ⊆ B and Bk+1 = [Bk]′ ⊆ B,since the σ-ideal B is perfect. By induction Bk ⊆ B for all k = 0, 1, . . .hence

⋃∞i=0 Bi ⊆ B. On the other hand, it is easy to see that

⋃∞i=0 Bi is a

perfect σ-ideal of R, so it should contain B. Thus, B =⋃∞

i=0 Bi.Recall that a subset M of a difference (σ-) ring R is called invariant if α(a) ∈

M whenever a ∈ M , α ∈ σ. Clearly, if A ⊆ R, then the set τ(a)|τ ∈ Tσ, a ∈ Ais invariant. This set will be denoted by A+.

Lemma 2.3.2 Let X and Y be two invariant subsets of a difference (σ-) ringR. Then XkYk ⊆ (XY )k for all k ∈ N.

PROOF. Let x ∈ X1 and y ∈ Y1. Then there exist τ1, . . . , τr ∈ Tσ andk1, . . . , kr ∈ N (r ≥ 1) such that x∗ = τ1(x)k1 . . . τr(x)kr belongs to [X].Similarly, there are β1, . . . , βs ∈ Tσ and l1, . . . , ls ∈ N (s ≥ 1) such thaty∗ = β1(y)l1 . . . βs(y)ls lie in [Y ]. Since the sets X and Y are invariant, x∗ andy∗ can be written as linear combinations of elements of X and Y , respectively,with coefficients in R. Then x∗y∗ ∈ [XY ].

Since, the element τ1(xy)k1 . . . τr(xy)krβ1(xy)l1 . . . βs(xy)ls is a multiple ofx∗y∗, it belongs to [XY ]. Therefore, xy ∈ [XY ]′ = (XY )1 whence X1Y1 ⊆(XY )1. Applying induction on k we obtain that XkYk ⊆ (XY )k for all k ∈ N.

Theorem 2.3.3 Let A and B be two subsets of R. Then(i) AkBk ⊆ (AB)k+1 for any k ∈ N. (By the product UV of two sets

U, V ⊆ R we mean the set UV = uv|u ∈ U, v ∈ V .)(ii) AB ⊆ AB.(iii) (AB)k ⊆ Ak

⋂Bk for any k ∈ N, k ≥ 1.

(iv) Ak

⋂Bk ⊆ (AB)k+1 for any k ∈ N.

(v) A⋂B = AB.PROOF. (i) Let x ∈ A+ and y ∈ B+. Then x = τ(a) and y = β(b) for some

a ∈ A, b ∈ B; τ, β ∈ Tσ. Since the element τ(xy)β(xy) is a multiple of τβ(ab),it belongs to [AB]. Therefore xy ∈ [AB]′ = (AB)1, so that A+B+ ⊆ (AB)1.Applying Lemma 2.3.2 to the invariant sets A+ and B+ we obtain that AkBk ⊆A+

k B+k ⊆ (A+B+)k ⊆ (AB)k+1 for k = 0, 1, . . . .

Statement (ii) is a direct consequence of (i). In order to prove (iii), noticethat [AB] ⊆ [A] ∩ [B], whence (AB)k ⊆ Ak

⋂Bk for k = 1, 2, . . . .

Statement (iv) can be obtained as follows. Let a ∈ Ak

⋂Bk (k ∈ N). By

part (i), a2 ∈ AkBk ⊆ (AB)k+1, hence a ∈ (AB)k+1. The last statement of thetheorem is an immediate consequence of (iii) and (iv).

As we have mentioned, the role of perfect difference ideals is similar to therole of radical ideals of commutative rings. The following proposition is an analogof the corresponding statement for radical ideals.

Proposition 2.3.4 Every perfect difference ideal of a difference ring R is theintersection of a family of prime difference ideals of R.


PROOF. Let I be a perfect difference ideal of R and x ∈ R\I. Let Σ denotethe the set of all perfect difference ideals J of R such that I ⊆ J and x /∈ J . ByZorn’s lemma Σ contains a maximal element P which is a prime difference ideal.Indeed, if ab ∈ P but a /∈ P, b /∈ P , then x ∈ P, a and x ∈ P, b. Thereforex ∈ P, a⋂P, b = P, ab = P (see Theorem 2.3.3(v)) that contradicts theinclusion P ∈ Σ. It follows that for every x ∈ R\I, there exists a prime differenceideal Px of R such that I ⊆ Px and x /∈ P . Thus, I =

⋂x∈R\I

Px.

Exercises 2.3.5 1. Let I be a perfect σ-ideal of a difference (σ-) ring R andM ⊆ R. Prove that I : M = a ∈ R|aM ⊆ I is a perfect σ-ideal of R. Usethis fact to give an alternative proof of the second statement of Theorem 2.3.3.[Hint: Let A,B ⊆ R. Consider the ideal AB : A and show that it containsB. Then consider AB : B.]

2. An element a of a difference (σ-) ring R is called a perfect unit of Rif a = R. Prove that the set S(R) of all perfect units of a σ-ring R is amultiplicative σ∗-subset of R.

It is easy to see that the radical r(I) of a difference ideal I is a differenceideal, and its inversive closure r(I)∗ (which coincides with the radical of theinversive closure of I) is a reflexive difference ideal contained in the perfectclosure I of the ideal I. The following definition introduces the class of idealsI satisfying the condition I = r(I)∗.

Definition 2.3.6 A difference ideal I of a difference (σ-) ring R is called com-plete if for every element a ∈ I, there exist τ ∈ Tσ, k ∈ N such that τ(a)k ∈ I.(In other words, a σ-ideal I of R is complete if I is the reflexive closure ofthe radical of I.)

Exercise 2.3.7 Show that a σ-ideal I of a difference (σ-) ring R is complete ifand only if the presence in I of a product of powers of transforms of an elementimplies the presence in I of a power of a transform of the element.

Definition 2.3.8 Let R be a difference ring with a basic set σ and I, J twoσ-ideals of R. The ideals I and J are called separated if I, J = R and stronglyseparated if [I, J ] = R. σ-ideals I1, . . . , Ir of R are called (strongly) separated inpairs if any two of the ideals Ii, Ij (1 ≤ i < j ≤ r) are (strongly) separated.

Exercise 2.3.9 Show that if difference ideals I1, . . . , Ir are strongly separated

in pairs, thenr⋂

i=1

Ii =r∏

i=1

Ii.

Exercise 2.3.10 Prove that the intersection of a finite family of complete dif-ference ideals is a complete difference ideal.

The following example shows that two separated difference ideals might notbe strongly separated.


Example 2.3.11 (R. Cohn). Let R = Qy be the algebra of σ-polynomialsin one σ-indeterminate y over Q (treated as an ordinary difference field whosebasic set σ consists of the identity isomorphism α). Let A = 1 + yα(y) andB = y + α(y) ∈ R. Then A,B = R, but [A,B] is a proper ideal of the ring R.(Moreover, even [A, B] is a proper ideal of R.)

Indeed, consider the σ-polynomial C = (1 − y)(1 + α(y)) Since 1 − y2 =A−yB ∈ A,B and Cα(C) is a multiple of α(1−y2), C ∈ A,B. Furthermore,the equalities C−(1−y)B = (1−y)2 and C+(1+α(y))B = (1+α(y))2 imply that1−y ∈ A,B and 1+y ∈ A,B, whence A,B = R. On the other hand, theideals [A] and [B] are not strongly separated, since ak = (−1)k|k = 0, 1, . . . is an algebraic solution of both [A] and [B] (that is, the replacement of everyαk(y) by (−1)k vanishes every σ-polynomial in [A] + [B] = [A,B]).

Lemma 2.3.12 Let I be a complete σ-ideal in a difference (σ-) ring R and letI = J1 ∩ J2 where J1 and J2 are two strongly separated perfect σ-ideals ofR. Then there exist a ∈ J1, b ∈ J2 such that a + b = 1, α(a) − a ∈ I, andα(b) − b ∈ I for every α ∈ σ.

PROOF. Since J1 + J2 = R, there exist x ∈ J1, y ∈ J2 such that x + y = 1.Obviously, xy ∈ I, hence there exist τ ∈ Tσ, k ∈ N such that (τ(x)τ(y))k ∈ I.Let u denote the sum of all terms in the expansion of (τ(x) + τ(y))2k−1 wherethe exponents of τ(x) are greater than or equal to k, and let v denote the sumof the remaining terms in this expansion. Since τ(x) + τ(y) = 1, u + v = 1.Furthermore, it is easy to see that u ∈ J1, v ∈ J2 and uv ∈ I.

For every α ∈ σ, we have α(u)α(v) ∈ I, hence uα(v)α(uα(v)) ∈ I, henceuα(v) ∈ I. Similarly, α(u)v ∈ I. Since the ideal I is complete, there existβ ∈ Tσ,m ∈ N such that β(u)m(αβ(v))m ∈ I and (αβ(u))mα(v)m ∈ I.

The equality v = 1 − u and the inclusion uv ∈ I imply that v2 = v −uv ≡ v (mod I), whence vm ≡ v (mod I) and β(v)m ≡ β(v)(mod I). Similarly,β(u)m ≡ β(u) (mod I).

Setting a = β(u) and b = β(v), we obtain two elements that have the de-sired properties. Indeed, as we have already seen, a ∈ J1, b ∈ J2, a + b = 1and α(a)b, aα(b) ∈ I for every α ∈ σ. The equality a + b = 1 implies thatα(a) − a = α(b) − b (α ∈ σ). Now, the inclusion a(b − α(b)) ∈ I yields(1−b)(a−α(a)) ∈ I. Since ab ∈ I and bα(a) ∈ I, we obtain that a−α(a) ∈ I andb − α(b) ∈ I.

Theorem 2.3.13 Let I be a complete difference ideal in a difference ringR with a basic set σ. Suppose that I is the intersection of some per-fect ideals J1, . . . , Js which are strongly separated in pairs. Then there existuniquely determined complete σ-ideals I1, . . . , Is such that I = I1

⋂ · · ·⋂ Is andIk = Jk (1 ≤ k ≤ s). In this representation, the ideals I1, . . . , Is are stronglyseparated in pairs. Furthermore, if the ideal I is reflexive, then so are I1, . . . , Is.

PROOF. We start with the case s = 2. Let I = J1

⋂J2 where J1 and J2

are two strongly separated perfect σ-ideals of R. By the last lemma, there existelements a ∈ J1, b ∈ J2 such that a + b = 1, α(a) − a ∈ I, and α(b) − b ∈ I


for every α ∈ σ. Let I1 = [I, a] and I2 = [I, b]. Then I1 = J1 and I2 = J2.Indeed, since J1 is a perfect σ-ideal containing I and a, I1 ⊆ J1. On the otherhand, if z ∈ J1, then zb ∈ J1J2 ⊆ J1 ∩ J2 ⊆ I, hence there exist γ ∈ Tσ

and l ∈ N such that γ(z)lγ(b)l ∈ I ⊆ I1. Substituting γ(b) = 1 − γ(a) andusing the inclusion γ(a) ∈ I1 we obtain that γ(z)l ∈ I1. Thus, z ∈ I1 whenceJ1 = I1 and the σ-ideal I1 is complete. Similarly, the σ-ideal I2 is completeand J2 = I2.

Let us show that I = I1

⋂I2. Clearly, I ⊆ I1 ∩ I2. Let w ∈ I1

⋂I2. Since

w ∈ I1 = [I, a], one can represent w as w = c0 + c1τ1(a) + · · · + cqτq(a) wherec0 ∈ I; c1, . . . , cq ∈ R; τ1, . . . , τq ∈ Tσ. Furthermore, since α(a) − a ∈ I forevery α ∈ σ (hence τ(a)−a ∈ I for every τ ∈ Tσ), we can write w as w = c+dawhere c ∈ I, d ∈ R.

Now, the inclusion ab ∈ I implies that bw ∈ I, and similar arguments showthat aw ∈ I. Therefore, w = (a+b)w ∈ I whence I = I1

⋂I2. Also, the equality

a + b = 1 implies that the σ-ideals I1 and I2 are strongly separated. (Noticethat the strong separation of I1 and I2 is equivalent to the strong separation ofJ1 and J2. Indeed, if J1 + J2 = R, then u + v = 1 for some u ∈ J1, v ∈ J2. Inthis case there exist τ ∈ Tσ, k ∈ N such that τ(u)k ∈ I1 and τ(v)k ∈ I2, so theexpansion of (τ(u) + τ(v))2k can be separated into a sum of two sets of termswhich are, respectively, in I1 and I2.)

To prove the uniqueness, let I ′1 and I ′2 be another pair of complete σ-idealsof R such that I = I ′1 ∩ I ′2 and I ′1 = J1, I ′2 = J2. Let x ∈ I ′1. Since b ∈ J2,there exist λ ∈ Tσ, p ∈ N such that λ(b)px ∈ I ′1I

′2 ⊆ I ⊆ I1. Substituting

λ(b) = 1 − λ(a) and using the inclusion λ(a) ∈ I1, we find that x ∈ I1. Thus,I ′1 ⊆ I1.

Conversely, let y ∈ I1. Then yµ(b) = y(1 − µ(a)) ∈ I ⊆ I ′1 for every µ ∈ Tσ,hence y(1 − µ(a)k) ∈ I ′1 for every µ ∈ Tσ, k ∈ N. Since a ∈ J1 and the σ-idealI ′1 is complete, one can choose µ and k so that µ(a)k ∈ I ′1. It follows that y ∈ I ′1hence I ′1 = I1. Similarly, I ′2 = I2.

To complete the proof in the case s = 2, assume that the σ-ideal I is reflexive.Let I∗1 and I∗2 denote the reflexive closures of I1 and I2, respectively. Obviously,I∗1 and I∗2 are complete σ∗-ideals of R and I∗i = Ji (i = 1, 2). Furthermore,I∗1 ∩I∗2 is the reflexive closure of I = I1∩I2 hence I∗1 ∩I∗2 = I. By the uniquenessof the decomposition of I, I∗i = Ii (i = 1, 2), that is, the σ-ideals I1 and I2 arereflexive.

Now, we are going to prove the general case of the theorem by induction ons. Suppose that the conclusion of the theorem is true for s = r − 1, r > 2, andconsider the case s = r. Let J = J2∩· · ·∩Jr, so that J = J1∩ J . Then the idealsJ1 and J are strongly separated. Indeed, since J1, . . . , Jr are strongly separatedin pairs, there exist ai ∈ J1, bi ∈ Ji such that ai + bi = 1 (i = 2, . . . , r). Thenb2b3 . . . br = (1−a2)(1−a3) . . . (1−ar) = 1−c where c ∈ J1. Since b2b3 . . . br ∈ J ,J + J1 = R.

By the first part of the proof (case s = 2), there exist complete σ-idealsI1 and M such that I = I1

⋂M , I1 + M = R, J1 = I1, and J = M.

By the induction hypothesis, M = I2

⋂ · · ·⋂ Ir where I2, . . . , Ir are completeand strongly separated in pairs σ-ideals of R such that Ii = Ji (i = 2, . . . , r).


It follows that I = I1

⋂ · · ·⋂ Ir where each Ii is complete, Ii = Ji (1 ≤ i ≤ r),and I1, . . . , Ir are strongly separated in pairs (as we have seen in the proof ofthe case s = 2, the fact that J1, . . . , Jr are strongly separated in pairs impliesthe strong separation of every pair Ik, Il, 1 ≤ k < l ≤ r).

Let us prove the uniqueness. Suppose that there is another decomposi-tion I = I ′1

⋂ · · ·⋂ I ′r where I ′i are complete σ-ideals such that I ′i = Ji

(i = 1, . . . , r). Since J1, . . . , Jr are strongly separated in pairs, the same is truefor I ′1, . . . , I

′r. Theorem 2.3.3(v) implies that I2 . . . Ir = J2

⋂ · · ·⋂ Jr whereI2 . . . Ir is the product of I2, . . . , Ir as ideals of R.

It is easy to show that the intersection of a finite family of strongly sepa-rated in pairs σ-ideals coincides with their product (see Exercise 2.3.9). There-fore, I2

⋂ · · ·⋂ Ir = J2

⋂ · · ·⋂ Jr = J . Similarly, I ′2⋂ · · ·⋂ I ′r = J . Thus,

I = I2

⋂ · · ·⋂ Ir and I ′ = I ′2⋂ · · ·⋂ I ′r are complete σ-ideals such that I =

I1

⋂I = I ′1

⋂I ′ and I = I ′ = J . By the uniqueness in the case s = 2, we

obtain that I1 = I ′1. Since I1 plays no special role in the decomposition of I,Ii = I ′i for i = 1, . . . , r.

The fact that all I1, . . . , Is (s ≥ 3) are reflexive if I is reflexive can be provedin the same way as in the case s = 2.

Definition 2.3.14 A difference ideal I of a difference (σ-) ring R is calledmixed if the inclusion ab ∈ I (a, b ∈ R) implies that aα(b) ∈ I for every α ∈ σ.

Mixed difference ideals deserve consideration because of the following obvi-ous property. A σ-ideal I of a difference (σ-) ring R is mixed if and only if forany nonempty set S ⊆ R, I : S is a σ-ideal of R.

Exercises 2.3.15 Let R be a difference ring with a basic set σ.1. Show that every perfect σ-ideal of R is mixed.2. Prove that every mixed σ-ideal of the ring R is complete [Hint: Let

I be a mixed σ-ideal of R. Show that an inclusion τ1(a)k1 . . . τp(a)kp ∈ I(p ≥ 1, τi ∈ Tσ, ki ∈ N for i = 1, . . . , p, a ∈ R) implies that τ1 . . . τp(a)k ∈ Iwhere k = maxk1, . . . , kp.]

Examples 2.3.16 Let K be an ordinary difference ring with a translation αand let Ky be the ring of difference polynomials in one difference indetermi-nate y over F .

1. The σ-ideal I = [y2] of the difference ring Ky is complete, since itsradical [y] is prime and, therefore, coincides with y2. At the same time, I isnot mixed, since it does not contain yα(y).

2. The σ-ideal J = [y2, yαy, yα2y, . . . ] is mixed since it consists of all σ-polynomials in which every term is at least of second degree in the indetermi-nates y, αy, α2y, . . . . At the same time, J is not perfect, since y ∈ J, y /∈ J .

3. Let us consider the σ-ideals M = [yαy] and Jk = [yk, yαy] (k = 1, 2, . . . ) ofthe ring of σ-polynomials Ky. It is easy to see that the ideals Jk are complete(the radical of Jk is [y] = Jk), but M is not complete. (Indeed, y ∈ M but


(αjy)k /∈ M for every j, k ∈ N). Let us show that M =⋂∞

k=1 Jk. Clearly, M iscontained in the intersection. In order to prove the opposite inclusion, notice,first, that if a σ-polynomial belongs to Jk = [yk] + [yαy], then each its term is

either in [yk] or in [yαy]. Let A ∈∞⋂

k=1

Jk. Then we can write A as A = B + C

where B is the sum of all terms of A that lie in [yαy], and C is the sum of theremaining terms of A. Let m be a positive integer that exceeds the total degreeof C in the indeterminates y, αy, α2y, . . . . Since A ∈ Jm and B ∈ [yαy], C ∈ Jm

and no term of C belongs to [yαy]. Also, no term of C belongs to [ym], since mis greater than the total degree of C. Thus, C = 0 hence A = B ∈ M .

4. Consider two σ-polynomials, A = 1 + y(αy), B = y + αy ∈ Ky. As it isshown in Example 2.3.11, the σ-ideals [A] and [B] are separated, but not stronglyseparated. It follows that I = [A,B] and the inversive closure of the radical of Iare proper σ-ideals of Ky but I = Ky. Thus, I is not complete, hence itcannot be represented as an intersection of a finite family of complete σ-ideals(see Exercise 2.3.10)

Definition 2.3.17 A difference ideal I of a difference ring R is called inde-composible if it has no representation as an intersection of two proper differenceideals which properly contain I.

Example 2.2.7 shows that the ideal I = [y2 − 1] of the ring of differencepolynomials Ky over an ordinary difference field K is indecomposible. At thesame time, its perfect closure can be represented as an intersection of two perfectideals properly containing I. Indeed, by Theorem 2.3.3(v), y−1∩y+1 =y2 − 1. Another example of this kind is given in the following exercise.

Exercise 2.3.18 Let K be an ordinary difference field with a basic set σ = αand let Ky, z be the ring of σ-polynomials in two σ-indeterminates y and zover K. Prove that the yz = y ∩ z, but the σ-ideal [yz] of Ky, z isindecomposible. [Hint: Use the fact that every term of a σ-polynomial A ∈ [yz]has a factor of the form (αky)(αlz) where k, l ∈ N.]

Let R be a difference ring with a basic set σ. Then the set Φ of all properσ-ideals of R is not empty (it contains (0)). By the Zorn’s lemma, the set Φcontains maximal elements (with respect to inclusion) that are called σ-maximalideals of R. Clearly, a reflexive closure of a proper σ-ideal is a proper σ∗-ideal,hence every σ-maximal ideal is reflexive.

The next example shows that a σ-maximal ideal is not necessarily prime.

Example 2.3.19 Let R denote the polynomial ring Q[x] in one indeterminatex and let α be an automorphism of R such that α(x) = −x and α(a) = a forevery a ∈ Q. Treating R as an ordinary difference ring with the basic set σ = αone can easily check that M = (x2 − 1) is a σ-maximal ideal of R. Indeed, ifg ∈ R \M , then g can be written as g = ax+ b+m where a, b ∈ Q, a2 + b2 = 0,and m ∈ M . If b = 0, then (1/2b)(g + α(g)) ≡ 1 (modM), while if b = 0, a = 0,then (x/2a)(g − α(g)) ≡ x2 ≡ 1 (modM). In both cases, [M, g] = R.


At the same time, the ideal M is not prime: x + 1 /∈ M,x − 1 /∈ M , butx2 − 1 ∈ M .

The following notion is a difference analog of the concept of a prime ideal.

Definition 2.3.20 A difference ideal Q of a difference (σ-) ring R is calledσ-prime if for any two difference ideals I and J of the ring R, the inclusionIJ ⊆ Q implies I ⊆ Q or J ⊆ Q.

Exercises 2.3.21 Let R be a difference ring with a basic set σ.1. Prove that a σ-ideal Q of R is σ-prime if and only if for any two elements

a, b ∈ R, the inclusion [a][b] ⊆ Q implies a ∈ Q or b ∈ Q.2. Show that every maximal σ-ideal of R is σ-prime.3. Prove that the following statements about a reflexive σ-ideal I are

equivalent.(i) I is σ-prime and perfect.(ii) I is σ-prime and mixed.(iii) I is a prime reflexive difference ideal.4. Prove that the difference ideal M in Example 2.3.19 is not complete. Thus,

there are σ-prime ideals (and even maximal σ-ideals) that are not complete (inparticular, they are not mixed and, of course, not perfect).

The following example shows that not all perfect difference (σ-) ideals areσ-prime.

Example 2.3.22 Let R denote the polynomial ring Q[x, y] in two indetermi-nates x and y. Let α be an automorphism of R such that α(f(x, y)) = f(2x, 2y)for any polynomial f(x, y) ∈ R. Treating R as an ordinary difference ring withthe basic set σ = α, one can easily check that I = (xy) is a perfect σ-idealwhich is not σ-prime. (Indeed, [x][y] ⊆ I, but x /∈ I and y /∈ I.)

2.4 Autoreduced Sets of Differenceand Inversive Difference Polynomials.Characteristic Sets

Let K be a difference field with a basic set σ = α1, . . . , αn, T = Tσ, andR = Ky1, . . . , ys the algebra of difference polynomials in σ-indeterminatesy1, . . . , ys over K. Then R can be viewed as a polynomial ring in the set of inde-terminates TY = τyi|τ ∈ T, 1 ≤ i ≤ s over K (here and below we often writeτyi instead of τ(yi)). Elements of this set are called terms. If τ = αk1

1 . . . αknn ∈ T

(k1, . . . , kn ∈ N), then the number ord τ =n∑

ν=1

kν is called the order of τ . The

order ord u of a term u = τyi ∈ TY is defined as the order of τ . As usual, if

2.4. AUTOREDUCED SETS 129

τ, τ ′ ∈ T , we say that τ ′ divides τ (and write τ ′|τ) if τ = τ ′τ ′′ for some τ ′′ ∈ T .If u = τyi and v = τ ′yj are two terms in TY , we say that u divides v (and writeu|v) if i = j and τ |τ ′. In this case we also say that v is a transform of u.

By a ranking of the family of indeterminates y1, . . . , ys we mean a well-ordering ≤ of the set of terms TY that satisfies the following two conditions:

(i) u ≤ τu for any u ∈ TY, τ ∈ T . (We denote the order on TY by the usualsymbol ≤ and write u < v if u ≤ v and u = v.)

(ii) If u, v ∈ TY and u ≤ v, then τu ≤ τv for any τ ∈ T .

A ranking of the family y1, . . . , ys is also referred to as a ranking of the setof terms TY . It is said to be orderly if the inequality ord u < ord v (u, v ∈ TY )implies u < v. An important example of an orderly ranking is the standardranking defined as follows: u = αk1

1 . . . αknn yi ≤ v = αl1

1 . . . αlnn yj ∈ TY if and

only if the (n+2)-tuple (n∑

ν=1

kν , i, k1, . . . , kn) is less than or equal to the (n+2)-

tuple (n∑

ν=1

lν , j, l1, . . . , ln) with respect to the lexicographic order on Nn+2.

In what follows, we assume that an orderly ranking of TY is fixed.Let A ∈ Ky1, . . . , ys. The greatest (with respect to the given ranking)

element of TY that appears in the σ-polynomial A is called the leader of A; it is

denoted by uA. If A is written as a polynomial in uA, A =d∑

i=0

IiuiA (d = deguA

A

and the σ-polynomials I0, . . . , Id do not contain uA), then Id is called the initialof the σ-polynomial A; it is denoted by IA.

If A is a σ-polynomial in the ring Ky1, . . . , ys, then the order of its leaderuA is called the order of of the σ-polynomial A; it is denoted by ordA. If τyj

and τ ′yj (1 ≤ j ≤ s) are terms of the greatest and smallest orders, respectively,that appear in a σ-polynomial A and contain yj , then ord τyj is called the orderof A with respect to yj and denoted by ordyj

A. The difference ord τ − ord τ ′

is called the effective order of A with respect to yj ; it is denoted by EordyjA.

If no transforms of yj appear in A, both ordyjA and Eordyj

A are defined tobe 0.

If τ1yi and τ2yk (1 ≤ i, k ≤ s) are terms of the greatest and smallest ordersamong all terms in A, then ord τ1 − ord τ2 is called the effective order of A anddenoted by EordA. It is easy to see that for any τ ∈ T , Eord(τA) = EordAand Eordyj

(τA) = EordyjA (1 ≤ j ≤ s).

Let A and B be two σ-polynomials in Ky1, . . . , ys. We say that A has lowerrank than B (or A is less than B) and write A < B, if either A ∈ K, B /∈ K oruA < uB, or uA = uB = u and deguA < deguB. If neither A < B nor B < A, wesay that A and B have the same rank and write rk A = rk B. The σ-polynomialA is said to be reduced with respect to B if A does not contain any power of atransform τuB (τ ∈ Tσ) whose exponent is greater than or equal to deguB

B. IfΣ is any subset of Ky1, . . . , ys \ K, then a σ-polynomial A ∈ Ky1, . . . , ys,


is said to be reduced with respect to Σ if A is reduced with respect to everyelement of Σ.

A set Σ ⊆ Ky1, . . . , ys is called an autoreduced set if either Σ = ∅ orΣ⋂

K = ∅ and every element of Σ is reduced with respect to all other elementsof Σ. It is easy to see that distinct elements of an autoreduced set have distinctleaders. Furthermore, Lemma 1.5.1 shows that every autoreduced set is finite.Indeed, an infinite autoreduced set would contain an infinite sequence of σ-polynomials A1, A2, . . . such that uA1 |uA2 , uA2 |uA3 , . . . . This fact immediatelyleads to a contradiction, since we would have a decreasing sequence of naturalnumbers deguA1

A1 > deguA2A2 > . . . .

Theorem 2.4.1 Let A = A1, . . . , Ap be an autoreduced set in a ring of σ-polynomials Ky1, . . . , ys over a difference field K with a basic set σ. LetI(A) = B ∈ Ky1, . . . , ys| either B = 1 or B is a product of finitely manyσ-polynomials of the form τ(IAi

) (τ ∈ Tσ, i = 1, . . . , p). Then for any C ∈Ky1, . . . , ys, there exist σ-polynomials J ∈ I(A) and C0 ∈ Ky1, . . . , yssuch that C0 is reduced with respect to A and JC ≡ C0(mod [A]) (that is,JC − C0 ∈ [A]).

PROOF. If C is reduced with respect to A, the statement is obvious (onecan take C0 = C and J = 1). Therefore, we can assume that C is not reducedwith respect to A. Let ui, di, and Ii denote the leader of Ai, degui

Ai, and theinitial of Ai, respectively (i = 1, . . . , p). Then C contains some power (τui)k ofa term τui (τ ∈ T , 1 ≤ i ≤ p) such that k ≥ di. Such a term τui of the highestpossible rank will be called the A-leader of the σ-polynomial C.

Let Σ denote the set of all σ-polynomials C for which the statement ofthe theorem is false. Suppose that Σ = ∅, and let D be a σ-polynomial in Σwhose A-leader v has the lowest possible rank and whose degree d = degvD isthe lowest among all σ-polynomials in Σ with the A-leader v. Obviously, D canbe written as D = D1v

d + D2 where the σ-polynomial D1 does not contain v,and degvD2 < d. Furthermore, v = τui for some τ ∈ T (1 ≤ i ≤ p), v is theleader of the σ-polynomial τAi, degv(τAi) = degui

Ai = di, and IτAi= τIi.

Let us consider the σ-polynomial E = (τIi)D − vd−di(τAi)D1. It is easyto see that if E contains a term w such that v < w, then w appears in D.Furthermore, in this case degwD ≥ degwE.

Since degvE < d, E /∈ Σ, hence there exist σ-polynomials F and J1 suchthat F is reduced with respect to A, J1 ∈ I(A), and J1E ≡ F (mod [A]). Thus,J1((τIi)D − vd−di(τAi)D1) ≡ F (mod [A]), so that JD ≡ F (mod [A]) whereJ = J1(τIi) ∈ I(A). We have arrived at a contradiction which implies that theset Σ is empty. This completes the proof of the theorem.

The σ-polynomial C0 in the last theorem is called the remainder of the σ-polynomial C with respect to A. If A = A, we say that C0 is a remainder ofC with respect to the σ-polynomial A.

The reduction process, that is, a transition from a given σ-polynomial C toa σ-polynomial C0 satisfying the conditions of the theorem, can be performedin many ways. Let us describe one of them.


First, we exclude from C all powers of transforms of uA1 whose exponentsare greater than or equal to deguA1

A1. It can be done as follows. Suppose thatv = τuA1 is the greatest (with respect to the given ranking of TY ) transformof uA1 such that C contains powers of v whose exponents are greater than orequal to d1 = deguA1

A1. Let m be the greatest exponent of such a power of v.Then the σ-polynomial C can be written as C = Imvm + Im−1v

m−1 + · · · + I0

where the σ-polynomials I0, . . . , Im do not contain v and, moreover, if u is anytransform of uA1 such that v < u, then I0, . . . , Im contain no power of u whoseexponent is greater than or equal to d1. It follows that the σ-polynomial C ′ =τ(IA1)C − Imvm−d1τA1 can contain only those powers of v whose exponentsdo not exceed m − 1. It is also clear that if u is any transform of uA1 andv < u, then C ′ contains no power of u whose exponent is greater than orequal to d1. Furthermore, C ′ ≡ C(mod [A]). Applying the same procedure to C ′

instead of C and continuing this process, we obtain a σ-polynomial C such thatC ≡ C(mod [A]), degvC < d1, and if u is any transform of uA1 such that v < u,then C contains no power of u whose exponent is greater than or equal to d1.

Let w be the greatest transform of uA1 in C such that degwC ≥ d1. It is clearthat w < v. Repeating the foregoing procedure we obtain a σ-polynomial C1

such that C1 ≡ C(mod [A]) and if u is any transform of uA1 , then C1 containsno power of u whose exponent is greater than or equal to d1.

At the next stage we repeat the same procedure to exclude from C1 all powersof transforms of uA2 whose exponents are greater than or equal to d2 = deguA2

.Since A1 and A2 are reduced with respect to each other, the σ-polynomial C2

obtained at this stage satisfies the condition C2 ≡ C(mod [A]) and contains nopowers of transforms of uAi

whose exponents are greater than or equal to di

(i = 1, 2).Subsequently applying this procedure to the polynomials A3, . . . , Ap we ar-

rive at a σ-polynomial C0 that satisfies conditions of the theorem.If C0 is a σ-polynomial that satisfies conditions of Theorem 2.4.1 (with a

given σ-polynomial C), we say that C reduces to C0 modulo A.In what follows, the elements of an autoreduced set are always written in

the order of increasing rank. (Thus, if A = A1, . . . , Ap is an autoreduced setin Fy1, . . . , ys, we assume that A1 < · · · < Ap.)

Definition 2.4.2 Let A = A1, . . . , Ap and B = B1, . . . , Bq be two autore-duced sets in the algebra of difference polynomials Fy1, . . . , ys. We say that Ahas lower rank than B and write rkA < rk B if one of the following conditionsholds:

(i) there exists k ∈ N, 1 ≤ k ≤ minp, q, such that rk Ai = rk Bi fori = 1, . . . , k − 1 and Ak < Bk;

(ii) p > q and rk Ai = rk Bi for i = 1, . . . , q.

Theorem 2.4.3 In every nonempty set of autoreduced subsets of Ky1, . . . , ysthere exists an autoreduced set of lowest rank.


PROOF. Let Φ be a nonempty set of autoreduced subsets of Ky1, . . . , ys.Let Φ0,Φ1, . . . be a sequence of subsets of Φ such that Φ0 = Φ and for every i =1, 2, . . . , Φi = A = A1, . . . , Aj ∈ Φi−1 | j ≥ i and Ai is of the lowest possiblerank. Obviously, Φ0 ⊇ Φ1 ⊇ . . . , and the ith σ-polynomials of autoreducedsets in Φi have the same leader vi. If Φi = ∅ for i = 1, 2, . . . , then v1, v2, . . . isan infinite sequence of terms such that no vi is a transform of any other vj . Onthe other hand, the existence of such a sequence of terms contradicts Lemma1.5.1, whence there is the smallest i > 0 such that Φi = ∅. It is easy to see thatany element of the nonempty set Φi−1 is an autoreduced set in Φ of lowest rank.

If J is a nonempty subset (in particular, an ideal) of the ring Ky1, . . . , ys,then the family of all autoreduced subsets of J is not empty (the empty setis autoreduced; also if 0 = A ∈ J , then A = A is an autoreduced set). Itfollows from the last theorem that J contains an autoreduced subset of lowestrank. Such a subset is called a characteristic set of J . The following propositiondescribes some properties of characteristic sets of difference polynomials.

Proposition 2.4.4 Let K be a difference field with a basic set σ, J a differenceideal of the algebra of σ-polynomials Ky1, . . . , ys, and Σ a characteristic setof J . Furthermore, let I =

∏A∈Σ

IA. Then

(i) The σ-ideal J does not contain nonzero difference polynomials reducedwith respect to Σ. In particular, if A ∈ Σ, then IA /∈ J .

(ii) If the σ-ideal J is prime and reflexive, then J = [Σ] : Λ(Σ) where Λ(Σ) is thefree commutative (multiplicative) semigroup generated by the set τ(I)|τ ∈ Tσ.

PROOF. If a nonzero σ-polynomial B ∈ J is reduced with respect to Σ,then B and the set A ∈ Σ |uA < uB form an autoreduced subset of J whoserank is lower than the rank of Σ. This contradiction with the choice of Σ provesstatement (i).

In order to prove the second statement, consider a σ-polynomial C ∈ [Σ] :Λ(Σ). By the definition of the last set, there exists a σ-polynomial D ∈ Λ(Σ)such that CD ∈ [Σ] ⊆ J . Since the ideal J is prime and D /∈ J (otherwiseIA ∈ J for some A ∈ Σ that contradicts statement (i) ), we have C ∈ J .Thus, [Σ] : Λ(Σ) ⊆ J . Conversely, let C ∈ J . By Theorem 2.4.1, there existσ-polynomials E ∈ Λ(Σ) and F such that F is reduced with respect to Σ andEC ≡ F (mod[Σ]). It follows that F ∈ J , hence F = 0 (by the first part of theproposition) and C ∈ [Σ] : Λ(Σ).

Let K be an inversive difference field with a basic set σ = α1, . . . , αn andΓ = Γσ the free commutative group generated by σ. Let Z− denote the setof all nonpositive integers and let Z(n)

1 ,Z(n)2 , . . . ,Z(n)

2n be all distinct Cartesianproducts of n factors each of which is either N or Z− (we assume that Z1 = N).As in Section 1.5, these sets will be called ortants of Zn. For any j = 1, . . . , 2n,


we set Γj = γ = αk11 . . . αkn

n ∈ Γ|(k1, . . . , kn) ∈ Z(n)j . Furthermore, if γ =

αk11 . . . αkn

n ∈ Γ, then the number ord γ =n∑

i=1

|ki| will be called the order of γ.

Let Ky1, . . . , ys∗ be the algebra of σ∗-polynomials in σ∗-indetermiantesy1, . . . , ys over K and let Y denote the set γyi|γ ∈ Γ, 1 ≤ i ≤ s whoseelements are called terms (here and below we often write γyi for γ(yi)). Bythe order of a term u = γyj we mean the order of the element γ ∈ Γ. SettingYj = γyi|γ ∈ Γj , 1 ≤ i ≤ s (j = 1, . . . , 2n) we obtain a representation of the

set of terms as a union Y =2n⋃

j=1

Yj .

Definition 2.4.5 A term v ∈ Y is called a transform of a term u ∈ Y if andonly if u and v belong to the same set Yj (1 ≤ j ≤ 2n) and v = γu for someγ ∈ Γj. If γ = 1, v is said to be a proper transform of u.

Definition 2.4.6 A well-ordering of the set of terms Y is called a ranking of thefamily of σ∗-indeterminates y1, . . . , ys (or a ranking of the set Y ) if it satisfiesthe following conditions. (We use the standard symbol ≤ for the ranking; it willbe always clear what order is denoted by this symbol.)

(i) If u ∈ Yj and γ ∈ Γj (1 ≤ j ≤ 2n), then u ≤ γu.(ii) If u, v ∈ Yj (1 ≤ j ≤ 2n), u ≤ v and γ ∈ Γj, then γu ≤ γv.

A ranking of the σ∗-indeterminates y1, . . . , ys is is called orderly if for anyj = 1, . . . , 2n and for any two terms u, v ∈ Yj , the inequality ord u < ord vimplies that u < v (as usual, v < w means v ≤ w and v = w). As an exampleof an orderly ranking of the σ∗-indeterminates y1, . . . , ys one can consider thestandard ranking defined as follows: u = αk1

1 . . . αknn yi ≤ v = αl1

1 . . . αlnn yj if

and only if the (2n+2)-tuple (n∑

ν=1

|kν |, |k1|, . . . , |kn|, k1, . . . , kn, i) is less than or

equal to the (2n+2)-tuple (n∑

ν=1

|lν |, |l1|, . . . , |ln|, l1, . . . , ln, j) with respect to the

lexicographic order on Zn+2.In what follows, we assume that an orderly ranking ≤ of the set of σ∗-

indeterminates y1, . . . , ys is fixed. If A ∈ Ky1, . . . , ys∗, then the greatest (withrespect to the ranking ≤) term from Y that appears in A is called the leader ofA; it is denoted by uA. If d = deguA, then the σ∗-polynomial A can be writtenas A = Idu

d + Id−1ud−1 + · · · + I0 where Ik(0 ≤ k ≤ d) do not contain u. The

σ∗-polynomial Id is called the initial of A; it is denoted by IA.The ranking of the set of σ∗-indeterminates y1, . . . , ys generates the following

relation on Ky1, . . . , ys∗. If A and B are two σ∗-polynomials, then A is saidto have rank less than B (we write A < B) if either A ∈ K,B /∈ K or A,B ∈Ky1, . . . , ys∗ \ K and uA < uB , or uA = uB = u and deguA < deguB. IfuA = uB = u and deguA = deguB, we say that A and B are of the same rankand write rk A = rk B.


Let A,B ∈ Ky1, . . . , ys∗. The σ∗-polynomial A is said to be reduced withrespect to B if A does not contain any power of a transform γuB (γ ∈ Γσ) whoseexponent is greater than or equal to deguB

B. If Σ ⊆ Ky1, . . . , ys \ K, then aσ-polynomial A ∈ Ky1, . . . , ys, is said to be reduced with respect to Σ if A isreduced with respect to every element of the set Σ.

A set Σ ⊆ Ky1, . . . , ys∗ is said to be autoreduced if either it is empty orΣ⋂

F = ∅ and every element of Σ is reduced with respect to all other elementsof Σ. As in the case of σ-polynomials, distinct elements of an autoreduced sethave distinct leaders and every autoreduced set is finite. The following statementis an analog of Theorem 2.4.1.

Theorem 2.4.7 Let A = A1, . . . , Ar be an autoreduced subset in the ringKy1, . . . , ys∗ and let D ∈ Ky1, . . . , ys∗. Furthermore, let I(A) denote the setof all σ∗-polynomials B ∈ Ky1, . . . , ys such that either B = 1 or B is a productof finitely many polynomials of the form γ(IAi

) where γ ∈ Γσ, i = 1, . . . , r.Then there exist σ-polynomials J ∈ I(A) and D0 ∈ Ky1, . . . , ys such that D0

is reduced with respect to A and JD ≡ D0(mod [A]∗).

The proof of this result and the process of reduction of inversive differ-ence polynomials with respect to an autoreduced set of such polynomials aresimilar to the proof of Theorem 2.4.1 and the corresponding procedure for σ-polynomials described after that theorem. We leave the detail proof of Theorem2.4.7 to the reader as an exercise.

The transition from a σ∗-polynomial D to the σ∗-polynomial D0 (called aremainder of D with respect to A) can be performed in the same way as inthe case of σ-polynomials (see the description of the corresponding reductionprocess after Theorem 2.4.1). We say that D reduces to D0 modulo A.

As in the case of σ-polynomials, the elements of an autoreduced set inFy1, . . . , ys∗ will be always written in the order of increasing rank. If A =A1, . . . , Ar and B = B1, . . . , Bs are two autoreduced sets of σ∗-polynomials,we say that A has lower rank than B and write rkA < rk B if either there ex-ists k ∈ N, 1 ≤ k ≤ minr, s, such that rk Ai = rk Bi for i = 1, . . . , k − 1 andAk < Bk, or r > s and rk Ai = rk Bi for i = 1, . . . , s.

Repeating the proof of Theorem 2.4.3, one obtains that every family of au-toreduced subsets of Ky1, . . . , ys∗ contains an autoreduced set of lowest rank.In particular, if ∅ = J ⊆ Fy1, . . . , ys∗, then the set J contains an autoreducedset of lowest rank called a characteristic set of J . The following result is the ver-sion of Proposition 2.4.4 for inversive difference polynomials. It can be provedin the same way as the statement for difference polynomials.

Proposition 2.4.8 Let K be an inversive difference field with a basic set σ, J aσ∗-ideal of the algebra of σ-polynomials Ky1, . . . , ys∗, and Σ a characteristicset of J . Then

(i) The ideal J does not contain nonzero σ∗-polynomials reduced with respectto Σ. In particular, if A ∈ Σ, then IA /∈ J .

(ii) If J is a prime σ∗-ideal, then J = [Σ] : Υ(Σ) where Υ(Σ) denotes the setof all finite products of elements of the form γ(IA) (γ ∈ Γσ, A ∈ Σ).


Let K be a difference field with a basic set σ and Ky1, . . . , ys the al-gebra of σ-polynomials in σ-indeterminates y1, . . . , ys over K. A σ-ideal I ofKy1, . . . , ys is called linear if it is generated (as a σ-ideal) by homogeneous

linear σ-polynomials, that is, σ-polynomials of the formm∑

i=1

aiτiykiwhere ai ∈

K, τi ∈ Tσ, 1 ≤ ki ≤ s for i = 1, . . . , m. If the σ-field K is inversive, then a σ∗-ideal of an algebra of σ∗-polynomials Ky1, . . . , ys∗ is called linear if it is gener-ated (as a σ∗-ideal) by homogeneous linear σ∗-polynomials, i.e., σ∗-polynomials

of the formm∑

i=1

aiγiyki(ai ∈ K, γi ∈ Γσ, 1 ≤ ki ≤ s for i = 1, . . . , m).

Proposition 2.4.9 Let K be a difference field with a basic set σ. Thenevery proper linear difference ideal of an algebra of difference polynomialsKy1, . . . , ys is prime. Similarly, if the σ-field K is inversive, then everyproper linear σ∗-ideal of an algebra of σ∗-polynomials Ky1, . . . , ys∗ isprime.

PROOF. We shall prove that if R = K[x1, x2, . . . ] is a ring of polynomials ina countable set of indeterminates x1, x2, . . . over a field K, then a proper ideal Lgenerated by a set of linear polynomials F = fi|i ∈ I is prime. (It is easy to seethat this result implies both statements of our proposition.) Indeed, if the idealL is not prime, then there exist two polynomials g, h ∈ R such that g /∈ L, h /∈ L,but gh ∈ L, that is, gh =

∑mk=1 λkfk for some polynomials λ1, . . . , λm ∈ R and

f1, . . . , fm ∈ F . Let xj | j ∈ J be the finite set of all indeterminates x1, x2, . . .that appear in polynomials g, h, fk, and λk (1 ≤ k ≤ m). Then the ideal of thepolynomial ring R′ = K[xj | j ∈ J] generated by f1, . . . , fm is not prime. Onthe other hand, if an ideal P of a polynomial ring A = K[y1, . . . , yn] in finitelymany indeterminates y1, . . . , yn over a field K is generated by linear polynomialsf1, . . . , fr, then P is prime. Indeed, without lost of generalization, we can assumethat f1 = y1 + a12y2 + . . . , a1nyn, f2 = yi2 + a2,i2+1yi2+1 + . . . a2nyn,. . . , fr =yir

+ ar,ir+1yir+1 + . . . arnyn where 1 < i2 < · · · < ir ≤ n (such a set ofgenerators can be obtained from any set of linear generators of P by the standardprocess of Gauss elimination). If g1g2 ∈ P , but gj ∈ A \ P (j = 1, 2), then onecan successively divide gj by f1 with respect to y1, then divide the obtainedremainder by f2 with respect to yi2 , etc. This divisions result in two polynomialsh1 and h2 such that gj−hj ∈ P and hj does not contain y1, yi2 , . . . , yir

(j = 1, 2).It follows that the element h1h2 does not contain this indeterminates as well.However, h1h2, as an element of P , is a linear combination of f1, . . . , fr withcoefficients in A and therefore should contain at least one of y1, yi2 , . . . , yir

.Thus, the ideal P is prime and this completes the proof of our proposition.

Definition 2.4.10 Let K be a difference field with a basic set σ and A an au-toreduced set in Ky1, . . . , ys that consists of linear σ-polynomials (respectively,let K be a σ∗-field and A an autoreduced set in Ky1, . . . , ys∗ that consists oflinear σ∗-polynomials). The set A is called coherent if the following two con-ditions hold.


(i) If A ∈ A and τ ∈ Tσ (respectively, γ ∈ Γσ), then τA (respectively, γA)reduces to zero modulo A.

(ii) If A,B ∈ A and v = τ1uA = τ2uB is a common transform of the leadersuA and uB (τ1, τ2 ∈ Tσ or τ1, τ2 ∈ Γσ if we consider the case of σ∗-polynomials),then the σ-polynomial (τ2IB)(τ1A) − (τ1IA)(τ2B) reduces to zero modulo A.

Theorem 2.4.11 Let K be a difference field with a basic set σ and J a linearσ-ideal of the algebra of σ-polynomials Ky1, . . . , ys (respectively, let K be aσ∗-field and J a linear σ∗-ideal of Ky1, . . . , ys∗). Then any characteristic setof J is a coherent autoreduced set of linear σ- (respectively, σ∗-) polynomials.

Conversely, if A ⊆ Ky1, . . . , ys (respectively, A ⊆ Ky1, . . . , ys∗) is anycoherent autoreduced set consisting of linear σ- (respectively, σ∗-) polynomials,then A is a characteristic set of the linear σ-ideal [A] (respectively, of the linearσ∗-ideal [A]∗).

PROOF. We shall prove the statement for σ-polynomials. The proof in thecase of inversive difference polynomials is similar.

Let Σ be a characteristic set of a linear σ-ideal J of Ky1, . . . , ys. By Propo-sition 2.4.4, J contains no nonzero σ-polynomials reduced with respect to Σ,hence Σ is a coherent autoreduced subset of J .

Conversely, let A be a coherent autoreduced set in Ky1, . . . , ys consistingof linear σ-polynomials. In order to prove that A is a characteristic set of theideal J = [A] we shall show that J contains no nonzero σ-polynomial reducedwith respect to A.

Suppose that a nonzeroσ-polynomialB ∈ J is reducedwith respect toA. Since

J = [A], B =p∑

i=1

CiτiAi for some σ-polynomials Ai ∈ A Ci ∈ Ky1, . . . , ys,and for some τi ∈ Tσ (1 ≤ i ≤ n). Obviously, for any i = 1, . . . , n, τiuAi

isthe leader of τiAi. (In the case of σ∗-polynomials (when τi ∈ Γσ) one can alsoassume τiuAi

= uτiAiwithout loss of generality; it follows from the fact that A is

coherent.) Let v be the greatest term in the set τ1uA1 , . . . , τpuAp. Without loss

of generality we can assume that there exists some positive integer q, 2 ≤ q ≤ p,such that τiuAi

< v for i = 1, . . . , q − 1 and τiuAi= v if q ≤ i ≤ p. Denoting the

coefficient of the leader v = τpuApof the σ-polynomial τpAp by ap, we can write

apB =q−1∑i=1

apCiτiAi +p−1∑i=q

Ci(apτiAi − (τiIAi)τpAp) +

p∑i=q

(τiIAi)CiτpAp

where the second sum is a σ-polynomial free of v. Since the set A is coherent,each σ-polynomial apτiAi − τiIAi

τpAp (q ≤ i ≤ p − 1) can be reduced to zero.Therefore, the last equality implies that apB can be written as

apB =p1∑

i=1

C(1)i τ

(1)i A

(1)i + C(1)

p τpAp (2.4.1)


where C(1)i ∈ Ky1, . . . , ys, τ

(1)i ∈ Tσ, A

(1)i ∈ A (1 ≤ i ≤ p1), C

(1)p ∈

Ky1, . . . , ys, and uτ(1)i

A(1)i = τ

(1)i u

A(1)i

< v = τpuAp. Since B is reduced

with respect to A, B is free of the leader uτpAp, hence this leader appears in

some σ-polynomials C(1)i (1 ≤ i ≤ p1). Therefore, there exist σ-polynomials

C(2)1 , . . . , C

(2)p1 reduced with respect to A and an integer k ∈ N, k ≥ 1, such that

akpC

(1)i ≡ C

(2)i (mod(τpAp) ) (i = 1, . . . , p1).

Multiplying both sides of (2.4.1) by akp we obtain that

ak+1p B ≡

p1∑i=1

C(2)i τ

(1)i A

(1)i (mod(τpAp)).

Moreover, since B and each C(2)i τ

(1)i A

(1)i (1 ≤ i ≤ p1) are free of τpuAp

, weactually have the equality

ak+1p B =

p1∑i=1

C(2)i τ

(1)i A

(1)i .

Let v1 be the greatest term in the set τ1uA(1)1

, . . . , τp1uA(1)p1. Obviously,

v1 < v and the σ-polynomial B1 =p1∑

i=1

C(2)i τ

(1)i A

(1)i is reduced with respect to

A. Therefore, one can apply the previous arguments and obtain a σ-polynomial

B2 =p2∑

i=1

C(3)i τ

(2)i A

(2)i such that B2 is reduced with respect to A, A

(2)i ∈ A,

τ(2)i ∈ Tσ, C

(3)i ∈ Ky1, . . . , ys, and u

τ(2)i A

(2)i

= τ(2)i u

A(2)i

< v1 for i = 1, . . . , p2

(p2 is some positive integer). Continuing in the same way, we obtain elementsC ∈ Ky1, . . . , ys, τ ∈ Tσ, and A ∈ A such that the σ-polynomial B = C(τA)is reduced with respect to A. On the other hand, uτA = τuA, so B cannot bereduced with respect to A. This contradiction shows that A is a characteristicset of ideal [A].

Corollary 2.4.12 Let K be an inversive difference field with a basic set σ andlet be a preorder on Ky1, . . . , ys∗ such that A1 A2 if and only if uA2 isa transform of uA1 (in the sense of Definition 2.4.5). Furthermore, let A be alinear σ∗-polynomial in Ky1, . . . , ys∗ \ K and ΓσA = γA|γ ∈ Γσ. Then theset of all minimal (with respect to ) elements of ΓσA is a characteristic set ofthe σ∗-ideal [A]∗.

PROOF. Let A be the set of all minimal (with respect to ) elements ofthe set γA | γ ∈ Γ. It is easy to check that A is an autoreduced coherent set(we leave the verification of the conditions of Definition 2.4.10 to the readeras an exercise). By Theorem 2.4.11, A is a characteristic set of the σ∗-ideal[A]∗.


Theorem 2.4.11 implies the following method of constructing a characteristicset of a proper linear σ∗-ideal I in the ring of σ∗-polynomials Ky1, . . . , ys∗(a similar method can be used for building a characteristic set of a linear σ-ideal in the ring of difference polynomials Ky1, . . . , ys). Suppose that I =[A1, . . . , Ap]∗ where A1, . . . , Ap are linear σ∗-polynomials and A1 < · · · < Ap.It follows from Theorem 2.4.11 that one should find a coherent autoreduced setΦ ⊆ Ky1, . . . , ys∗ such that [Φ]∗ = I. Such a set can be obtained from the setA = A1, . . . , Ap via the following two-step procedure.

Step 1. Constructing an autoreduced set Σ ⊆ I such that [Σ]∗ = I.If A is autoreduced, set Σ = A. If A is not autoreduced, choose the smallest

i (1 ≤ i ≤ p) such that some σ∗-polynomial Aj , 1 ≤ i < j ≤ p, is not reducedwith respect to Ai. Replace Aj by its remainder with respect to Ai (obtained bythe procedure described after Theorem 2.4.1) and arrange the σ∗-polynomialsof the new set A1 in ascending order. Then apply the same procedure to theset A1 and so on. After each iteration the number of σ∗-polynomials in the setdoes not increase, one of them is replaced by a σ∗-polynomial of lower or equalrank, and the others do not change. Therefore, the process terminates after afinite number of steps when we obtain a desired autoreduced set Σ.

Step 2. Constructing a coherent autoreduced set Φ ⊆ I.Let Σ0 = Σ be an autoreduced subset of I such that [Σ]∗ = I. If Σ is

not coherent, we build a new autoreduced set Σ1 ⊆ I by adding to Σ0 newσ∗-polynomials of the following types.

(a) σ∗-polynomials (γ1IB1)γ2B2 − (γ2IB2)γ1B1 constructed for every pairB1, B2 ∈ Σ such that the leaders uB1 and uB1 have a common transformv = γ1uB1 = γ2uB2 and (γ1IB1)γ2B2 − (γ2IB2)γ1B1 is not reducible to zeromodulo Σ0.

(b) σ∗-polynomials of the form γA (γ ∈ Γσ, A ∈ Σ0) that are not reducibleto zero modulo Σ0.

It is clear that rk Σ1 < rk Σ0. Applying the same procedure to Σ1 andcontinuing in the same way, we obtain autoreduced subsets Σ0,Σ1, . . . of Isuch that rk Σi+1 < rk Σi for i = 0, 1, . . . . Obviously, the process terminatesafter finitely many steps, so we obtain an autoreduced set Φ ⊆ I such thatΦ = Σk = Σk+1 = . . . for some k ∈ N. It is easy to see that Φ is coherent, so itis a characteristic set of the ideal I.

We conclude this section with two helpful results on difference and inversivedifference polynomials.

Theorem 2.4.13 Let K be an inversive difference field with a basic set σ =α1, . . . , αn, Ky1, . . . , ys∗ the ring of σ∗-polynomials in σ∗-indeterminatesy1, . . . , ys over K, and A an irreducible σ∗-polynomial in Ky1, . . . , ys∗ \ K. Fur-thermore, let M be a nonzero σ∗-polynomial in the ideal [A]∗ of Ky1, . . . , ys∗

written in the form M =l∑

i=1

CiAi (l ≥ 1) where Ci ∈ Ky1, . . . , ys∗ (1 ≤ i ≤ l)

and Ai = γiA for some distinct elements γ1, . . . , γl ∈ Γ. Finally, let ui denote


the leader of the σ∗-polynomial Ai (i = 1, . . . , l). Then there exists ν ∈ N,1 ≤ ν ≤ l, such that deguν

M ≥ deguνAν .

PROOF. Suppose, that the statement of the theorem is not true. Let m bethe minimal positive integer for which there exists a σ∗-polynomial

M = C1A1 + · · · + CmAm (2.4.2)

in the σ∗-ideal [A]∗ such that deguiM < degui

Ai for i = 1, . . . , m. (Ci ∈Ky1, . . . , ys∗, Ai = βiA for some distinct elements β1, . . . , βm ∈ Γ, and ui

is the leader of Ai, 1 ≤ i ≤ m.) Note that all the leaders ui are distinct. Indeed,suppose that the σ∗-polynomials βiA and βjA (1 ≤ i, j ≤ m, βi = βj) havethe same leader u = γyk (γ ∈ Γ, 1 ≤ k ≤ s). Suppose that γ belongs to aset Γq (1 ≤ q ≤ 2n) in the considered above representation Γ =

⋃2n

i=1 Γi. It iseasy to see that one can choose λ, µ ∈ Γq such that all terms of λA belong toYq (we use the notation of Definition 2.4.5) and µβi, µβj ∈ Γq. Then the σ∗-polynomials (µβi)(λA) and (µβj)(λA) have the same leader µλu. On the otherhand, since all terms of λA, including the leader v of this σ∗-polynomial, lie inYq and µβi, µβj ∈ Γq, µβiv is the leader of (µβi)(λA) and µβjv is a leader of(µβj)(λA). Therefore, µβiv = µβjv that contradicts the fact that βi = βj .

Thus, we may assume that u1 < · · · < um. Furthermore, every σ∗-polynomialAi (1 ≤ i ≤ m) can be written as Ai = Iiu

dii + o(udi

i ) where di = deguiAi, Ii is

the initial of Ai, and o(udii ) is a σ∗-polynomial such that degui

o(udii ) < di.

Since ui < um for i = 1, . . . , m−1, the σ∗-polynomials A1, . . . , Am−1 do notcontain term um. Furthermore, without loss of generality we may assume thatAm does not divide Ci for i = 1, . . . , m−1 (otherwise, one can regroup terms in(2.4.2) and obtain a representation of M as a sum of fewer than m terms withthe same property; the existence of such a representation would contradict theminimality of m).

Since Am is irreducible, the resultant of the polynomials Ci and Am withrespect to um is a nonzero polynomial that does not contain um. Thus, forevery i = 1, . . . , m − 1, there exists a number qi ∈ N and a σ∗-polynomialC ′

i ∈ Ky1, . . . , ys∗ which is reduced with respect to Am and satisfies thecondition

IqimCi ≡ C ′

i (mod (Am)) (2.4.3)

Therefore, one can multiply both sides of equality (2.4.2) by a sufficientlybig power Iq

m and obtain that

IqmM ≡ C ′

1A1 + · · · + C ′m−1Am−1 (mod (Am)) (2.4.4)

for some σ∗-polynomials C ′1, . . . , C

′m−1 reduced with respect to Am. Since um is

not contained in Im and degumM < degum

Am, we arrive at the equality

IqmM = C ′

1A1 + · · · + C ′m−1Am−1. (2.4.5)

If the σ∗-polynomial Im does not contain u1, . . . , um−1, then degui(Iq

mM) < di

(1 ≤ i ≤ m − 1) that contradicts the choice of M . Therefore, Im contains


some of the leaders u1, . . . , um−1. Let ur be the greatest such a leader. Sincethe σ∗-polynomial A is irreducible and deg Im < deg A, the σ∗-polynomialsIqm and Ar are relatively prime, hence the resultant R1 of these polynomials

with respect to the indeterminate ur is distinct from zero and does not containur. Furthermore, the resultant can be written as R1 = P1I

qm + Q1Ar where

P1, Q1 ∈ Ky1, . . . , ys∗.Let up be the greatest of the leaders u1, . . . , ur contained in R1. Since R1

contains a term v, which does not appear in Am or Ar (v can be chosen asthe lowest term contained in Ap), the σ∗-polynomials Ap and R1 are relativelyprime. The resultant R2 of these σ∗-polynomials a with respect to the term up

can be written as

R2 = P2R1 +Q2Ap = P2(P1Iqm +Q1Ar)+Q2Ap (P2, Q2 ∈ Ky1, . . . , ys∗).

Clearly, R2 = 0 and this resultant does not contain up, up+1, . . . , um. Continuingin the same way we obtain a σ∗-polynomial R = T0I

qm + T1Ar1 + · · · + TlArl

which does not contain u1, . . . , um (T0, . . . , Tl ∈ Ky1, . . . , ys∗, r1 = r > r2 =p > · · · > rl). Multiplying both sides of equation (2.4.5) by T0, we obtain that

T0IqmM = (R − T1Ar1 − · · · − TlArl

)M = T0C′1A1 + · · · + T0C

′m−1Am−1,

so thatRM = D1A1 + · · · + Dm−1Am−1

for some σ∗-polynomials D1, . . . , Dm−1, and

degui(RM) = degui

M < di, 1 ≤ i ≤ m − 1.

We have arrived at the contradiction with the minimality of the length of rep-resentation (2.4.2). The theorem is proved.

Let Ky1, . . . , ys∗ be the algebra of σ∗-polynomials in σ∗-indeterminatesy1, . . . , ys over an inversive difference (σ∗-) field K. Then every σ∗-polynomialA ∈ Ky1, . . . , ys∗ \ K has an irreducible factor that generates a σ∗-idealcontaining [A]∗. This simple observation leads to the following consequence ofTheorem 2.4.13.

Corollary 2.4.14 Let K be an inversive difference field with a basic set σ,Ky1, . . . , ys∗ the ring of σ∗-polynomials in σ∗-indeterminates y1, . . . , ys overK, and A an irreducible σ∗-polynomial in Ky1, . . . , ys∗ \K. Then 1 /∈ [A]∗.

It is easy to see that the arguments of the proof of Theorem 2.4.13 can beapplied to the case of difference polynomials over a difference field. (In this caseone even does not have to justify the fact that the leaders ui of the σ-polynomialsAi in (2.4.2) are distinct; this is obvious.) As a result we obtain the followingstatement.

Theorem 2.4.15 Let K be a difference field with a basic set σ, Ky1, . . . , ysthe ring of σ-polynomials in σ-indeterminates y1, . . . , ys over K, and A an ir-reducible σ-polynomial in Ky1, . . . , ys \ K. Let M be a nonzero σ-polynomial

2.5. RITT DIFFERENCE RINGS 141

in the ideal [A] of Ky1, . . . , ys written in the form M =l∑

i=1

CiAi (l ≥ 1)

where Ci ∈ Ky1, . . . , ys (1 ≤ i ≤ l) and Ai = τiA for some distinct elementsτ1, . . . , τl ∈ T . Furthermore, let ui denote the leader of the σ-polynomial Ai (i =1, . . . , l). Then there exists ν ∈ N, 1 ≤ ν ≤ l, such that deguν

M ≥ deguνAν .

Corollary 2.4.16 Let K be a difference field with a basic set σ, Ky1, . . . , ysthe ring of σ-polynomials in σ-indeterminates y1, . . . , ys over K, and A anirreducible σ-polynomial in Ky1, . . . , ys \ K. Then 1 /∈ [A].

With the assumptions of the last corollary, one can easily see that 1 /∈ (A)1where (A)1 = f ∈ Ky1, . . . , ys | τ1(f)k1 . . . τr(f)kr ∈ [A] for some τ1, . . . , τr ∈Tσ. (We used this notation in the description of the process of shuffling con-sidered after Definition 2.3.1.) On the other hand, if K is a partial differencefield, it is possible that A is an irreducible σ-polynomial in Ky1, . . . , ys \ Kand 1 ∈ A (see Example 7.3.1 below).

2.5 Ritt Difference Rings

Let R be a difference ring with a basic set σ and let Tσ be the free commutativesemigroup generated by σ. As before, the perfect closure of a set S ⊆ R isdenoted by S, while Sk denotes the set obtained at the k-th step of theprocess of its construction described at the beginning of section 2.3. (Recallthat for any M ⊆ R, we consider the set M ′ = a ∈ R|τ1(a)k1 . . . τr(a)kr ∈ Mfor some τi ∈ Tσ, ki,∈ N (1 ≤ i ≤ r) and define S0 = S, Sk = [Sk−1]′ fork = 1, 2, . . . .)

Definition 2.5.1 Let J be a subset of a difference ring R. A finite subset S ofJ is called a basis of J if S = J. If J = Sm for some m ∈ N, S is said tobe an m-basis of J . A difference ring in which every subset has a basis is calleda Ritt difference ring.

Proposition 2.5.2 A difference ring R is a Ritt difference ring if and only ifevery perfect difference ideal of R has a basis. If every perfect difference idealof R has an m-basis, then every set in R has an m-basis. (In this and similarstatements the number m is not fixed but depends on the set.)

PROOF. Obviously, one should just prove that if every perfect differenceideal has a basis (m-basis), then every subset of R has a basis (respectively,m-basis). Note, first, that if a difference (σ-) ring R contains a set without basis(m-basis), then R contains a maximal set without basis (m-basis). Indeed, letSλ|λ ∈ Λ be a linearly ordered family of subsets of R such that no Sλ has abasis (m-basis), and let S =

⋃λ∈Λ Sλ. If S has a basis (m-basis) B, then B is

a finite subset of some Sν (ν ∈ Λ). Therefore, B ⊆ Sν ⊆ S = B (orBm ⊆ (Sν)m ⊆ Sν ⊆ S = Bm if B is an m-basis for S) that contradicts thefact that Sν has no basis. Now, it remains to apply the Zorn’s lemma.


Suppose that every perfect difference ideal of R has a basis, but R is not aRitt difference ring. Then R should contain a maximal subset S that does nothave a basis. Let J = [S]∗ be the reflexive closure of the σ-ideal [S]. It is easy tosee that J does not have a basis. Indeed, if B is a basis of J , then there existsτ ∈ Tσ such that τ(B) ⊆ [S]. It follows that there exists a finite set C ⊆ S suchthat every element of τ(B) is a finite linear combination of elements of C andtheir transforms. In this case, C = S that contradicts the choice of S.

Since S is a maximal set without a basis, S = J , that is, S is a σ∗-ideal of R.We are going to show that this ideal is prime. Let ab ∈ S, but a /∈ S, b /∈ S. Bythe maximality of S, there exists a finite set B0 ⊆ S such that B0, a = S, aand B0, b = S, b. Using Theorem 2.3.3(v) we obtain that S ⊆ B0, a ∩B0, b ⊆ B0, ab ⊆ S hence B0, ab is a basis of S, contrary to the choiceof S. Thus, S is a prime σ∗-ideal of R. In particular, the difference ideal S isperfect that contradicts our assumption on the ring R.

In order to prove the statement about m-bases, assume that there are subsetsof R that do not have m-bases (but every perfect σ-ideal of R has an m-basis).As we have seen, one can choose a maximal subset S of this type. Proceedingas before (just using statement (iv) of Theorem 2.3.3 instead of its statement(v)), we obtain that S is a reflexive prime (hence, perfect) difference ideal withan m-basis. This contradiction with the choice of S completes the proof.

As a consequence of the proof of the last statement we obtain the followingresult.

Corollary 2.5.3 If a difference ring contains a set without basis (m-basis),then it contains a reflexive prime difference ideal without basis (m-basis) whichis maximal among sets without bases (m-bases).

Proposition 2.5.4 The following conditions on a difference (σ-) ring R areequivalent.

(i) R is a Ritt difference ring.(ii) R satisfies the ascending chain condition for perfect difference ideals.(iii) Every nonempty set of perfect σ-ideals of R contains a maximal element

under inclusion.(iv) Every prime reflexive difference ideal of R has a basis.

PROOF. (i) ⇒ (ii). Let J1 ⊆ J2 ⊆ . . . be a chain of perfect σ-ideals of R,and let J =

⋃∞i=1 Ji. Clearly, J is a perfect σ-ideal, so it has a basis B. Since B

is finite, there exists i ∈ N, i ≥ 1, such that B ⊆ Ji. Then Ji = B = J , henceJk = J for all k ≥ i.

(ii) ⇒ (iii). Let Φ be any nonempty collection of perfect σ-ideals of R. Chooseany J1 ∈ Φ. If J1 is a maximal element of Φ, (iii) holds, so we can assume thatJ1 is not maximal. Then there is some J2 ∈ Φ such that J1 J2. If J2 ismaximal in Φ, (iii) holds, so we can assume there is J3 ∈ Φ properly containingJ2. Proceeding in this way one sees that if (iii) fails we can produce an infinitestrictly increasing chain of elements of Φ, contrary to (ii).


(iii) ⇒ (vi). Let P be a prime σ∗-ideal of R, and let Σ be the set of all perfectσ-ideals I of R such that I ⊆ P and I has no basis. By (iii), if Σ = ∅, then Σcontains a maximal element J . If J = P , let x ∈ P \J . If B is a basis of J , thenB, x is a basis of J, x, hence J, x ∈ Σ. This contradicts the maximality ofJ , so J = P and the σ-ideal P has a basis.

The implication (vi)⇒ (i) is an immediate consequence of Corollary 2.5.3.

The following theorem is a strengthened version of Proposition 2.3.4 for Rittdifference rings.

Theorem 2.5.5 Every perfect difference ideal of a Ritt difference ring is theintersection of a finite number of reflexive prime difference ideals.

PROOF. Suppose that a Ritt difference (σ-) ring R contains a perfect idealthat cannot be represented as the intersection of finitely many prime σ∗-ideals.By Proposition 2.5.4, there exists a maximal perfect σ-ideal J with this prop-erty. Since J is not prime, there exist a, b ∈ R \ J such that ab ∈ J . UsingTheorem 2.3.3(v) we obtain that J ⊆ J, a ∩ J, b ⊆ J, ab = J , so thatJ = J, a ∩ J, b. Because of the maximality of J , both J, a and J, b canbe represented as intersections of finitely many prime σ∗-ideals. Therefore, Jhas such a representation that contradicts our assumption. This completes theproof of the theorem.

If J is a perfect difference ideal of a Ritt difference (σ-) ring R, then afinite set of prime σ∗-ideals, whose intersection is J , is not uniquely determined.For example, if σ consists of an identity automorphism, then the last theorembecomes a statement about the representation of a radical ideal of a commutativering as the intersection of finitely many prime ideals. It is well-known that sucha representation is not unique (say, (xy) = (x) ∩ (y) = (x) ∩ (y) ∩ (x, y) inthe polynomial ring Q[x, y]). However, the uniqueness holds for irredundantrepresentations in the following sense.

Definition 2.5.6 Let a perfect difference ideal J be represented as the intersec-tion of prime difference ideals P1, . . . , Pr. This representation is called irredun-dant (and J is said to be the irredundant intersection of P1, . . . , Pr ) if Pi Pj

for i = j.

Note, that since all the ideals P1, . . . , Pr in an irreduntant representation

J =r⋂

i=1

Pi are prime, Pi ⋂j =i

Pj for i = 1, . . . , r.

Theorem 2.5.7 Every perfect difference ideal in a Ritt difference ring is theirredundant intersection of a finite set of prime difference ideals. The ideals ofthis set are uniquely determined, each of them is reflexive.

PROOF. Let J be a perfect difference ideal of a Ritt difference (σ-) ringR. Our previous theorem shows that J can be represented as an intersection of


finitely many prime σ (even σ∗-) ideals Q1, . . . , Qr. If Qi ⊆ Qj for distinct i andj, then the ideal Qj can be removed from this intersection without changingit. Therefore, J is the intersection of the family of minimal elements of the setQ1, . . . , Qr under inclusion. Obviously, this intersection is irredundant.

Let J =r⋂

i=1

Pi and J =s⋂

j=1

P ′j be two irreduntant representations of J as

intersections of prime σ-ideals. Then P1 ⊇s⋂

j=1

P ′j hence P1 ⊇ P ′

j for some j, say

for j = 1. Similarly, P ′1 contains some Pi which must be P1, since P1 ⊇ P ′

1 ⊇ Pi.Thus, P1 = P ′

1. Similarly, one can arrange P ′1, . . . , P

′s in such a way that P2 =

P ′2, etc. We obtain that s = r and the set P1, . . . , Pr coincides with the set

P ′1, . . . , P

′r.

To prove the last statement of the theorem, assume that J =r⋂

i=1

Pi is a

representation of J as an irredundant intersection of prime σ-ideals. Taking thereflexive closures of both sides of the last equality and using the fact that J is

reflexive, we obtain that J =r⋂

i=1

P ∗i where P ∗

i denotes the reflexive closure of Pi

(1 ≤ i ≤ r). By the uniqueness of the irreduntant representation (obviously, allσ∗-ideals P ∗

i are prime and P ∗i ⊇ Pi) we obtain that P ∗

i = Pi for i = 1, . . . , r,so all members of the irreduntant representation are prime reflexive differenceideals.

If J =r⋂

i=1

Pi is a representation of a perfect difference ideal J as an ir-

redundant intersection of prime σ-ideals, then the ideals P1, . . . , Pr (uniquelydetermined by J) are called the essential prime divisors of J .

Theorems 2.3.13 and 2.5.7 lead to the following decomposition theorem forcomplete ideals in a Ritt difference ring.

Theorem 2.5.8 Let I be a proper complete difference ideal in a Ritt differencering R with a basic set σ. Then there exist proper complete σ-ideals J1, . . . , Jm

of R such that(i) I = J1

⋂ · · ·⋂ Jm,(ii) J1, . . . , Jm are strongly separated in pairs (hence, I = J1J2 . . . Jm),(iii) no Jk (1 ≤ k ≤ m) is the intersection of two strongly separated proper

σ-ideals.

The ideals J1, . . . , Jm are uniquely determined by the ideal I. If I is reflexive,so are Jk (1 ≤ k ≤ m). Finally, if the σ-ideal I is mixed, then J1, . . . , Jm arealso mixed.

PROOF. Let Q = I and let Φ = P1, . . . , Pr be the set of all essentialprime divisors of J . We say that Pi is linked to Pj if either i = j or i = j andthere exists a sequence of distinct elements Pi1 = Pi, Pi2 , . . . , Pis

= Pj ∈ Φ such


that Pikand Pik+1 are not strongly separated (k = 1, . . . , s − 1). It is easy to

see that we obtain an equivalence relation on the set Φ, so one can consider apartition of Φ into the union of pairwise disjoint subsets Φ1, . . . ,Φm such thattwo prime σ-ideals Pi, Pj ∈ Φ are linked if and only if they belong to the sameΦk (1 ≤ k ≤ m).

Let Qk =⋂

P∈ΦkP for k = 1, . . . , m. Then Q1, . . . , Qm are perfect σ-ideals

and Q = Q1∩· · ·∩Qm. As we showed in the proof of Theorem 2.3.13, the fact thatsome ideals L1, . . . , Ls (s ≥ 3) are strongly separated in pairs implies that theideals L1 and

⋂si=2 Li are strongly separated. Therefore, if i = j (1 ≤ i, j ≤ m)

then Qi and any ideal of the set Φj are strongly separated. It follows that Qi andQj =

⋂P∈Φj

P are strongly separated, so Q1, . . . , Qm are strongly separated inpairs.

By Theorem 2.3.13, there exist pairwise strongly separated complete σ-idealsJ1, . . . , Jm such that I = J1

⋂ · · ·⋂ Jm and Jk = Qk for k = 1, . . . , m. We aregoing to show that no Jk is the intersection of two strongly separated properσ-ideals. Indeed, suppose that some Ji, say J1, can be represented as J1 =J1

⋂J2 where J1 and J2 are proper σ-ideals strictly containing J1 such that

[J1, J2] = J1 + J2 = R. Then there exist a ∈ J1, b ∈ J2 such that a + b = 1.Then ab ∈ J1 ⊆ Q1, hence each essential prime divisor of Q1 contains either aor b.

If a sequence Pi1 , . . . , Pisprovides a link between two elements of Φ1, then

any two adjacent members of this sequence either both contain a or both containb (otherwise, they would be strongly separated). Therefore, either all elementsof Φ1 contain a or all such elements contain b. Without loss of generality, we canassume that all elements of Φ1 contain a. Then a ∈ Q1 hence there exists τ ∈ Tσ,l ∈ N, l ≥ 1, such that τ(a)l ∈ J1 ⊆ J2. Now the equality (τ(a) + τ(b))l = 1implies that 1 ∈ J2, contrary to our assumption. This completes the proof ofthe existence of the decomposition with properties (i) - (iii).

To prove the uniqueness, assume that I = J ′1

⋂ · · ·⋂ J ′d is another repre-

sentation of I as an intersection of complete σ-ideals satisfying conditions (i)- (iii). Let Q′

i = Ji (i = 1, . . . , d). As in the proof of Theorem 2.5.7, weobtain that Q′

1, . . . , Q′d are pairwise strongly separated perfect σ-ideals and

Q = Q′1

⋂ · · ·⋂Q′d. It follows from Theorem 2.3.13 that no Q′

i can be repre-sented as the intersection of two strongly separated proper perfect σ-ideals (theexistence of such a representation would imply the existence of a representationof J ′

1 as an intersection of two strongly separated complete σ-ideals).Let Pi1, . . . , Piki

be the essential prime divisors of Q′i (i = 1, . . . , d). Then

no Pij contains Pkl unless i = k, j = l. Indeed, if i = k, j = l, then Pij andPkl are essential prime divisors of the same Qi; if i = k, then Pij and Pkl

contain strongly separated ideals Q′i and Q′

k, respectively, so Pij + Pkl = R.Since Pij = R and Pkl = R, none of these two ideals can contain the other one.

Applying Theorem 2.5.7 to the equality⋂d

i=1

⋂ri

j=1 Pij = Q we obtain thatthe ideals Pij coincide (in some order) with P1, . . . , Pr. Without loss of general-ity, we can assume that Ψ1 = P1, . . . , Pt is the set of all essential prime divisorsof Q′

1 (1 ≤ t ≤ r). Then every Pi ∈ Ψ1 can be linked to P1 (otherwise, applying


the same procedure that was used for Qis, we obtain two proper strongly sepa-rated perfect σ-ideals with intersection Q′

1). Furthermore, since Q′1 is strongly

separated from each Q′i, 2 ≤ i ≤ r, each of the ideals Pt+1, . . . , Pr is strongly

separated from each element of the set Ψ1. It follows that Ψ1 = Φk and Q′1 = Qk

for some k (1 ≤ k ≤ m). Similarly, each Q′j is one of Qi and we obtain that

d = m and Q′i can be arranged in such a way that Q′

1 = Q1, . . . , Q′m = Qm.

By Theorem 2.3.3, if I is reflexive, then each Jk (1 ≤ k ≤ m) is also reflexive.Suppose that the σ-ideal I is mixed. We shall show that each Jk (1 ≤ m) isalso mixed in this case. Let ab ∈ J1 (without loss of generality we can assumek = 1) and let L = J2 ∩ · · · ∩ Jm. As we have seen, the ideals J1 and L arestrongly separated, hence there exist u ∈ J1, v ∈ L such that u + v = 1. Thenabv ∈ I, hence aα(b)v = aα(b)(1 − u) ∈ I ⊆ J1 for every α ∈ σ. It follows thataα(b) ∈ J1 for every α ∈ σ, so that the σ-ideal J1 is mixed. This completes theproof of the theorem.

Definition 2.5.9 The ideals J1, . . . , Jm whose existence is established byTheorem 2.5.8 are called the essential strongly separated divisors of the completeσ-ideal I.

The following result is a consequence of Theorem 2.5.8.

Corollary 2.5.10 Let R be a Ritt difference ring with a basic set σ and let Jbe a proper perfect σ-ideal of R. Then

(i) the essential strongly separated divisors of J are perfect σ-ideals;(ii) the ideal J can be represented as J = J1

⋂ · · ·⋂ Js where J1, . . . , Js arepairwise separated proper perfect σ-ideals such that no Jk (1 ≤ k ≤ s) is anintersection of two separated proper perfect σ-ideals. The ideals J1, . . . , Js areuniquely determined by the ideal J .

PROOF. If the ideal I in the conditions of Theorem 2.5.8 is perfect, thenI = I and the perfect ideals Q1, . . . , Qm obtained in the proof of the theoremare essential strongly separated divisors of I. This proves statement (i).

To prove (ii) assume that Φ = P1, . . . , Pr is the set of all essential primedivisors of J . We say that Pi is linked to Pj (1 ≤ i, j ≤ r) if there exista sequence Pi = Pi1 , Pi2 , . . . , Pid

= Pj of elements of Φ such that no two itssuccessive members are separated. As in the proof of Theorem 2.5.8 we obtaina partition of Φ into a union of pairwise disjoint subsets Φ1, . . . ,Φs such thatevery two elements of the same subset are linked to each other, while no elementof Φi is not linked to any element of Φj if i = j. Setting Jk =

⋂P∈Φk

P (k =1, . . . , s) we arrive at the representation J = J1 ∩ · · · ∩ Js. Now, repeating thearguments of the proof of Theorem 2.5.8 (where we showed that Q1, . . . , Qm

are pairwise strongly separated), we obtain that the perfect ideals J1, . . . , Js

are pairwise separated. (One should use the fact that if I, I1, . . . Iq are perfectσ-ideals and I is separated from each Ii, 1 ≤ i ≤ q, then I is separated fromI ′ =

⋂qi=1 Ii. This result is the immediate consequence of Theorem 2.3.3(v):

clearly, it is sufficient to prove the statement for q = 2, and for this case we haveR = I, I1

⋂I, I2 ⊆ I, I1I2 ⊆ I, I ′ ⊆ R.)


Let us show that no Jk (1 ≤ k ≤ m) is the intersection of two proper sepa-rated perfect σ-ideals. Suppose that some Jk, say J1, is such an intersection, sothat J1 = J ′ ⋂ J ′′ where J ′ and J ′′ are perfect σ-ideals such that J ′, J ′′ = R.If Ψ′ and Ψ′′ denote the sets of essential prime divisors of J ′ and J ′′, respectively,then J =

⋂P∈Ψ′ ⋃ Ψ′′ P whence Φ1 ⊆ Ψ′ ∪ Ψ′′. Since J ′ and J ′′ are separated,

every element of Ψ′ is separated from every element of Ψ′′. It follows that eitherΦ1 ⊆ Ψ′ or Φ1 ⊆ Ψ′′. (Otherwise, there would be a sequence of elements ofΦ1 that provides a link between some Pi ∈ Ψ′ ⋂Φ1 and Pj ∈ Ψ′′ ⋂Φ1; such asequence would contain two adjacent elements Pik

∈ Ψ′ and Pil∈ Ψ′′, contrary

to the fact that any two such elements are separated.) We obtain that eitherJ1 ⊇ J ′ or J1 ⊃ J ′′. In both cases J ′, J ′′ = R that contradicts our assumption.

To prove the uniqueness, suppose that J = J ′1

⋂ · · ·⋂ J ′t where J ′

1, . . . , J′t

satisfy the conditions of statement (ii). Let Ψ′i be the set of all essential prime

divisors of J ′i (1 ≤ i ≤ t). Then every two elements of the same Ψ′

i can be linked(otherwise, J ′

i can be represented as an intersection of two proper separatedperfect σ-ideals strictly containing J ′

i). Suppose that P ′ ∈ Φ1, so that P ′ is anessential prime divisor of J1. Since P ′ ⊇ ⋂P |P ∈ ⋃t

j=1 Ψj, P ′ contains someessential prime divisor Q′ of some J ′

j (Q′ ∈ Ψj). If P ′′ is another element ofΦ1 and P ′′ contains some essential prime divisor Q′′ of some J ′

k, then k = j(otherwise, P ′ and P ′′ would be separated). Thus, every element of Φ1 containsan element of the same set Ψj , whence J1 ⊇ J ′

j . Similarly, J ′j ⊇ Jl for some

l, 1 ≤ l ≤ s. It follows that l = 1 and J1 = J ′j . Continuing, we find that each

of Jk is some J ′i and each J ′

i is some Jk. Thus, s = t and the ideals J ′i can

be arranged in such a way that Jk = J ′k for k = 1, . . . , s. This completes the

proof. The ideals J1, . . . , Js constructed in Corollary 2.5.10 are called the essential

separated divisors of J .The following basis theorem for difference rings is due to R. Cohn. It gener-

alizes the Hilbert’s Basis Theorem for polynomial rings and can be consideredas a difference analog of the Ritt-Raudenbush theorem in differential algebra.

Theorem 2.5.11 Let R be a Ritt difference ring with a basic set σ and letS = Rη1, . . . , ηs be a σ-overring of R generated by a finite family of elementsη1, . . . , ηs. Then S is a Ritt σ-ring. Moreover, if every set in R has an m-basis, then every set in S has an m-basis. In particular, an algebra of differencepolynomials Ry1, . . . , ys in a finite set of difference indeterminates y1, . . . , ys

is a Ritt difference ring.If R is an inversive Ritt σ-ring and S∗ = Rη1, . . . , ηs∗ is a finitely

generated σ∗-overring of R, then S∗ is a Ritt σ∗-ring. If every set in R hasan m-basis, then every set in S∗ has an m-basis. In particular, an algebra ofσ∗-polynomials in a finite set of σ∗-indeterminates over R is a Ritt σ∗-ring.

PROOF. Obviously, it is sufficient to prove the theorem for the ring ofσ-polynomials S = Ry1, . . . , ys. Indeed, suppose that the statement of thetheorem is true in this case, and Ry1, . . . , ys∗ is an algebra of σ∗-polynomials ina finite set of σ∗-indeterminates y1, . . . , ys over an inversive (σ∗- ) Ritt ring R. If


J is a perfect σ∗-ideal of Ry1, . . . , ys∗, then J0 = J∩Ry1, . . . , ys is a perfectσ-ideal of the algebra of σ-polynomials Ry1, . . . , ys. It is easy to see that if Φis a basis (m-basis) of J0 in Ry1, . . . , ys, then Φ is a basis (respectively, m-basis) of J in Ry1, . . . , ys∗. The statement for an arbitrary finitely generateddifference (inversive difference) overring of a difference (inversive difference) ringR follows from the fact that such an overring is the image of the ring of σ-(respectively, σ∗-) polynomials under a ring σ-homomorphism.

Suppose that S = Ry1, . . . , ys is not a Ritt σ-ring. By Corollary 2.5.3, Scontains a reflexive prime difference ideal Q without basis which is a maximalelement in the set of all perfect σ-ideals of S without bases. Let Q0 = Q ∩ R.Since Q0 is a perfect σ-ideal in R, it has a basis: Q0 = Ψ for some finite setΨ ⊆ Q0.

Let Σ be the set of all σ-polynomials in Q that have no coefficients in Q0.It is easy to see that Σ ⊆ S \ R and Σ = ∅. Indeed, if Σ = ∅, then everyσ-polynomial f ∈ Q has a coefficient in Q0. If we subtract the correspondingmonomial from f , the difference f ′ will be an element of Q with a coefficient inQ0. Continuing in the same way, we obtain that all coefficients of f belong toQ0. In this case we would have Q = Ψ that contradicts the choice of Q.

Let us construct an autoreduced set A = A1, . . . , Ar ⊆ Σ as follows. LetA1 be a σ-polynomial of the lowest rank in Σ. Suppose that A1, . . . , Ai (i ≥ 1)has been constructed and they form an autoreduced set. If Σ contains no σ-polynomial reduced with respect to A1, . . . , Ai, we set A = A1, . . . , Ai.Otherwise, we choose a σ-polynomial Ai+1 of the lowest rank in the set ofall elements of Σ reduced with respect to A1, . . . , Ai. Obviously, the setA1, . . . , Ai, Ai+1 is autoreduced. After a finite number of steps this processterminates and we obtained an autoreduced set A = A1, . . . , Ar ⊆ Σ suchthat Σ contains no σ-polynomial reduced with respect to A. Furthermore, if aσ-polynomial A ∈ Q is reduced with respect A, then all its coefficients belongto Q0. (Indeed, if A′ is the sum of all monomials in A whose coefficients do notlie in Q0, then A′ ∈ Σ and A′ is reduced with respect to A, hence A′ = 0.)

Let Ij denote the initial of a σ-polynomial Aj (1 ≤ j ≤ r). Since each Ij isreduced with respect to A, Ij /∈ Q for j = 1, . . . , r. (Otherwise, Ij would havecoefficients in Q0, hence Aj would have coefficients in Q0 that contradicts theinclusion Aj ∈ Σ.)

Let I = I1I2 . . . Ir. Since Q is a prime σ-ideal, I /∈ Q. Furthermore, sinceQ is a maximal perfect σ-ideal without basis, the perfect σ-ideal generatedby Q and I has a basis: Q, I = Φ for some finite set Φ. Obviously, onecan choose Φ as a set B1, . . . , Bs, I for some B1, . . . , Bs ∈ Q. Let C ∈ Q.By Theorem 2.4.1, there exist a σ-polynomial D ∈ [A1, . . . , Ar] and elementsτ1, . . . , τq ∈ T ; k1, . . . , kq ∈ N such that the σ-polynomial C ′ = I ′C −D, whereI ′ = (τ1I)k1 . . . (τqI)kq , is reduced with respect to A. Since C ′ ∈ Q, all coeffi-cients of C ′ lie in Q0, whence C ′ ∈ Ψ. It follows that I ′C ∈ Ψ⋃A, hence(τ1(IC))k1 . . . (τq(IC))kq ∈ Ψ⋃A, hence IC ∈ Ψ⋃A.

The last inclusion implies that the set IQ = IA |A ∈ Q is contained inthe perfect σ-ideal Ψ⋃A. Therefore (see Theorem 2.3.3), Q = Q ∩ Q, I ⊆Q

⋂B1, . . . , Bs, I = IQ⋃

B1Q⋃ · · ·⋃BsQ ⊆ Ψ⋃A⋃B1, . . . , Bs ⊆Q.

2.6. VARIETIES OF DIFFERENCE POLYNOMIALS 149

We obtain that the finite set Ψ⋃A⋃B1, . . . , Bs is a basis of the perfect σ-

ideal Q that contradicts the choice of Q. This completes the proof of the theorem.(The statements for m-bases can be established with the same arguments; weleave the corresponding proof to the reader.)

2.6 Varieties of Difference Polynomials

Let K be a difference field with a basic set σ, Ky1, . . . , ys an algebra of σ-polynomials in σ-indeterminates y1, . . . , ys over K, Φ ⊆ Ky1, . . . , ys, and E afamily of σ-overfields of K. Furthermore, let ME(Φ) denote the set of all s-tuplesa = (a1, . . . , as) with coordinates from some field Ka ∈ E which are solutions ofthe set Φ (that is, f(a1, . . . , as) = 0 for any f ∈ Φ). Then the σ-field K is calledthe ground difference (or σ-) field, and ME(Φ) is said to be the E-variety definedby the set Φ (ME(Φ) is also called the E-variety of the set Φ over Ky1, . . . , ysor over K).

Now, let M be a set of s-tuples such that coordinates of every point a ∈ Mbelong to some field Ka ∈ E . If there exists a set Φ ⊆ Ky1, . . . , ys such thatM = ME(Φ), then M is said to be an E-variety over Ky1, . . . , ys (or anE-variety over K).

Definition 2.6.1 Let K be a difference field with a basic set σ, and let L =K(x1, x2, . . . ) be the field of rational fractions in a denumerable set of indeter-minates x1, x2, . . . over K. Furthermore, let L be the algebraic closure of L.Then the family U(K) of all σ-overfields of K which are defined on subfields ofL is called the universal system of σ-overfields of K. If Φ is a subset of the alge-bra of σ-polynomials Ky1, . . . , ys and U = U(K), then the U-variety MU (Φ)(also denoted by M(Φ)) is called the variety defined by the set Φ over K (orover Ky1, . . . , ys).

A set of s-tuples M over the σ-field K is said to be a difference variety(or simply variety) over Ky1, . . . , ys (or over K) if there exists a set Φ ⊆Ky1, . . . , ys such that M = MU(K)(Φ). In what follows, we assume that adifference field K with a basic set σ, an algebra of σ-polynomials Ky1, . . . , ys,and a family E of σ-overfields of K are fixed. E-varieties and varieties overKy1, . . . , ys will be called E-varieties and varieties, respectively.

Proposition 2.6.2 Let η = (η1, . . . , ηs) be an s-tuple over the σ-field K. Thenthere exists and s-tuple ζ = (ζ1, . . . , ζs) over K such that ζ is equivalent to ηand all ζi (1 ≤ i ≤ s) belong to some field from the universal system U(K).

PROOF. Let Tη = τ(ηi) | τ ∈ T, 1 ≤ i ≤ s and let B be a transcendencebasis of the set Tη over K. Then the set B is countable, so K(B) is K-isomorphicto a subfield L1 of the field of rational fractions L = K(x1, x2, . . . ) considered inthe definition of the universal system. Since the field extension K(Tη)/K(B) isalgebraic, the algebraic closure L of the field L contains a subfield L1, an over-field of L1, which is isomorphic to K(Tη) under an extension of the previouslydescribed isomorphism of L1 onto K(B).


Since K〈η1, . . . , ηs〉/K is a difference (σ-) field extension and K〈η1, . . . , ηs〉coincides with the field K(Tη), one can define a difference (σ-) field structure onL1 such that the difference field extensions K〈η1, . . . , ηs〉/K and L1/K are σ-isomorphic. Let ζi denote the image of ηi under this σ-isomorphism (1 ≤ i ≤ s).Then the s-tuple ζ = (ζ1, . . . , ζs) is equivalent to η = (η1, . . . , ηs) and ζ liesin a difference field of the universal system U(K) (it follows from the fact thatL1 ∈ U(K)).

If A1 and A2 are two E-varieties and A1 ⊆ A2 (A1 A2), then A1 is said tobe a E-subvariety (respectively, a proper E-subvariety) of A2. U(K)-subvarietiesof a variety A are called subvarieties of A. An E-variety (variety) A is calledreducible if it can be represented as a union of two its proper E-subvarieties(subvarieties). If such a representation does not exist, the E-variety (variety) Ais said to be irreducible.

Let an E-variety (variety)A be represented as a union of its irreducible E-subvarieties (subvarieties): A = A1

⋃ · · ·⋃Ak. This representation is calledirredundant if Ai Aj for i = j (1 ≤ i, j ≤ k).

The following theorem summarizes basic properties of E-varieties. As before,we assume that a family E of σ-overfields of K is fixed. Furthermore, if A isa set of s-tuples a with coordinates from a σ-field Ka ∈ E (we say that Ais a set of s-tuples from E over K), then ΦE(A) denotes the perfect σ-idealf ∈ Ky1, . . . , ys|f(a1, . . . , as) = 0 for any a = (a1, . . . , as) ∈ A of the ringKy1, . . . , ys. (We drop the index E when we talk about varieties.)

Theorem 2.6.3 (i) If Φ1 ⊆ Φ2 ⊆ Ky1, . . . , ys, then ME(Φ2) ⊆ ME(Φ1).(ii) If A1 and A2 are two sets of s-tuples from E over K and A1 ⊆ A2, then

ΦE(A2) ⊆ ΦE(A1).(iii) If A is an E-variety, then A = ME(ΦE(A)).(iv) If J1, . . . , Jk are σ-ideals of the ring Ky1, . . . , ys and J = J1

⋂ · · ·⋂ Jk,then ME(J) = ME(J1)

⋃ · · ·⋃ME(Jk).(v) If A1, . . . ,Ak are E-varieties over K and A = A1

⋃ · · ·⋃Ak, then A isan E-variety over K and ΦE(A) = ΦE(A1)

⋂ · · ·⋂ΦE(Ak).(vi) The intersection of any family of E-varieties is an E-variety.(vii) An E-variety A is irreducible if and only if ΦE(A) is a prime reflexivedifference ideal of Ky1, . . . , ys.(viii) Every E-variety A has a unique irreduntant representation as a unionof irreducible E-varieties, A = A1

⋃ · · ·⋃Ak. (The E-varieties A1, . . . ,Ak arecalled irreducible E-components of A.) Furthermore, Ai

⋃j =i Aj for i =

1, . . . , k.(ix) If A1, . . . ,Ak are irreducible E-components of an E-variety A, then the

prime σ∗-ideals ΦE(A1), . . . ,ΦE(Ak) are essential prime divisors of the perfectσ-ideal ΦE(A).

PROOF. The proof of statements (i) - (vi) is a quite easy exercise, and weleave it to the reader. Actually, these statements are similar to the corresponding


properties of varieties over polynomial rings (see the results on affine alge-braic varieties in section 1.2). To prove property (vii) suppose, first, thatan E-variety A is reducible: A = A1 ∪ A2 where A1 and A2 are properE-subvarieties of A. Applying properties (ii), (iii), and (v), we obtain thatΦ(A) = Φ(A1)

⋂Φ(A2) and Φ(A) Φ(Ai) (i = 1, 2). If f ∈ Φ(A1) \ Φ(A)

and g ∈ Φ(A2) \ Φ(A), then fg ∈ Φ(A), so the ideal Φ(A) is not prime. Onthe other hand, if the E-variety, A is irreducible, but the ideal Φ(A) is notprime, then there exist σ-polynomials f1, f2 /∈ Φ(A) such that f1f2 ∈ Φ(A). Inthis case A = ME(Φ(A)

⋃f1)⋃ME(Φ(A)

⋃f2) (see property (iv)), henceA = ME(Φ(A)

⋃fi) for i = 1 or i = 2. Since fi /∈ Φ(A) (i = 1, 2), we obtaina contradiction.

Let us prove properties (viii) and (ix). Let A be a E-variety and let Φ(A) =P1

⋂ · · ·⋂Pk be a representation of Φ(A) as an irredundant intersection ofperfect σ-ideals in Ky1, . . . , ys. By properties (iii) and (iv), we have A =ME(Φ(A)) = ME(P1)

⋃ · · ·⋃ME(Pk). Let us set Ai = ME(Pi) (1 ≤ i ≤ k)and show that each Ai is an irreducible E-variety. By property (v), Φ(A) =Φ(A1)

⋂ · · ·⋂Φ(Ak), and it is easy to see that Pi ⊆ Φ(Ai) for i = 1, . . . , k. Let fand g be two σ-polynomials such that f ∈ Φ(A1) and g ∈ P2

⋂ · · ·⋂Pk, g /∈ P1.Then g ∈ Φ(A2)

⋂ · · ·⋂Φ(Ar), hence fg ∈ Φ(A) ⊆ P1. Since g /∈ P1, we havef ∈ P1. Thus, Φ(A1) = P1, so the E-variety A1 is irreducible (we refer to prop-erty (vii)). Clearly, the same arguments can be applied to any E-variety Ai, so wecan conclude that all E-varieties A1, . . . ,Ak are irreducible. If Ai ⊆ Aj for i = j,then Pi = Φ(Ai) ⊆ Φ(Aj) = Pj that contradicts the fact that P1

⋂ · · ·⋂Pk

is an irreduntant representation of the perfect σ-ideal Φ(A). Therefore, A =A1

⋃ · · ·⋃Ak is an irredundant representation of A as a union of irreducibleE-varieties. To prove the uniqueness, suppose that A = A′

1

⋃ · · ·⋃A′l is another

such a representation. Then A1 = A⋂A1 = (A′1

⋂A1)⋃ · · ·⋃(A′

l

⋂A1). Byproperty (vi), every intersection A′

j

⋂A1 (1 ≤ j ≤ l) is an E-variety. Since A1

is irreducible, A1 = A′j

⋂A1 for some j, 1 ≤ j ≤ l. It follows that A1 ⊆ A′j .

Similarly A′j ⊆ A′

i for some i, 1 ≤ i ≤ k hence A1 ⊆ Ai, hence i = 1 andA1 = A′

j . Using the same arguments one can show that every Ai coincides withsome A′

j(i). This completes the proof of property (viii). Furthermore, the argu-ments of this proof show that if A1, . . . ,Ak are irreducible E-components of anE-variety A, then Φ(A1), . . . ,Φ(Ak) are essential prime divisors of the perfectσ-ideal Φ(A). The theorem is proved.

Theorem 2.6.3 implies that A → ΦE(A) is an injective mapping of the setof all E-varieties over Ky1, . . . , ys into a set of all perfect σ-ideals of the ringKy1, . . . , ys. If E is the universal system of σ-overfields of K, then this map-ping is bijective. More precisely, Theorem 2.6.3 implies the following statementabout varieties.

Theorem 2.6.4 (i) If J is a perfect σ-ideal of the ring Ky1, . . . , ys, thenΦ(M(J)) = J .

(ii) M(J) = ∅ if and only if J = Ky1, . . . , ys.


(iii) The mappings A → Φ(A) and P → M(P ) are two mutually inversemappings that establish one-to-one correspondence between the set of all varietiesover K and the set of all perfect σ-ideals of the ring Ky1, . . . , ys.(iv) The correspondence A → Φ(A) maps irreducible components of an ar-

bitrary variety B onto essential prime divisors of the perfect σ-ideal Φ(B) inKy1, . . . , ys. In particular there is a one-to-one correspondence between irre-ducible varieties over K and prime σ-ideals of the σ-ring Ky1, . . . , ys.

If A is an irreducible variety over Ky1, . . . , ys, then a generic zero of thecorresponding prime ideal Φ(A) is called a generic zero of the variety A.

The following result is a version of the Hilbert’s Nullstellensatz for differencefields.

Theorem 2.6.5 Let K be a difference field with a basic set σ and Ky1, . . . , ysan algebra of σ-polynomials in σ-indeterminates y1, . . . , ys over K. Let f ∈Ky1, . . . , ys,Φ ⊆ Ky1, . . . , ys, and M(Φ) the variety defined by the set Φover K. Then the following conditions are equivalent.

(i) Every s-tuple in M(Φ) is a solution of the σ-polynomial f .(ii) f ∈ Φ.

PROOF. The implication (ii)=⇒ (i) is obvious, so we just have to provethat (i) implies (ii). If f is annulled by every s-tuple in M(Φ), then M(Φ) =M(Φ ∪ f), hence the variety of the perfect σ-ideal Φ coincides with thevariety of the perfect σ-ideal Φ∪f. Applying Theorem 2.6.4 we obtain thatΦ = Φ ∪ f, whence f ∈ Φ.

Two varieties A1 and A2 over the ring of difference polynomials Ky1, . . . , ysare said to be separated if A1

⋂A2 = ∅.If a variety A is represented as a union of pairwise separated varieties

A1, . . . ,Ak and no Ai is the union of two nonempty separated varieties, thenA1, . . . ,Ak are said to be essential separated components of A. (All varieties areconsidered over the same ring Ky1, . . . , ys.)

Exercise 2.6.6 Let A1 and A2 be two varieties over an algebra of σ-polynomialsKy1, . . . , ys over a difference (σ-) field K. Prove that A1 and A2 are separatedif and only if the perfect σ-ideals Φ(A1) and Φ(A2) are separated.

Exercise 2.6.7 Let J1 and J2 be two ideals of an algebra of σ-polynomialsKy1, . . . , ys over a difference (σ-) field K. Prove that J1 and J2 are separatedif and only if the varieties M(J1) and M(J2) are separated.

The following statement is due to R. Cohn.

Theorem 2.6.8 Let K be a difference field with a basic set σ and Ky1, . . . , ysan algebra of σ-polynomials in σ-indeterminates y1, . . . , ys over K. Then:


(i) Every non-empty variety A over Ky1, . . . , ys can be represented as aunion of a uniquely determined family of its essential separated components.Each of these components is a union of some irreducible components of thevariety A.

(ii) If A1, . . . ,Ak are essential separated components of a variety A, thenΦ(A1), . . . ,Φ(Ak) are essential separated divisors of the perfect σ-ideal Φ(A)in Ky1, . . . , ys.

PROOF. (i) It is clear that the irreducible components of a variety A canbe grouped to form its essential separated components A1, . . . ,Ak. If A =A′

1

⋃ · · ·⋃A′l is another representation of A as a union of its essential pairwise

separated components, then A′1 ⊆ A1

⋃ · · ·⋃Ak. Let B′ be an irreducible com-ponent of A′

1, and let B1, . . . ,Bp be all irreducible components of the varietiesA1, . . . ,Ak. Then B′ ⊆ B1

⋃ · · ·⋃Bp hence B′ ⋂Bi = ∅ for some i (1 ≤ i ≤ p)hence B′ ⊆ Bi. Thus, every irreducible component of A′

1, is contained in anirreducible component of Aj for some index j, 1 ≤ j ≤ k. (If two irreduciblecomponents of A′

1 are contained in different varieties Ai and Aj (i = j), then A′1

can be represented as a union of two its proper subvarieties contained in Ai and⋃j =i Ai, respectively. This contradicts the fact that A′

1 is an essential separatedcomponent of A.) Without loss of generality we can assume that j = 1, thatis, A′

1 ⊆ A1. By the same arguments, A1 ⊆ A′t for some index t (1 ≤ t ≤ l),

hence A1 = A′1. Continuing in the same way, we obtain that k = l and Ai = A′

i

(1 ≤ i ≤ l) after some renumeration of A1, . . . ,Ak.(ii) It is easy to see that if A1, . . . ,Ak are essential separated components of

A, then the perfect difference ideals Φ(Ai) (1 ≤ i ≤ k) are separated in pairs. ByTheorem 2.6.3(v), Φ(A) =

⋂ki=1 Φ(Ai). Now, in order to prove that Φ(Ai) are

essential separated divisors of Φ(A), we just need to notice that no Φ(Ai) (1 ≤i ≤ k) can be represented as Φ(Ai) = Φ′

i

⋃Φ′′

i where Φ′i and Φ′′

i are separatedproper perfect σ-ideals of Ky1, . . . , ys strictly containing Φ(Ai). Indeed, ifsuch a representation exists, then Ai = MU (Φ(Ai)) = MU (Φ′

i)⋃MU (Φ′′

i ) (seeTheorem 2.6.3(iii), (iv)). In this case the varieties MU (Φ′

i) and MU (Φ′′i ) would

not be separated, so the perfect ideals Φ′i and Φ′′

i would not be separated, aswell. This completes the proof of the theorem.

Now, let K be an inversive difference field with a basic set σ, Ky1, . . . , ys∗an algebra of σ∗-polynomials in σ∗-indeterminates y1, . . . , ys over K, and E aset of σ∗-overfields of K. If Φ ⊆ Ky1, . . . , ys∗, then the set ME(Φ) consistingof all s-tuples with coordinates in some σ∗-field in E that are solutions of everyσ∗-polynomial in Φ is called an E-variety over Ky1, . . . , ys∗ determined by theset Φ. Let A be a set of s-tuples over K such that all coordinates of each s-tuplea ∈ A belong to some σ∗-field Ka ∈ E . The set A is said to be an E-variety overKy1, . . . , ys∗ if there exists a set Φ ⊆ Ky1, . . . , ys∗ such that A = ME(Φ).

Let L = K(x1, x2, . . . ) be the field of rational fractions in a denumerableset of indeterminates x1, x2, . . . over the σ∗-field K and let L be the algebraicclosure of L. Then the family U∗(K) consisting of all σ∗-overfields of K definedon subfields of L is called the universal system of σ∗-overfields of K. As in the


case of non-inversive difference fields, one can prove that if η = (η1, . . . , ηs) isany s-tuple over the σ∗-field K, then there exists an s-tuple ζ = (ζ1, . . . , ζs) suchthat ζ is equivalent to η, and all coordinates of the point ζ lie in some σ∗-fieldKζ ∈ U∗(K) (the proof is similar to the proof of Proposition 2.6.2). A U∗(K)-variety over Ky1, . . . , ys∗ is called a variety over this ring of σ∗-polynomials.

The concepts of E-subvariety, proper E-subvariety, subvariety, and propersubvariety over Ky1, . . . , ys∗, as well as the notions of reducible and irreducibleE-varieties and varieties, are precisely the same as in the case of s-tuples over analgebra of (non-inversive) difference polynomials. If a E-variety (variety) A overKy1, . . . , ys∗ is represented as a union of its E-subvarieties (subvarieties), A =A1

⋃ · · ·⋃Ak, and Ai Aj for i = j (1 ≤ i, j ≤ k), then this representation iscalled irredundant.

All properties of E-varieties and varieties over an algebra of difference polyno-mials listed in Theorems 2.6.3, 2.6.4, 2.6.5 and 2.6.6 remain valid for E-varietiesand varieties over Ky1, . . . , ys∗. The formulations and proofs of the corre-sponding statements are practically the same, and we leave them to the readeras an exercise. One should just replace the ring Ky1, . . . , ys by Ky1, . . . , ys∗and treat ME(Φ) (Φ ⊆ Ky1, . . . , ys∗) and ΦE(A) (A is a set of s-tuples aover K whose coordinates belong to some σ∗-field Ka ∈ E) as the set a =(a1, . . . , as)|a1, . . . , as belong to some σ∗-field Ka ∈ E and f(a1, . . . , as) = 0 forall f ∈ Φ and the perfect σ-ideal f ∈ Ky1, . . . , ys∗|f(a1, . . . , as) = 0 for anya = (a1, . . . , as) ∈ A of the ring Ky1, . . . , ys∗, respectively. (If E = U∗(K),then ΦE(A) and ME(Φ) are denoted by M(Φ) and Φ(A), respectively.) By ageneric zero of an irreducible variety A over Ky1, . . . , ys∗ we mean a genericzero of the corresponding perfect σ-ideal Φ(A) of the ring Ky1, . . . , ys∗. Theconcept of separated varieties over Ky1, . . . , ys∗ is introduced in the same wayas in the case of varieties over an algebra of difference polynomials. Obviously,the statements of Theorem 2.6.8 remain valid for varieties of inversive differencepolynomials as well.

Chapter 3

Difference Modules

3.1 Ring of Difference Operators. DifferenceModules

Let R be a difference ring with a basic set σ = α1, . . . , αn and let T denotethe free commutative semigroup generated by the elements α1, . . . , αn. Recallthat we define the order of an element τ = αk1

1 . . . αknn ∈ T (k1, . . . , kn ∈ N)

as the number ord τ =∑n

i=1 ki and set Tr = τ ∈ T |ord τ = r, T (r) = τ ∈T |ord τ ≤ r for any r ∈ N.

Definition 3.1.1 An expression of the form∑

τ∈T aτ τ , where aτ ∈ R for anyτ ∈ T and only finitely many elements aτ are different from 0, is called a dif-ference (or σ-) operator over the difference ring R. Two σ-operators

∑τ∈T aττ

and∑

τ∈T bττ are considered to be equal if and only if aτ = bτ for any τ ∈ T .

The set of all σ-operators over the σ-ring R will be denoted by D. This setcan be equipped with a ring structure if we define

∑τ∈T aττ +

∑τ∈T bτ τ =∑

τ∈T (aτ + bτ )τ , a∑

τ∈T aτ τ =∑

τ∈T (aaτ )τ , (∑

τ∈T aττ)τ1 =∑

τ∈T aτ (ττ1),τ1a = τ1(a)τ1 for any

∑τ∈T aττ,

∑τ∈T bττ ∈ D, a ∈ R, τ1 ∈ T and extend the

multiplication by distributivity. Then D becomes a ring called the ring of dif-ference (or σ-) operators over R.

The order of a σ-operator A =∑

τ∈T aτ τ ∈ D is defined as the numberordA = maxord τ |aτ = 0. If for any r ∈ N we set D(r) = ∑τ∈T aτ τ ∈D | ord τ = r for every τ ∈ T and D(r) = 0 for any r ∈ Z, r < 0, then thering of difference operators can be considered as a graded ring (with positivegrading): D =

⊕r∈Z D(r). The ring D can be also treated as a filtered ring

equipped with the ascending filtration (Dr)r∈Z such that Dr = 0 for any negativeinteger r and Dr = A ∈ D|ordA ≤ r for any r ∈ N. (This filtration is calledstandard.) Below, while considering D as a graded or filtered ring, we meanthe gradation with the homogeneous components D(r) (r ∈ Z) or the filtration(Dr)r∈Z, respectively.

155

156 CHAPTER 3. DIFFERENCE MODULES

Definition 3.1.2 Let R be a difference ring with a basic set σ = α1, . . . , αnand let D be the ring of σ-operators over R. Then a left D-module is called adifference R-module or a σ-R-module. In other words, an R-module M is calleda difference (or σ-) R-module, if the elements of the set σ act on M in such away that α(x + y) = α(x) + α(y), α(βx) = β(αx), and α(ax) = α(a)α(x) forany x, y ∈ M ; α, β ∈ σ; a ∈ R.

If R is a difference (σ-) field, then a σ-R-module M is also called a differencevector space over R or a vector σ-R-space.

We say that a difference R-module M is finitely generated, if it is finitelygenerated as a left D-module. By a graded difference (or σ-)R-module we alwaysmean a graded left module over the ring of σ-operators D =

⊕r∈Z D(r). If

M =⊕

q∈Z M (q) is a graded σ-R-module and M (q) = 0 for all q < 0, we saythat M is positively graded and write M =

⊕q∈N M (q).

Let R be a difference ring with a basic set σ and let D be the ring ofσ-operators over R equipped with the standard filtration (Dr)r∈Z. In what fol-lows, by a filtered σ-R-module we always mean a left D-module equipped withan exhaustive and separated filtration. In other words, by a filtration of a σ-R-module M we mean an ascending chain (Mr)r∈Z of R-submodules of M suchthat DrMs ⊆ Mr+s for all r, s ∈ Z, Mr = 0 for all sufficiently small r ∈ Z, and⋃

r∈Z Mr = M .If (Mr)r∈Z is a filtration of a σ-R-module M , then gr M will denote the

associate graded D-module with homogeneous components grrM = Mr+1/Mr

(r ∈ Z). Since the ring grD =⊕

r∈Z Dr+1/Dr is naturally isomorphic to D, weidentify these two rings.

Definition 3.1.3 Let R be a difference ring with a basic set σ and let M and Nbe two σ-R-modules. A homomorphism of R-modules f : M → N is said to bea difference (or σ-) homomorphism if f(αx) = αf(x) for any x ∈ M , α ∈ σ. Asurjective (respectively, injective or bijective) difference homomorphism is calleda difference (or a σ-) epimorphism (respectively, a difference monomorphism ora difference isomorphism).

If M and N are equipped with filtrations (Mr)r∈Z and (Nr)r∈Z, respectively,and a σ-homomorphism f : M → N has the property that f(Mr) ⊆ Nr for anyr ∈ Z, then f is said to be a homomorphism of filtered σ-R-modules.

The following proposition will be used in the next section where we studythe dimension of difference modules.

Proposition 3.1.4 Let R be an Artinian commutative ring, α an injectiveendomorphism of R, and

0 −→ Ki−→ M

δ−→ N −→ 0

an exact sequence of finitely generated R-modules, where i is an injection andδ is an additive mapping of M onto N such that δ(ax) = α(a)δ(x) for anya ∈ R, x ∈ M . Then N is a finitely generated α(R)-module and

lR(K) + lα(R)(N) = lR(M) (3.1.1)

3.2. DIMENSION POLYNOMIALS OF DIFFERENCE MODULES 157

PROOF. If elements x1, . . . , xs (s ∈ N+) generate the R-module M , then

N = δ(M) = δ

(s∑

i=1

Rxi

)=

s∑i=1

α(R)δ(xi), so δ(x1), . . . , δ(xs) is a finite system

of generators of the α(R)-module N . Let us show that if

K = K0 ⊃ K1 ⊃ · · · ⊃ Kt = 0 and N = N0 ⊃ N1 ⊃ · · · ⊃ Np = 0

are composition series of the R-module K and α(R)-module N , respectively,then

M = δ−1(N0) ⊃ δ−1(N1) ⊃ · · · ⊃ δ−1(Np) = i(K0) ⊃ i(K1) ⊃ · · · ⊃ i(K1) = 0(3.1.2)

is a composition series of the R-module M . (Since δ(ax) = α(a)δ(x) for anya ∈ R and x ∈ M , δ−1(Nj) (1 ≤ j ≤ p) are R-submodules of M). It is clear thatthe R-modules i(Kr−1)/i(Kr) (1 ≤ r ≤ t) are simple. Thus, it remains to showthat all R-modules Lj = δ−1(Nj−1)/δ−1(Nj) (1 ≤ j ≤ p) are simple. For everyj = 1, . . . , p, let us consider the mapping δj : Lj −→ Nj−1/Nj such that δj(ξ +δ−1(Nj)) = δ(ξ) + Nj for any element ξ + δ−1(Nj) ∈ Lj (ξ ∈ δ−1(Nj−1)). It iseasy to see that δj is well-defined, additive and bijective. Furthermore, δj(aξ) =α(a)ξ for any elements ξ = ξ+δ−1(Nj) ∈ Lj and a ∈ R, so that if P is any properα(R)-submodule of Nj−1/Nj , then δ−1

j (P ) is a proper R-submodule of Lj . Sinceall α(R)-modules Nj−1/Nj are simple, all R-modules Lj are simple. Therefore,the ascending chain (3.1.2) is a composition series of the R-module M , whencelR(K) + lα(R)(N) = lR(M).

3.2 Dimension Polynomials of DifferenceModules

We start this section with a generalization of the classical theorem on Hilbertpolynomial (see Theorem 1.3.7(ii)) to the case of graded modules over a ring ofdifference operators.

Theorem 3.2.1 Let R be an Artinian difference ring with a basic set σ =α1, . . . , αn and let M =

⊕q∈N M (q) be a finitely generated positively graded

σ-R-module. Then(i) The length lR(M (q)) of every R-module M (q) is finite.(ii) There exists a polynomial φ(t) in one variable t with rational coefficients

such that φ(q) = lR(M (q)) for all sufficiently large q ∈ N.(iii) deg φ(t) ≤ n − 1 and the polynomial φ(t) can be written as φ(t) =

n−1∑i=0

ai

(t + i

i

)where a0, a1, . . . , an−1 ∈ Z.

PROOF. Let D denote the ring of σ-operators over the ring R. Without lossof generality, one can assume that M is generated as a left D-module by a finite


set of homogeneous elements x1, . . . , xk where xi ∈ M (si) (1 ≤ i ≤ k) for somes1, . . . , sk ∈ N. Then every element of a homogeneous component M (q) (q ∈ N)can be represented as a finite sum

∑ki=1 uixi where ui ∈ D(q−si) if q ≥ si, and

ui = 0 if q < si. Since every element of D(j) (j ∈ N) is a finite linear combinationwith coefficients in R of the monomials αj1

1 . . . αjnn with

∑nν=1 jν = j and there

are exactly(

j + n − 1n − 1

)such monomials, M (q) is generated over R by the finite

set αj11 . . . αjn

n xi|1 ≤ i ≤ k and∑n

ν=1 jν = q−si. Since the ring R is Artinian,lR(M (q)) is finite for all q ∈ N.

In order to prove statements (ii) and (iii), we proceed by induction on n =Card σ. If n = 0, then the ring of σ-operators D coincides with R. In this case,if y1, . . . , ym is a system of homogeneous generators of the D-module M , thenthe degree of any nonzero homogeneous element of M is equal to the degreed(yi) of some generator yi (1 ≤ i ≤ m). Therefore, M (q) = 0 and lR(M (q)) = 0for all q > max1≤i≤md(yi), so for n = 0 statement (ii) is true.

Now, suppose that Card σ = n (σ = α1, . . . , αn) and statements (ii)and (iii) are true for all σ-rings with Card σ = n − 1. Let σ′ = α1, . . . , αn−1and let D′ be the ring of σ′-operators over R (treated as a σ′-ring). Further-more, let B be the ring of σ-operators over the σ-ring αn(R) and B′ the ring ofσ′-operators over the σ′-ring αn(R). We consider D′, B, and B′ as graded ringswhose homogeneous components are defined in the same way as the homoge-neous components of the ring D.

Let θ be the mapping of the σ-R-module M into itself such that θ(x) = αnxfor any x ∈ M and let K = Kerθ, N = Imθ. Then K and N can be consideredas a positively graded D- and D′- modules, respectively: K =

⊕q∈N K(q) and

N =⊕

q∈N N (q) where K(q) = Ker(θ|M(q)) and N (q) = θ(M (q)) (q ∈ N). Sincefor every q ∈ N the sequence

0 −→ K(q) ν−→ M (q) θ−→ N (q) −→ 0

(ν is the embedding) satisfies the conditions of Proposition 3.1.4 (with δ = θ),we have

lR(M (q)) = lR(K(q)) + lαn(R)(N (q)). (3.2.1)

Let L(q) = Rθ(M (q)) for every q ∈ N. It is clear that L(q) ⊆ M (q+1) andαi(R)θ(M (q)) ⊆ αi(R)θ(M (q+1)) ⊆ Rθ(M (q+1)) = L(q+1) (q ∈ N, i = 1, . . . , n),so one can consider positively graded D-modules L =

⊕q∈N L(q) and A =⊕

q∈N A(q) where A(q) = M (q+1)/L(q) for any q ∈ N. (The action of an elementαi ∈ σ on the graded module A is defined in such a way that αi(ξ + L(q)) =αi(ξ) + L(q+1) for any element ξ + L(q) ∈ A(q) (q ∈ N). Clearly, this action iswell-defined and αiA

(q) ⊆ A(q+1) for any q ∈ N, i = 1, . . . , n).Since M (q+1) (q ∈ N) is a finitely generated R-module, the R-modules L(q)

and A(q) are also finitely generated. Therefore, lR(L(q)) < ∞, lR(A(q)) < ∞and

lR(M (q+1)) = lR(L(q)) + lR(A(q)) (3.2.2)

for all q ∈ N.


Now, let us consider the graded B-module θ(L) =⊕

q∈N αn(R)θ2(M (q)). Letθq be the restriction of the mapping θ on the R-module L(q) and B(q) = Kerθq

(q ∈ N). Then B(q) ⊆ L(q) = Rθ(M (q)) ⊆ M (q+1). Applying Proposition 3.1.4to the exact sequence

0 −→ B(q) −→ L(q) θq−→ αn(R)θ2(M (q)) −→ 0

we obtain that the lengths of homogeneous components of the graded D-modulesB =

⊕q∈N B(q) and L =

⊕q∈N L(q) satisfy the equality

lR(L(q)) = lR(B(q)) + lαn(R)(αn(R)θ2(M (q))). (3.2.3)

LettingC(q) denote the finitely generatedαn(R)-moduleN (q)/αn(R)θ2(M (q))(q ∈ N) we see that the mappings C(q) −→ C(q+1), such that

ξ + αn(R)θ2(M (q)) → αiξ + αn(R)θ2(M (q+1))

(1 ≤ i ≤ n, q ∈ N, ξ ∈ N (q)), are well-defined and can be considered as actionsof the elements of σ on the direct sum C =

⊕q∈N C(q) with respect to which

C can be viewed as a graded B-module. Now, the canonical exact sequence ofαn(R)-modules

0 −→ αn(R)θ2(M (q)) −→ N (q) = θ(M (q)) −→ N (q)/αn(R)θ2(M (q)) −→ 0

implies that

lαn(R)(N (q)) = lαn(R)(αn(R)θ2(M (q))) + lαn(R)(C(q)). (3.2.4)

Combining equalities (3.2.1) - (3.2.4) we obtain that

lR(M (q+1)) − lR(M (q)) = lR(L(q)) + lR(A(q)) − lR(K(q)) − lαn(R)(N (q))

= lR(B(q)) +[lαn(R)(N (q)) − lαn(R)(C(q))

]+ lR(A(q)) − lR(K(q)) − lαn(R)(N (q))

whence

lR(M (q+1))− lR(M (q)) = lR(A(q))+ lR(B(q))− lαn(R)(C(q))− lR(K(q)). (3.2.5)

Since the D-modules A,B,K and the B-module C are annihilated by themultiplication by αn, the first three modules are finitely generated gradedD′-modules, while C is a finitely generated graded B′-module. By the inductivehypothesis, there exist polynomials φ1(t), φ2(t), φ3(t), and φ4(t) in one variable twith rational coefficients such that φ1(q) = lR(A(q)), φ2(q) = lR(B(q)), φ3(q) =lαn(R)(C(q)), and φ4(q) = lR(K(q)) for all sufficiently large q ∈ N, say, for allq > q0 (q0 ∈ N). Moreover, deg φj(t) ≤ n−2 (j = 1, 2, 3, 4) and this polynomial


can be written as φj(t) =n−2∑i=0

a(j)i

(t + i

i

)where a

(j)0 , a

(j)1 , . . . , a

(j)n−2 ∈ Z (1 ≤

j ≤ 4).Setting φ0(t) = φ1(t) + φ2(t) − φ3(t) − φ4(t) we obtain that φ0(q) =

lR(M (q+1)) − lR(M (q)) for all q ≥ q0 and deg φ0(t) ≤ n − 2. Further-

more, the polynomial φ0(t) can be written as φ0(t) =n−2∑i=0

bi

(t + i

i

)where

bi = a(1)i + a

(2)i − a

(3)i − a

(4)i (0 ≤ i ≤ n − 2).

Let c = lR(M (q0)). Then lR(M (q)) = lR(M (q0)) +q−1∑j=q0

[lR(M (j+1)) −

lR(M (j))] = c +q−1∑j=q0

φ0(j) = c +n−2∑i=0

bi

q−1∑j=q0

(j + i

j

). Applying the basic

combinatorial identity(

x + 1m

)=

(x

m

)+

(x

m − 1

)and its consequence

m∑k=0

(x + k

k

)=

(x + m + 1

m

)(see (1.4.2) and (1.4.3) ), we obtain that

lR(M (q)) = c +n−2∑i=0

bi

[(i + q

i + 1

)−

(i + q0

i + 1

)]= c +

n−2∑i=0

bi

(i + q + 1

i + 1

)

−n−2∑i=0

bi

(i + q

i

)−

n−2∑i=0

(i + q0

i + 1

)=

n−1∑i=0

b′i

(i + q

i

)where b′n−1 = bn−2, b′i = bi−1 − bi for i = n − 2, . . . , 1, and b′0 = c − b0 −n−2∑i=0

(i + q0

i + 1

).

Thus, the polynomial φ(t) =n−1∑i=0

b′i

(t + i

i

)satisfies conditions (ii) and (iii)

of the theorem. This completes the proof.

Definition 3.2.2 A filtration (Mr)r∈Z of a σ-R-module M is called excellentif all R-modules Mr (r ∈ Z) are finitely generated and there exists r0 ∈ Z suchthat Mr = Dr−r0Mr0 for any r ∈ Z, r ≥ r0.

Remark. Let us consider the ring R as a filtered ring with the trivial fil-tration (Rr)r∈Z such that Rr = R for all r ≥ 0 and Rr = 0 for any r < 0. LetP be an R-module and let (Pr)r∈Z be a non-descending chain of R-submodulesof P such that

⋃r∈Z Pr = P and Pr = 0 for all sufficiently small r ∈ Z. Then

P can be treated as a filtered R-module with the filtration (Pr)r∈Z and the leftD-module D⊗

R P can be considered as a filtered σ-R-module with the filtration((D⊗

R P )r)r∈Z where (D⊗R P )r is the R-submodule of the tensor product

D⊗R P generated by the set u⊗

x|u ∈ Di and x ∈ Pr−i (i ∈ Z, i ≤ r).


In what follows, while considering D⊗R P as a filtered σ-R-module (P is an

exhaustively and separately filtered module over the σ-ring R with the trivialfiltration) we shall always mean the filtration ((D⊗

R P )r)r∈Z.

Theorem 3.2.3 Let R be an Artinian difference ring with a basic set σ =α1, . . . , αn and let (Mr)r∈Z be an excellent filtration of a σ-R-module M .Then there exists a polynomial ψ(t) in one variable t with rational coeffi-cients such that ψ(r) = lR(Mr) for all sufficiently large r ∈ Z. Furthermore,

deg ψ(t) ≤ n and the polynomial ψ(t) can be written as ψ(t) =n∑

i=0

ci

(t + i

i

)where c0, c1, . . . , cn ∈ Z.

PROOF. Since the filtration (Mr)r∈Z is excellent, there exists r ∈ Zsuch that Ms = Ds−rMr0 for all s > r. Let us consider Mr as a filteredR-module with the filtration (Mr

⋂Mq)q∈Z and let π be the natural mapping

D⊗R Mr −→ M such that π(u

⊗x) = ux for any u ∈ D, x ∈ Mr. It is

easy to see that π((D⊗R Mr)q) = Mq for any r, q ∈ Z, q > r, so that π is

a surjective homomorphism of filtered D-modules (D⊗R Mr is treated as a

filtered D-module with respect to the filtration introduced above). It follows(see, Proposition 1.3.16) that the appropriate sequence of graded D-modulesgr(D⊗

R Mr)gr π−−→ gr M −→ 0 is exact. Taking into account the natural epi-

morphism D⊗R gr Mr

ρ−→ gr(D⊗R Mr) we obtain that the sequence of graded

D-modulesD

⊗R

gr Mrgr π ρ−−−−−→ gr M −→ 0

is exact. Since the D-module D⊗R gr Mr is finitely generated (because Mr

is a finitely generated R-module), gr M is also a finitely generated D-module.Applying Theorem 3.2.1, we obtain that there exists a polynomial φ(t) in onevariable t with rational coefficients such that φ(s) = lR(grsM) for all sufficientlylarge s ∈ Z (say, for all s > s0, where s0 ∈ Z), deg φ ≤ n − 1, and φ(t)

can be represented as φ(t) =n−1∑i=0

ai

(t + i

i

)where a0, a1, . . . , an−1 ∈ Z. Since

lR(Mr) = lR(Ms0−1) +r∑

s=s0

lR(grsM) for all r ∈ Z, r > s0, Corollary 1.4.7

shows that there exists a polynomial ψ(t) =n∑

i=0

ci

(t + i

i

)(c0, c1, . . . , cn ∈ Z)

such that ψ(r) = lR(Mr) for all r ∈ Z, r > s0. This completes the proof.

Definition 3.2.4 Let (Mr)r∈Z be an excellent filtration of a σ-R-module Mover an Artinian σ-ring R. Then the polynomial ψ(t), whose existence is esta-blished by Theorem 3.2.3, is called the difference (or σ-) dimension or char-acteristic polynomial of the module M associated with the excellent filtration(Mr)r∈Z.


Example 3.2.5 Let R be a difference field with a basic set σ = α1, . . . , αnand let D be the ring of σ-operators over R. Then the ring D can be consideredas a filtered σ-R-module with the excellent filtration (Dr)r∈Z. If r ∈ N, thenthe elements αk1

1 . . . αknn , where k1, . . . , kn ∈ N and

∑ni=1 ki ≤ r, form a basis

of the vector R-space Dr. Therefore, lR(Dr) = dimRDr = Card(k1, . . . , kn)

∈ Nn|k1 + · · · + kn ≤ r =(

r + n

n

)(see Proposition 1.4.9) whence ψD(t) =(

t + n

n

)is the characteristic polynomial of the ring D associated with the fil-

tration (Dr)r∈Z.

Let R be a difference ring with a basic set σ = α1, . . . , αn, D the ringof σ-operators over R, M a filtered σ-R-module with a filtration (Mr)r∈Z, andR[x] the ring of polynomials in one indeterminate x over R. Let D′ denote thesubring

∑r∈N

Dr

⊗R

Rxr of the ring D⊗R R[x] and M ′ denote the left D′-module∑

r∈N

Mr

⊗R

Rxr. (Note that the ring structure on D⊗R R[x] is induced by the

ring structure of D[x] with the help of the natural isomorphism of D-modulesD⊗

R R[x] ∼= D[x].)

Lemma 3.2.6 With the above notation, let all components of the filtration(Mr)r∈Z be finitely generated R-modules. Then the following conditions areequivalent:

(i) The filtration (Mr)r∈Z is excellent;(ii) M ′ is a finitely generated D′-module.

PROOF. Suppose that the filtration (Mr)r∈Z is excellent, that is, thereexists r0 ∈ Z such that Mr = Dr−r0Mr0 for any r ∈ Z, r ≥ r0. For any nonzeromodule Mi with i ∈ Z, i ≤ r0, let eij |1 ≤ j ≤ si be some finite system of gen-erators of Mi over R. Since there is only finitely many nonzero modules Mi withi ≤ r0, we obtain a finite system of elements E = eij |i ≤ r0, 1 ≤ j ≤ si. Let usshow that the finite set E′ = eij

⊗xk|eij ∈ E, 0 ≤ k ≤ r0 generates M ′ as a

D′-module by showing that every element of the form m⊗

axr (m ∈ Mr, a ∈ R)can be written as a linear combination of elements of E′ with coefficients in D′.(All tensor products are considered over the ring R, so we often use the symbol⊗

instead of⊗

R.) If r ≤ r0, then m =∑sr

j=1 cjerj for some c1, . . . , csr∈ R

whence m⊗

axr =(∑sr

j=1 cjerj

)⊗axr =

∑sr

j=1(cj

⊗a)(erj

⊗xr) (cj

⊗a ∈

D0

⊗Rx0 ⊆ D′ for j = 1, . . . , sr). Now, let r > r0. Since Mr = Dr−r0Mr0 , the

element m can be represented as a finite sum m =∑q

i=1 uimi where ui ∈ Dr−r0

and mi ∈ Mr0 (1 ≤ i ≤ r0). Since the elements er0j (1 ≤ j ≤ sr0) generateMr0 over R, there exist elements cij ∈ R 1 ≤ i ≤ q, 1 ≤ j ≤ sr0) such thatmi =

∑sr0j=1 cijer0j for i = 1, . . . , q.


Therefore, m⊗

axr = (∑q

i=1 uimi)⊗

axr =(∑q

i=1 ui

∑sr0j=1 cijer0j

)⊗axr

=∑sr0

j=1 u′jer0j

⊗axr where u′

j =∑q

i=1 uicij (1 ≤ j ≤ sr0). Thus, m⊗

axr =∑sr0j=1 u′

jer0j

⊗axr =

∑sr0j=1(u

′j

⊗axr−r0)(er0j

⊗xr0) where the elements

u′j

⊗axr−r0 lie in Dr−r0

⊗Rxr−r0 ⊆ D′. This completes the proof of the

implication (i)⇒(ii).Now, suppose that M ′ is a finitely generated D′-module. Without loss of

generality, we can assume that M ′ is generated by a finite set of homoge-neous elements ek

⊗xr(k)|1 ≤ k ≤ s for some positive integer s, r(k) ∈ N

for any k. Let r0 = maxr(k)|1 ≤ k ≤ s. If r ≥ r0 and m ∈ Mr, then theelement m

⊗xr can be represented as m

⊗xr =

∑sk=1 wk(ek

⊗xr(k)) with

w1, . . . , ws ∈ D′. Without loss of generality, we can assume that the elementswk are homogeneous, wk = vk

⊗xr−r(k) for some vk ∈ Dr−rk

(1 ≤ k ≤ s).(If wk are not homogeneous, one can use their homogeneous components as co-efficients in a representation of m

⊗xr as a linear combination of elements of

the form ek

⊗xr(k).) In this case, m

⊗xr =

∑sk=1(vk

⊗xr−r(k))(ek

⊗xr(k)) =

(∑s

k=1 vkek)⊗

xr. Since xr is a basis of the free R-module Rxr and ek ∈ Mr0

(1 ≤ k ≤ s), the last equality implies that m =∑s

k=1 vkek whence Mr =Dr−rk

Mr0 , so that the filtration (Mr)r∈Z is excellent.

Lemma 3.2.7 Let R be a Noetherian difference ring with a basic set σ =α1, . . . , αn whose elements act on the ring R as automorphisms. Then therings of σ-operators D and the ring D′ considered above are left Noetherian.

PROOF. First of all, note that the ring D is isomorphic to the ring of skewpolynomials R[z1, . . . , zn;α1, . . . , αn]. By Theorem 1.7.20, the latter ring is leftNoetherian, so the ring D is also left Noetherian.

Let us consider automorphisms β1, . . . , βn of the ring R[x] such thatβi(

∑mj=0 ajx

j) =∑m

j=0 βi(aj)xj (1 ≤ i ≤ n) for every polynomial∑m

j=0 ajxj in

R[x]. Furthermore, for every i = 1, . . . , n, let di denote the skew βi-derivation ofthe ring R[x] such that di(f) = xβi(f) − xf for any f ∈ R[x], and let S denotethe ring of skew polynomials R[x][z1, . . . , zn; d1, . . . , dn;β1, . . . , βn]. Then therings D′ and S are isomorphic, the corresponding isomorphism φ : D′ −→ Sacts on the homogeneous components Dr

⊗R Rxr (r ∈ Z) of the ring D′ as

follows: if ω⊗

axr = ωa⊗

xr ∈ Dr

⊗R Rxr (a ∈ R, ω ∈ Dr) and ωa =∑

i1,...,inai1,...,in

αi11 . . . αin

n (ai1,...,in∈ R for all indices i1, . . . , in and the sum is

finite), then φ(ω⊗

axr) =∑

i1,...,inai1,...,in

xr−i1−···−in(z1 + x)i1 . . . (zn + x)in .It is easy to see that the mapping φ is well-defined and bijective (such a mappingis usually called a homogenezation of the ring D′). By Theorem 1.7.20, the ringS is left Noetherian, so the ring D′ is also left Noetherian.

Theorem 3.2.8 Let R be a Noetherian difference ring with a basic set σ =α1, . . . , αn whose elements act on the ring R as automorphisms. Furthermore,let ρ : N −→ M be an injective homomorphism of filtered σ-R-modules and letthe filtration of the module M be excellent. Then the filtration of N is alsoexcellent.


PROOF. Let (Mr)r∈Z and (Nr)r∈Z be the given filtrations of the modulesM and N , respectively. Since the filtration (Mr)r∈Z is excellent, every R-moduleMr (r ∈ Z) is finitely generated and, therefore, Noetherian. Since ρ is an injectivehomomorphism of filtered modules, Nr (r ∈ Z) is isomorphic to a submodule ofMr, whence Nr is finitely generated over R.

Let D denote the ring of difference (σ-) operators over R with the usualfiltration (Dr)r∈Z and let the ring D′ =

∑r∈Z Dr

⊗R Rxr and the D′-modules

M ′ =∑

r∈Z Mr

⊗R Rxr, N ′ =

∑r∈Z Nr

⊗R Rxr, be defined as above. It fol-

lows from Lemmas 3.2.6 and 3.2.7 that M ′ is a finitely generated D′-moduleand the ring D′ is left Noetherian. Since the mapping ρ is compatible with thefiltrations, Nr ⊆ Mr for any r ∈ Z whence N ′ is a D′-submodule of M ′. Itfollows that N ′ is a finitely generated D′-module hence the filtration (Nr)r∈Z isexcellent.

Let R be a difference field with a basic set σ = α1, . . . , αn, D the ringof σ-operators over R, and M a finitely generated σ-R-module with generators

x1, . . . xm (i.e., M =m∑

i=1

Dxi). Then it is easy to see that the vector R-spaces

Mr =m∑

i=1

Dxi (r ∈ Z) form an excellent filtration of M . If (M ′r)r∈Z is another

excellent filtration of M , then there exists k ∈ Z such that DsM′k = M ′

k+s andDsMk = Mk+s for all s ∈ N. Since

⋃r∈Z

Mr =⋃r∈Z

M ′r = M , there exists p ∈ N

such that Mk ⊆ M ′k+p and M ′

k ⊆ Mk+p hence Mr ⊆ M ′r+p and M ′

r ⊆ Mr+p forall r ∈ Z, r ≥ k.

Thus, if ψ(t) and ψ1(t) are the characteristic polynomials of the σ-R-moduleM associated with the excellent filtrations (Mr)r∈Z and (M ′

r)r∈Z, respectively,then ψ(r) ≤ ψ1(r + p) and ψ1(r) ≤ ψ(r + p) for all sufficiently large r ∈ Z. Itfollows that deg ψ(t) = deg ψ1(t) and the leading coefficients of the polynomialsψ(t) and ψ1(t) are equal. Since the degree of a characteristic polynomial of Mdoes not exceed n, ∆nψ(t) = ∆nψ1(t) ∈ Z. (The n-th finite difference ∆nf(t)of a polynomial f(t) is defined as usual: ∆f(t) = f(t + 1) − f(t),∆2f(t) =∆(∆f(t)), . . . . Clearly, deg ∆kf(t) ≤ deg f(t)−k for any positive integer k, andif f(t) ∈ Z for all sufficiently large r ∈ Z, then every polynomial ∆kf(t) has thesame property. In particular, ∆kf(t) ∈ Z for all k ≥ deg f(t)). We arrive at thefollowing result.

Theorem 3.2.9 Let R be a difference field with a basic set σ = α1, . . . , αn,let M be a σ-R-module, and let ψ(t) be a characteristic polynomial associatedwith an excellent filtration of M . Then the integers ∆nψ(t), d = deg ψ(t), and∆dψ(t) do not depend on the choice of the excellent filtration.

Definition 3.2.10 Let R be a difference field with a basic set σ = α1, . . . , αn,M a finitely generated σ-R-module, and ψ(t) a characteristic polynomial associ-ated with an excellent filtration of M . Then the numbers ∆nψ(t), d = deg ψ(t),and ∆dψ(t) are called the difference (or σ-) dimension, difference (or σ-) type,


and typical difference (or σ-) dimension of M , respectively. These characteristicsof the σ-R-module M are denoted by δ(M) or σ-dimKM , t(M) or σ-typeKM ,and tδ(M) or σ-tdimKM , respectively.

The following two theorems give some properties of the difference dimension.

Theorem 3.2.11 Let R be an inversive difference field with a basic set of au-tomorphisms σ = α1, . . . , αn and let

0 −→ Ni−→ M

j−→ P −→ 0

be an exact sequence of finitely generated σ-R-modules. Then δ(N) + δ(P ) =δ(M).

PROOF. Let (Mr)r∈Z be an excellent filtration of the σ-R-module M andlet Nr = i−1(i(N)

⋂Mr), Pr = j(Mr) for any r ∈ Z. Clearly, (Pr)r∈Z is an

excellent filtration of the σ-R-module P , and Theorem 3.2.8 shows that (Nr)r∈Z

is an excellent filtration of N .Let ψN (t), ψM (t), and ψP (t) be the characteristic polynomials of the σ-R-

modules N , M , and P , respectively, associated with our excellent filtrations. Forany r ∈ Z, the exactness of the sequence 0 → Nr → Mr → j(Mr) → 0 impliesthe equality dimRNr +dimRj(Mr) = dimRMr, so that ψN (t)+ψP (t) = ψM (t).

Therefore, δ(M) = ∆nψM (t) = ∆n(ψN (t)+ψP (t)) = ∆nψN (t)+∆nψP (t)) =δ(N) + δ(P ).

Theorem 3.2.12 Let R be a difference field with a basic set of automorphismsσ = α1, . . . , αn, D the ring of σ-operators over R, and M a finitely generatedσ-R-module. Then δ(M) is equal to the maximal number of elements of Mlinearly independent over D.

PROOF. First of all, let us show that δ(M) = 0 if and only if every elementof M is linearly dependent over the ring D (i. e., for any z ∈ M , there exists aσ-operator u ∈ D such that uz = 0).

Suppose first that δ(M) = 0 and some element x ∈ M is linearly independentover D. Then the mapping φ : D −→ M , such that φ(u) = ux for any u ∈ D, isan injective homomorphism of left D-modules. Applying Theorem 3.2.11 to theexact sequence of finitely generated σ-R-modules

0 −→ D φ−→ M → M/φ(D) −→ 0

we obtain that δ(D) = δ(M) − δ(M/φ(D)) ≤ δ(M) = 0. On the other hand,

Example 3.2.5 shows that δ(D) = ∆n

(t + n

n

)= 1. Therefore, if δ(M) = 0, then

every element of M is linearly dependent over D.Conversely, suppose that every element of the σ-R-module M is linearly

dependent over D. Let elements ξ1, . . . , ξk generate M as a left D-module andlet Ni = Dξi (1 ≤ i ≤ k). Furthermore, for any i = 1, . . . , k, let φi denote the


D-homomorphism D −→ Ni such that φi(u) = uξi for any u ∈ D. Since xi

(1 ≤ i ≤ k) is linearly dependent over D, Ker φi = 0 and δ(Ker φi) ≥ 1. (As wehave already proved, if δ(Ker φi) = 0, then one could find two nonzero elementsu ∈ D and v ∈ Ker φi such that uv = 0. This is impossible, since the ring Ddoes not have zero divisors.) Applying Theorem 3.2.11 to the exact sequence offinitely generated σ-R-modules 0 −→ Ker φi −→ D −→ Ni −→ 0, we obtainthat 0 ≤ δ(Ni) = δ(D) − δ(Ker φi) = 1 − δ(Ker φi) ≤ 0 whence δ(Ni) = 0

for i = 1, . . . , k. It follows that 0 ≤ δ(M) = δ(k∑

i=1

Ni) ≤k∑

i=1

δ(Ni) = 0, so

that δ(M) = 0. (The inequality δ(k∑

i=1

Ni) ≤k∑

i=1

δ(Ni) is a consequence of the

fact that δ(P + Q) ≤ δ(P ) + δ(Q) for any two finitely generated σ-R-modulesP and Q. The last inequality, in turn, can be obtain by applying Theorem 3.2.11to the canonical exact sequence 0 −→ P −→ P + Q −→ P + Q/P −→ 0 : onecan easily see that δ(P + Q) = δ(P ) + δ(P + Q/P ) = δ(P ) + δ(Q/P

⋂Q) ≤

δ(P ) + δ(Q) ). Thus, δ(M) = 0 if and only if every element of M is linearlydependent over D.

Now, let p be the maximal number of elements of the σ-R-module M whichare linearly independent over the ring D. Let x1, . . . , xp be any system of

elements of M that are linearly independent over D, and let F =p∑

i=1

Dxi. Then

(p∑

i=1

Drxi)r∈Z is an excellent filtration of the σ-R-module F . It follows from

Example 3.2.5 that the characteristic polynomial associated with this filtration

has the form ψ(t) = p

(t + n

n

)whence δ(F ) = ∆nψ(t) = p. Furthermore, the

fact that every element of the σ-R-module M/F is linearly dependent over thering D implies that δ(M/F ) = 0. Applying Theorem 3.2.11 to the exact sequence0 −→ F −→ M −→ M/F −→ 0, we obtain that δ(M) = δ(F ) + δ(M/F ) =δ(F ) = p. This completes the proof.

3.3 Grobner Bases with Respect to SeveralOrderings and Multivariable DimensionPolynomials of Difference Modules

Let K be a difference field with a basic set σ = α1, . . . , αn, T the commutativesemigroup of all power products αk1

1 . . . αknn (k1, . . . , kn ∈ N) and D the ring of

σ-operators over K. Let us fix a partition of the set σ into a disjoint union ofits subsets:

σ = σ1 ∪ · · · ∪ σp (3.3.1)

3.3. MULTIVARIABLE DIMENSION POLYNOMIALS 167

where p ∈ N, and σ1 = α1, . . . , αn1, σ2 = αn1+1, . . . , αn1+n2, . . . , σp =αn1+···+np−1+1, . . . , αn (ni ≥ 1 for i = 1, . . . , p ;n1 + · · · + np = n).

For any element τ = αk11 . . . αkn

n ∈ T (k1, . . . , kn ∈ N), the numbers ordiτ =∑n1+···+ni

ν=n1+···+ni−1+1 kν (1 ≤ i ≤ p) are called the orders of τ with respect to σi (weassume that n0 = 0, so the indices in the sum for ord1τ change from 1 to n1). Asbefore, the order of the element τ is defined as ord τ =

∑nν=1 ki =

∑pi=1 ordiτ.

We shall consider p orders <1, . . . , <p on the set T defined as follows:τ = αk1

1 . . . αknn <i τ ′ = αl1

1 . . . αlnn if andonly if thevector (ordiτ, ord τ, ord1τ, . . . ,

ordi−1τ, ordi+1τ, . . . , ordpτ, kn1+···+ni−1+1, . . . , kn1+···+ni, k1, . . . , kn1+···+ni−1 ,

kn1+···+ni+1, . . . , kn) is less than the vector (ordiτ′, ord τ ′, ord1τ

′, . . . , ordi−1τ′,

ordi+1τ′, . . . , ordpτ

′, ln1+···+ni−1+1, . . . ,ln1+···+ni, l1, . . . , ln1+···+ni−1 , ln1+···+ni+

1, . . . , ln) with respect to the lexicographic order on Nn+p+1. It is easy to seethat the set T is well-ordered with respect to each of the orders <1, . . . , <p.

Let a σ-operator A ∈ D be written in the form A =∑d

i=1 aiui where 0 =ai ∈ K,ui ∈ T (1 ≤ i ≤ d) and ui = uj whenever i = j. Then for everyk = 1, . . . , p, the highest with respect to <k term ui is called the <k-leader ofA; it is denoted by u

(k)A .

If r1, . . . , rp are non-negative integers, then T (r1, . . . , rp) will denote the setof all elements τ ∈ T such that ordiτ ≤ ri (i = 1, . . . , p). The vector K-subspaceof D generated by the set T (r1, . . . , rp) will be denoted by Dr1,...,rp

.Setting Dr1,...,rp

= 0 for any (r1, . . . , rp) ∈ Zp \ Np, we obtain a fam-ily Dr1,...,rp

|(r1, . . . , rp) ∈ Zp of vector K-subspaces of D which is calledthe standard p-dimensional filtration of the ring D. It is easy to see thatDr1,...,rp

⊆ Ds1,...,spif (r1, . . . , rp) ≤P (s1, . . . , sp), where ≤P denotes the

product order on Zp (recall that this is a partial order on Zp such that(a1, . . . , ap) ≤P (b1, . . . , bp) if and only if ai ≤ bi for i = 1, . . . , p). Furthermore,Di1,...,ip

Dr1,...,rp= Dr1+i1,...,rp+ip

for any (r1, . . . , rp), (i1, . . . , ip) ∈ Np.

Definition 3.3.1 Let M be a vector σ-K-space (that is, a left D-module). Afamily Mr1,...,rp

|(r1, . . . , rp) ∈ Zp is said to be a p-dimensional filtration of Mif the following four conditions hold.

(i) Mr1,...,rp⊆ Ms1,...,sp

for any p-tuples (r1, . . . , rp), (s1, . . . , sp) ∈ Zp suchthat (r1, . . . , rp) ≤P (s1, . . . , sp).

(ii)⋃

(r1,...,rp)∈Zp

Mr1,...,rp= M .

(iii) There exists a p-tuple (r(0)1 , . . . , r

(0)p ) ∈ Zp such that Mr1,...,rp

= 0 if ri <

r(0)i for at least one index i (1 ≤ i ≤ p).(iv) Dr1,...,rp

Ms1,...,sp⊆ Mr1+s1,...,rp+sp

for any p-tuples (r1, . . . , rp), (s1, . . . ,sp) ∈ Zp.

If every vector K-space Mr1,...,rpis finite-dimensional and there exists an

element (h1, . . . , hp) ∈ Zp such that Dr1,...,rpMh1,...,hp

= Mr1+h1,...,rp+hpfor

any (r1, . . . , rp) ∈ Np, the p-dimensional filtration Mr1,...,rp|(r1, . . . , rp) ∈ Zp

is called excellent.


It is easy to see that if z1, . . . , zk is a finite system of generators of avector σ-K-space M , then ∑k

i=1 Dr1,...,rpzi|(r1, . . . , rp) ∈ Zp is an excellent

p-dimensional filtration of M .Let K be a difference field with a basic set σ = α1, . . . , αn, T the free

commutative semigroup generated by σ, and D the ring of σ-operators over K.Furthermore, we assume that a partition (3.3.1) of the set σ is fixed.

A vector σ-K-space which is a free module over the ring of difference oper-ators D is called a free σ-K-module or a free vector σ-K-space. If a free vectorσ-K-space E is generated (as a free D-module) by a finite system of elementse1, . . . , em, we say that E is a finitely generated free vector σ-K-space andcall e1, . . . , em free generators of E. In this case the elements of the form τeν

(τ ∈ T, 1 ≤ ν ≤ m) are called terms while the elements of the semigroup T arecalled monomilas. The set of all terms is denoted by Te. It is easy to see thatthis set generates E as a vector space over the field K.

By the order of a term τeν with respect to σi (1 ≤ i ≤ p) we mean the orderof the monomial τ with respect to σi. A term u′ = τ ′eµ is said to be a multipleof a term u = τeν if µ = ν and τ ′ is a multiple of τ in the semigroup T . In thiscase we write u|u′. (Clearly, u|u′ if and only if there exists τ ′′ ∈ T such thatu′ = τ ′′u. Then we write τ ′′ = u′

u .)The least common multiple of two terms u = τ1ei and v = τ2ej is defined as

follows:

lcm(u, v) =

0, if i = j,lcm(τ1, τ2)ei if i = j.

We shall consider p orderings of the set Te that correspond to the orderingsof the semigroup T introduced at the beginning of this section. These orderingsare denoted by the same symbols <1, . . . , <p and defined as follows: if τeµ, τ ′eµ ∈Te, then τeµ <i τ ′eν if and only if τ <i τ ′ in T or τ = τ ′ and µ < ν.

Since the set Te is a basis of the vector K-space E, every nonzero elementf ∈ E has a unique (up to the order of the terms in the sum) representation inthe form

f = a1τ1ei1 + · · · + alτleil(3.3.2)

where τ1ei1 , . . . , τleilare distinct elements of Te, and a1, . . . , al are nonzero

elements of K.

Definition 3.3.2 Let f be a nonzero element of the D-module E written inthe form (3.3.2.) and let τνeiν

(1 ≤ ν ≤ p) be the greatest term of the setτ1ei1 , . . . , τleil

with respect to an order <k (1 ≤ k ≤ p). Then the term τνeiν

is called the k-leader of the element f ; it is denoted by u(k)f . (Of course, it is

possible that u(j)f = u

(j′)f for some distinct numbers j and j′.) The non-negative

integer ordku(k)f is called the kth order of f and denoted by ordkf (k = 1, . . . , p).

The coefficient of u(k)f in f is said to be the k-leading coefficient of f ; it is denoted

by lck(f).


Definition 3.3.3 Let f, g ∈ E, g = 0, and let k, i1, . . . , il be distinct elements ofthe set 1, . . . , p. Then f is said to be (<k, <i1 , . . . , <il

)-reduced with respectto g if f does not contain any multiple τu

(k)g (τ ∈ T ) such that ordiν

(τu(iν)g ) ≤

ordiνu

(iν)f (ν = 1, . . . , l).

An element f ∈ E is said to be (<k, <i1 , . . . , <il)-reduced with respect to a

set G ⊆ E, if f is (<k, <i1 , . . . , <il)-reduced with respect to every element of G.

Let us consider p − 1 new symbols z1, . . . , zp−1 and the free commutativesemigroup Γ of all power products γ = αk1

1 . . . αknn zl1

1 . . . zlp−1p−1 with non-negative

integer exponents. Let Γe = γej |γ ∈ Γ, 1 ≤ j ≤ m = Γ × e1, . . . , en. Forany nonzero element f ∈ E, let di(f) = ordiu

(i)f − ordiu

(1)f (2 ≤ i ≤ p) and let

ρ : E → Γe be defined by ρ(f) = zd2(f)1 . . . z

dp(f)p−1 u

(1)f .

Definition 3.3.4 With the above notation, let N be a D-submodule of E. A fi-nite set of nonzero elements G = g1, . . . , gr ⊆ N is called a Grobner basisof N with respect to the orders <1, . . . , <p if for any 0 = f ∈ N , thereexists gi ∈ G such that ρ(gi)|ρ(f) in Γe.

It is clear that every Grobner basis of N with respect to the orders <1, . . . , <p

is a Grobner basis of N with respect to <1 in the usual sense. Therefore (seeTheorem 1.8.17), every Grobner basis of N with respect to the orders <1, . . . , <p

generates N as a left D-module.A set g1, . . . , gr ⊆ E is said to be a Grobner basis with respect to the orders

<1, . . . , <p if G is a Grobner basis of N =∑r

i=1 Dgi with respect to <1, . . . , <p .

Definition 3.3.5 Given f, g, h∈E, with g = 0, we say that the element f(<k,

<i1 , · · · <il)-reduces to h modulo g in one step and write f

g−−−−−−−−−→<k,<i1 ,...,<il

h if

and only if f contains some term w with a coefficient a such that u(k)g |w,

h = f − a

(w

u(k)g

(lck(g))

)−1w

u(k)g

g

and ordiν

(w

u(k)g

u(iν)g

)≤ ordiν

u(iν)f (1 ≤ ν ≤ l).

Definition 3.3.6 Let f, h ∈ E and let G = g1, . . . , gr be a finite set of non-zero elements of E. We say that f (<k, <i1 , . . . , <il

)-reduces to h modulo G

and write fG−−−−−−−−−→

<k;<i1 ,...,<il

h if and only if there exists a sequence of elements

g(1), g(2), . . . g(q) ∈ G and a sequence of elements h1, . . . , hq−1 ∈ E such that

fg(1)

−−−−−−−−−→<k,<i1 ,...,<il

h1g(2)

−−−−−−−−−→<k,<i1 ,...,<il

. . .g(q−1)

−−−−−−−−−→<k,<i1 ,...,<il

hq−1g(q)

−−−−−−−−−→<k,<i1 ,...,<il

h .


Theorem 3.3.7 With the above notation, let G = g1, . . . , gr ⊆ E be aGrobner basis with respect to the orders <1, . . . , <p on Te. Then for any f ∈ E,

there exist elements g ∈ E and Q1, . . . , Qr ∈ D such that f − g =r∑

i=1

Qigi and

g is (<1, . . . , <p)-reduced with respect to the set G.

PROOF. If f is (<1, . . . , <p)-reduced with respect to G, the statement isobvious (one can set g = f). Suppose that f is not (<1, . . . , <p)-reduced withrespect to G. Let u

(j)i = u

(j)gi (1 ≤ i ≤ r, 1 ≤ j ≤ p) and let ai be the coefficient

of the term u(1)i in gi (i = 1, . . . , r). In what follows, a term wh, that appears

in an element h ∈ E, will be called a G-leader of h if wh is the greatest (withrespect to the order <1) term among all terms τu

(1)i (τ ∈ T, 1 ≤ i ≤ r) that

appear in h and satisfy the condition ordj(τu(j)i ) ≤ ordju

(j)h for j = 2, . . . , p.

Let wf be the G-leader of the element f and let cf be the coefficient of wf inf . Then wf = τu

(1)i for some τ ∈ T , 1 ≤ i ≤ r, such that ordj(τu

(j)i ) ≤ ordju

(j)f

for j = 2, . . . , p. Without loss of generality we may assume that i corresponds tothe maximum (with respect to the order <1) 1-leader u

(1)i in the set of all such

1-leaders of elements of G. Let us consider the element f ′ = f − cf (τ(ai))−1τgi.Obviously, f ′ does not contain wf and ordj(u

(j)f ′ ) ≤ ordju

(j)f for j = 2, . . . , p.

Furthermore, f ′ cannot contain any term of the form τ ′u(1)k (τ ′ ∈ T , 1 ≤ k ≤

r), that is greater than wf (with respect to <1) and satisfies the conditionordj(τ ′u(j)

k ) ≤ ordju(j)f ′ for j = 2, . . . , p. Indeed, if the last inequality holds, then

ordj(τ ′u(j)k ) ≤ ordju

(j)f , so that the term τ ′u(1)

k cannot appear in f . This term

cannot appear in τgi either, since u(1)τgi = τu

(1)gi = wf <j τ ′u(1)

k . Thus, τ ′u(1)i

cannot appear in f ′, whence the G-leader of f ′ is strictly less (with respectto the order <1) than the G-leader of f . Applying the same procedure to theelement f ′ and continuing in the same way, we obtain an element g ∈ E suchthat f − g is a linear combination of elements g1, . . . , gr with coefficients in Dand g is (<1, . . . , <p)-reduced with respect to G.

The process of reduction described in the proof of the last theorem can berealized by the following algorithm which can be used for the reduction withrespect to any finite set of elements of the free D-module E.

Algorithm 3.3.8 (f, r, g1, . . . , gr; g; Q1, . . . , Qr)Input: f ∈ E, a positive integer r, G = g1, . . . , gr ⊆ E where gi = 0

for i = 1, . . . , r

Output: Element g ∈ E and elements Q1, . . . , Qr ∈ D such thatg = f − (Q1g1 + · · · + Qrgr) and g is reduced with respect to G

BeginQ1 := 0, . . . , Qr := 0, g := f

While there exist i, 1 ≤ i ≤ r, and a term w, that appears in g with anonzero coefficient c(w), such that u

(1)gi |w and


ordj( w

u(1)gi

u(j)gi ) ≤ ordju

(j)g for j = 2, . . . , p do

z:= the greatest (with respect to <1) of the terms w that satisfythe above conditions.k:= the smallest number i for which u

(1)gi is the greatest (with

respect to <1) 1-leader of an element gi ∈ G such thatu

(1)gi |z and ordj( z

u(1)gi

ugi) ≤ ordju

(j)g for j = 2, . . . , p.

Qk := Qk + c(z)(

z

u(1)gk

(lc1(gk))−1

z

u(1)gk

gk

g := g − c(z)(

z

u(1)gk

(lc1(gk))−1

z

u(1)gk

gk

End

The following example illustrates Algorithm 3.3.8.

Example 3.3.9 Let K = Q(x, y) be the field of rational fractions in two vari-ables x and y over Q treated as a difference field with a basic set σ = α1, α2where α1f(x, y) = f(x + 1, y) and α2f(x, y) = f(x, y + 1) for every f(x, y) ∈Q(x, y). We consider a partition σ = σ1

⋃σ2 of the set σ where σ1 = α1 and

σ2 = α2.Let D be the ring of σ-operators over K, let E be the free left D-module

with free generators e1 and e2, and let G = g1, g2 where

g1 = x2α1α2e1 + xα22e2 − xyα1e1 + y2e2,

g2 = 2xα21α2e2 + α1α

22e1 + yα2e1.

We are going to to apply Algorithm 3.3.8 to reduce the element

f = xα21α

22e1 + yα1α

23e2 − xyα1α2e1

with respect to G.Notice that u

(1)g1 = α1α2e1, u

(2)g1 = α2

2e2, u(1)g2 = α2

1α2e2, and u(2)g2 = α1α

22e1.

Furthermore, g1 and g2 are (<1, <2)-reduced with respect to each other. Indeed,u

(1)g2 divides no term of g1, so g1 is (<1, <2)-reduced with respect to g2. The

only term of g2 which is a multiple of u(1)g1 is w = α1α

22e1. Since w = α2u

(1)g1

and ord2(α2u(2)g1 ) = ord2(α3

2) = 3 > ord2u(2)g2 = 2, we obtain that g2 is (<1, <2)-

reduced with respect to g1.According to Algorithm 3.3.8 the first step of the (<1, <2)-reduction of the

element f with respect to G is as follows.

f → f1 = f − x1

(x + 1)2α1α2g1

= f − x

(x + 1)2[(x + 1)2α2

1α22e1 + (x + 1)α1α

32e2

− (x + 1)(y + 1)α21α2e1 + (y + 1)2α1α2e2

]=

(y − x

x + 1

)α1α

32e2 +

x(y + 1)x + 1

α21α2e1 − x(y + 1)2

(x + 1)2α1α2e2 .


(Notice that ord2(α1α2u(2)g1 ) ≤ ord2u

(2)f ; both orders are equal to 3.)

Since u(1)f1

= α21α2e1 = α1u

(1)g1 and the order of α1u

(2)g1 = α1α

22e2 with respect

to σ2 does not exceed ord2(u(2)f1

) = 3, the second step of the (<1, <2)-reduction is

f1 → f2 = f1 − x(y + 1)x + 1

1(x + 1)2

α1g1

=xy(y + 1)(x + 1)2

α21e1 +

(y − x(y + 1)

x + 1

)α1α

32e2 − x(y + 1)

(x + 1)2α1α

22e2

− x(y + 1)2

(x + 1)2α1α2e2 − xy2(y + 1)

(x + 1)3α1e2 .

Since no term of f2 is divisible by u(1)g1 = α1α2e1 or by u

(1)g2 = α2

1α2e2, theelement f2 is (<1, <2)-reduced with respect to G.

The proof of Theorem 3.3.7 shows that if G is a Grobner basis of aD-submodule N of E, then a reduction step described in Definition 3.3.5(with some g ∈ G) can be applied to every nonzero element f ∈ N . As a resultof such a step, we obtain an element of N whose G-leader is strictly less (withrespect to <1) than the G-leader of f . This observation leads to the followingstatement.

Proposition 3.3.10 Let G = g1, . . . , gr be a Grobner basis of a D-submoduleN of E with respect to the orders <1, . . . , <p. Then

(i) f ∈ N if and only if fG−−−−−−−−→

<1,<2,...,<p

0 .

(ii) If f ∈ N and f is (<1, <2, . . . , <p)-reduced with respect to G, then f ∈ 0.

Definition 3.3.11 Let f and g be two nonzero elements in the free D-moduleE and let k ∈ 1, . . . , p. Then the element

Sk(f, g) =

(lcm(u(k)

f , u(k)g )

u(k)f

(lck(f))

)−1lcm(u(k)

f , u(k)g )

u(k)f

f

−(

lcm(u(k)f , u

(k)g )

u(k)g

(lck(g))

)−1lcm(u(k)

f , u(k)g )

u(k)g

g

is called the kth S-polynomial of f and g.

With the above notation, one can obtain the following two statements thatgeneralize the corresponding properties of the classical Grobner basis.

Proposition 3.3.12 Let f, g1, . . . , gr be nonzero elements in E (r ≥ 1) and let

f =r∑

i=1

ciωigi where ωi ∈ T , ci ∈ K (1 ≤ i ≤ r). Let k ∈ 1, . . . , p and for any


ν, j ∈ 1, . . . , r, let u(k)νj = lcm(u(k)

gν , u(k)gj ). Furthermore, suppose that ω1u

(k)g1 =

· · · = ωru(k)gr = u, u

(k)g <k u and there is a nonempty set I ⊆ 1, . . . , p \ k

such that ωiu(l)gi ≤l u

(l)f for all i ∈ 1, . . . , r, l ∈ I. Then there exist elements

cνj ∈ K (1 ≤ ν ≤ s, 1 ≤ j ≤ t) such that

f =s∑

ν=1

t∑j=1

cνjθνjSk(gν , gj)

where θνj =u

u(k)νj

and

θνju(k)Sk(gν ,gj)

<k u, θνju(l)Sk(gν ,gj)

≤l u(l)f (1 ≤ ν ≤ s, 1 ≤ j ≤ t, l ∈ I).

PROOF. For every i = 1, . . . , r, let di = lck(ωigi) = ωi(lck(gi)). Sinceω1u

(k)g1 = · · · = ωru

(k)gr = u and u

(k)f <k u,

∑ri=1 cidi = 0. Let hi = d−1

i ωigi =(ωi(lck(gi))−1ωigi (i = 1, . . . , r). Then lck(hi) = 1 and

f =r∑

i=1

ciωigi =r∑

i=1

cidihi = c1d1(h1 − h2) + (c1d1 + c2d2)(h2 − h3)

+ · · · + (c1d1 + · · · + cr−1dr−1)(hr−1 − hr).

(The last sum should end with the term (c1d1+· · ·+crdr)hr in order to representan identity, but this term is equal to zero.)

For every ν, j ∈ 1, . . . , r, ν = j, let τνj =u

(k)νj

u(k)gν

, γνj =u

(k)νj

u(k)gj

, and θνj =u

u(k)νj

(since u(k)gν |u and u

(k)gj |u, the term u

(k)νj divides u). Then

θνjSk(gν , gj) = θνj

[(τνj(lck(gν)))−1τνjgν − (γνj(lck(gj)))−1γνjgj

]= [θνj(τνj(lck(gν)))]−1 u

u(k)gν

gν − [θνj(γνj(lck(gj)))]−1 u

u(k)gj

gj

= [ων(lck(gν))]−1ωνgν − [ωj(lck(gj))]

−1ωjgj

= hν − hj .

It follows that f = c1d1θ12Sk(g1, g2) + (c1d1 + c2d2)θ23Sk(g2, g3) + · · · +

(r−1∑i=1

cidi)θr−1,rSk(gr−1, gr), θi,i+1u(k)Sk(gi,gi+1)

= u(k)θi,i+1Sk(gi,gi+1)

= u(k)hi−hi+1

<k

u (since u(k)hi

= u(k)hi+1

and lck(hi) = lck(hi+1) = 1), and we have the inequalities

θi,i+1u(l)Sk(gi,gi+1)

≤l u(l)f for all i = 1, . . . , r − 1, l ∈ I. This completes the proof.

Theorem 3.3.13 With the above notation, let G = g1, . . . , gr be a Grobnerbasis of a D-submodule N of E with respect to each of the following sequences of


orders: <p; <p−1, <p; . . . ; <k+1, . . . , <p (1 ≤ k ≤ p−1). Furthermore, supposethat

Sk(gi, gj)G−−−−−−−−−−→

<k,<k+1,...,<p

0 for any gi, gj ∈ G.

Then G is a Grobner basis of N with respect to <k, <k+1, . . . , <p.

PROOF. First, let us prove that under the conditions of the theorem everyelement f ∈ N can be represented as

f =r∑

i=1

higi (3.3.3)

where h1, . . . , hr ∈ D,

max<ku(k)

hiu(k)

gi| 1 ≤ i ≤ r = u

(k)f (3.3.4)

andordj(u

(j)hi

u(j)gi

) ≤ ordju(j)f (j = k + 1, . . . , p). (3.3.5)

(The symbol max<kin (3.3.4) means the maximum with respect to the term

order <k.)We proceed by induction on p− k. If p− k = 0, that is k = p, our statement

is a classical result of the theory of Grobner bases (see Theorem 1.8.18; the factthat we consider modules over D rather than modules over a polynomial ringdoes not play an essential role). Let k < p and let f ∈ N . By the inductionhypothesis, f can be written as

f =r∑

i=1

Higi (3.3.6)

where H1, . . . , Hr ∈ D,

max<k+1u(k+1)Hi

u(k+1)gi

| 1 ≤ i ≤ r = u(k+1)f (3.3.7)

and

ordj(u(j)Hi

u(j)gi

) ≤ ordju(j)f (j = k + 2, . . . , p; i = 1, . . . , r). (3.3.8)

Let us choose among all representations of f in the form (3.3.6) with condi-tions (3.3.7) and (3.3.8) a representation with the smallest (with respect to <k)possible term

u = max<ku(k)

Hiu(k)

gi| 1 ≤ i ≤ r.

Setting di = lck(Hi) (1 ≤ i ≤ r) and breaking the sum in (3.3.6) in two parts,we can write

f =r∑

i=1

Higi =∑

u(k)Hi

u(k)gi

=u

Higi +∑

u(k)Hi

u(k)gi

<ku

Higi


or

f =∑

u(k)Hi

u(k)gi

=u

diu(k)Hi

gi +∑

u(k)Hi

u(k)gi

=u

(Hi − diu(k)Hi

)gi +∑

u(k)Hi

u(k)gi

<ku

Higi (3.3.9)

Notice that if u = u(k)f , then the expansion (3.3.9) satisfies conditions (3.3.3) -

(3.3.5). Indeed, in this case u(k+1)f = max<k+1u(k+1)

Hiu

(k+1)gi | 1 ≤ i ≤ r (see

(3.3.7)), hence maxordk+1(u(k+1)Hi

u(k+1)gi ) | 1 ≤ i ≤ r ≤ ordk+1u

(k+1)f and

ordj(u(j)Hi

u(j)gi ) ≤ ordju

(j)f for j = k + 2, . . . , p; i = 1, . . . , r ((see (3.3.8)).

Suppose that u(k)f <k u. Since u = max<k

u(k)Hi

u(k)gi | 1 ≤ i ≤ r, we have

u(k)

Hi−diH(k)i

<k u (1 ≤ i ≤ r), so the expansion (3.3.9) implies that the k-leader

of the first sum in (3.3.9) does not exceed u with respect to <k. Furthermore,it is clear that u

(k)Hi

u(k)gi = u for any term in the sum

f =∑

u(k)Hi

u(k)gi

=u

diu(k)Hi

gi (3.3.10)

and for every j = k + 1, . . . , p,

ordju(j)

f≤ maxi∈Iordj(u

(k)Hi

u(j)gi

) ≤ maxi∈Iordj(u(j)Hi

u(j)gi

) ≤ ordju(j)f

where I denotes the set of all indices i ∈ 1, . . . , r that appear in (3.3.10).Let u

(k)νj = lcm(u(k)

gν , u(k)gj ) for any ν, j ∈ I, ν = j and let τνj = u

u(k)νj

∈ T .

(Since u = u(k)Hi

u(k)gi for every i ∈ I, u

(k)νj |u.)

By Proposition 3.3.12, there exist elements cνj ∈ K such that

f =∑ν,j

cνjτνjSk(gν , gj) (3.3.11)

whereu

(k)τνjSk(gν ,gj)

<k u(k)

f= u

andordju

(j)τνjSk(gν ,gj)

≤ ordju(j)

f(j = k + 1, . . . , p).

Since Sk(gν , gj)G−−−−−−−−−−→

<k,<k+1,...,<p

0, there exist qiνj ∈ D such that

Sk(gν , gj) =r∑

i=1

qiνjgi

and

u(k)qiνj

u(k)gi

≤k u(k)Sk(gν ,gj)

,

ordl(u(l)qiνj

u(l)gi

) ≤l ordlu(l)Sk(gν ,gj)


for l = k + 1, . . . , p. Thus, for any indices ν, j in the sum (3.3.11) we have

τνjSk(gν , gj) =r∑

i=1

(τνjqiνj)gi

whereu(k)

τνjqiνju(k)

gi= τνju

(k)qiνj

u(k)gi

≤k τνju(k)Sk(gν ,gj)

<k u.

It follows that

f =∑ν,j

cνj

r∑i=1

(τνjqiνj)gi =r∑

i=1

⎛⎝∑ν,j

cνjτνjqiνj

⎞⎠ gi =r∑

i=1

Higi (3.3.12)

whereHi =

∑ν,j

cνjτνjqiνj (1 ≤ i ≤ r)

andu

(k)

Hiu(k)

gi<k u (1 ≤ i ≤ r).

Furthermore, for any l = k + 1, . . . , p, we have

ordl(u(l)

Hiu(l)

gi) ≤ maxν,j

ordl

(τνju

(l)

Hiu(l)

gi

)≤ maxν,j

max

ordl

(τνj

u(k)νj

u(k)gν

u(l)gν

), ordl

(τνj

u(k)νj

u(k)gj

u(l)gj

)

= maxν,j

max

ordl

(u

u(k)gν

u(l)gν

), ordl

(u

u(k)gj

u(l)gj

)

≤ ordlu = ordlu(k)

f≤ ordlu

(l)

f,

so that representation (3.3.12) satisfies the condition

ordl(u(l)

Hiu(l)

gi) ≤ ordlu

(l)

f(3.3.13)

for i = 1, . . . , r. Substituting (3.3.12) into (3.3.9) we obtain

f =r∑

i=1

Higi +∑

u(k)Hi

u(k)gi

=u

(Hi − diu(k)Hi

)gi +∑

u(k)Hi

u(k)gi

<ku

Higi (3.3.14)

where, denoting each Hi−diu(k)Hi

in the second sum by H ′i, we have the following

conditions:(i) u

(k)

Hiu

(k)gi <k u

(ii) u(k)H′

iu

(k)gi <k u for any term with index i in the second sum in (3.3.14).

(iii) u(k)Hi

u(k)gi <k u for any term with index i in the third sum in (3.3.14).


Also, for every l = k + 1, . . . , p, the inequality (3.3.13) implies thatordl(u

(l)

Hiu

(l)gi ) ≤ ordlu

(l)

f. Therefore,

ordl(u(l)

Hiu(l)

gi) ≤ maxordl(u

(k)Hi

u(l)gi

) ≤ maxordl(u(l)Hi

u(l)gi

) ≤ ordlu(l)f

where the maxima are taken over the set of all indices i that appear in the firstsum in (3.3.9). Furthermore, inequality (3.3.8) implies that for every index i inthe second sum in (3.3.14), one has

ordl(u(l)H′

iu(l)

gi) ≤ ordl(u

(k)Hi

u(l)gi

) ≤ ordlu(l)f (l = k + 1, . . . , p)

and for every index i in the third sum in (3.3.14) we have

ordl(u(l)Hi

u(l)gi

) ≤ ordlu(l)f (l = k + 1, . . . , p).

Thus, (3.3.14) is a representation of f in the form (3.3.6) with conditions (3.3.7)

and (3.3.8) such that if one writes (3.3.14) as f =r∑

i=1

H ′igi (combining the sums

in (3.3.14)), then maxu(k)

H′1u(k)

g1, . . . , u

(k)

H′r

u(k)gr

<k u and one has conditions of

the types (3.3.7) and (3.3.8). We have arrived at a contradiction with our choiceof representation (3.3.6) of f with conditions (3.3.7), (3.3.8) and the smallest(with respect to <k) possible value of maxu(k)

H′i

u(k)gi | 1 ≤ i ≤ r = u. Thus,

every element f ∈ N can be represented in the form (3.3.3) with conditions(3.3.4) and (3.3.5).

The last theorem allows one to construct a Grobner basis of a D-moduleN ⊆ E with respect to <1, . . . , <p starting with a Grobner basis of N withrespect to <p .

Exercise 3.3.14 Write down the algorithm for constructing a Grobner basisof a D-module N ⊆ E with respect to <1, . . . , <p starting with arbitrary finitesystem of generators of N over D. (Use Theorem 3.3.13 to build an algorithmconsisting of a sequence of p analogs of the Buchberger Algorithm 1.8.12). Thenprove that the constructed algorithm terminates at a desired Grobner basis (usethe idea of the proof of the termination of Algorithm 1.8.12).

Theorem 3.3.15 Let D be the ring of difference (σ-) operators over a difference(σ-) field K, M a vector σ-K-space generated (as a left D-module) by a finiteset f1, . . . , fm, and E a free left D-module with free generators e1, . . . , em.Let π : E −→ M be the natural D-epimorphism (π(ei) = fi for i = 1, . . . , m),N = Ker π, and G = g1, . . . , gd a Grobner basis of N with respect to <1, . . . ,

<p. Furthermore, for any (r1, . . . , rp) ∈ Zp, let Mr1...rp=

m∑i=1

Dr1...rpfi and let


Vr1...rp= u ∈ Te|ordiu ≤ ri for i = 1, . . . , p, and u = τu

(1)g for any τ ∈ T, g ∈

G, Wr1...rp=u ∈ Te \ Vr1...rp

|ordiu ≤ ri for i = 1, . . . , p and for every τ ∈T,

g ∈ G such that u = τu(1)g , there exists i ∈ 2, . . . , p such that ordiτu

(i)g > ri,

and Ur1...rp= Vr1...rp

⋃Wr1...rp

.Then for any (r1, . . . , rp) ∈ Np, the set π(Ur1...rp

) is a basis of the vectorK-space Mr1...rp

.

PROOF. Let us prove, first, that every element τfi (1 ≤ i ≤ m, τ ∈T (r1, . . . , rp)), which does not belong to π(Ur1...rp

), can be written as a fi-nite linear combination of elements of π(Ur1...rp

) with coefficients in K (sothat the set π(Ur1...rp

) generates the vector K-space Mr1...rp). Indeed, since

τfi /∈ π(Ur1...rp), τei /∈ Ur1...rp

whence τei = τ ′u(1)gj for some τ ′ ∈ T , 1 ≤ j ≤ d,

such that ordν(τ ′u(ν)gj ) ≤ rν (ν = 2, . . . , p).

Let us consider the element gj = aju(1)gj +. . . (aj ∈ K, aj = 0), where dots are

placed instead of the sum of the other terms of gj with nonzero coefficients (ob-viously, those terms are less than u

(1)gj with respect to the order <1). Since gj ∈

N = Ker π, π(gj) = ajπ(u(1)gj ) + · · · = 0, whence π(τ ′gj) = ajπ(τ ′u(1)

gj ) + · · · =ajπ(τei)+· · · = ajτfi+· · · = 0, so that τfi is a finite linear combination with co-efficients in K of some elements τlfl (1 ≤ l ≤ m) such that τl ∈ T (r1, . . . , rp) andτlel <1 τ ′u(1)

gj . (ord1τl ≤ r1, since τlel <1 τei and τ ∈ T (r1, . . . , rp); ordν τl ≤ rν

(ν = 2, . . . , p), because τlel ≤ν u(ν)τ ′gj

= τ ′u(ν)gj and ordν(τ ′u(ν)

gj ) ≤ rν .) Thus, wecan apply the induction on τej (τ ∈ T, 1 ≤ j ≤ m) with respect to the order<1 and obtain that every element τfi (τ ∈ T (r1, . . . , rp), 1 ≤ j ≤ m) can bewritten as a finite linear combination of elements of π(Ur1...rp

) with coefficientsin the field K.

Now, let us prove that the set π(Ur1...rp) is linearly independent over K.

Suppose thatk∑

i=1

aiπ(ui) = 0 for some u1, . . . , uk ∈ Ur1...rp, a1, . . . , ak ∈ K.

Then h =k∑

i=1

aiui is an element of N which is (<1, . . . , <p)-reduced with respect

to G. Indeed, if a term u = τej appears in h (so that u = ui for some i =1, . . . , k), then either u is not a multiple of any u

(1)gν (1 ≤ ν ≤ d) or u = τu

(1)gν

for some τ ∈ T, 1 ≤ ν ≤ d, such that ordµ(τu(µ)gν ) > rµ ≥ ordµu

(µ)h for some

µ, 2 ≤ µ ≤ p. By Proposition 3.3.10, h = 0, whence a1 = · · · = ak = 0. Thiscompletes the proof of the theorem.

Now we are ready to prove the main result of this section, the theorem ona multivariable dimension polynomial associated with a difference vector spacewith an excellent p-dimensional filtration.

Theorem 3.3.16 Let K be a difference field with a basic set σ = α1, . . . , αnand let D be the ring of difference operators over K equipped with the standard


p-dimensional filtration corresponding to partition (3.3.1) of the set σ. Further-more, let ni = Card σi (i = 1, . . . , p) and let Mr1...rp

|(r1, . . . , rp) ∈ Zp be anexcellent p-dimensional filtration of a vector σ-K-space M . Then there exists apolynomial φ(t1, . . . , tp) ∈ Q[t1, . . . , tp] such that

(i) φ(r1, . . . , rp) = dimKMr1...rpfor all sufficiently large (r1, . . . , rp) ∈ Zp;

(ii) degtiφ ≤ ni (1 ≤ i ≤ p), so that deg φ ≤ n and the polynomial φ(t1, . . . , tp)

can be represented as

φ(t1, . . . , tp) =n1∑

i1=0

. . .

np∑ip=0

ai1...ip

(t1 + i1

i1

). . .

(tp + ip

ip

)

where ai1...ip∈ Z for all i1, . . . , ip.

PROOF. Since the p-dimensional filtration Mr1...rp|(r1, . . . , rp) ∈ Zp is

excellent, there exists an element (h1, . . . , hp) ∈ Zp such that Dr1,...,rpMh1,...,hp

= Mr1+h1,...,rp+hpfor any (r1, . . . , rp) ∈ Np. Furthermore, Mh1,...,hp

is a finite-dimensional vector K-space and any its basis generates M as a left D-module,

so M =m∑

i=1

Dyi for some elements y1, . . . , ym ∈ Mh1,...,hp.

Let E be a free D-module with a basis e1, . . . , em, let N be the kernel ofthe natural σ-epimorphism π : E → M (π(ei) = yi for i = 1, . . . , m), and letthe set Ur1...rp

(r1, . . . , rp ∈ N) be the same as in the conditions of Theorem3.3.15. Furthermore, let G = g1, . . . , gd be a Grobner basis of N with respectto <1, . . . , <p. By Theorem 3.3.15, for any r1, . . . , rp ∈ N, π(Ur1,...,rp

) is a basisof the vector K-space Mr1,...,rp

. Therefore, dimKMr1,...,rp= Card π(Ur1,...,rp

) =CardUr1,...,rp

. (It was shown in the second part of the proof of Theorem 3.3.15that the restriction of the mapping π on Ur1,...,rp

is bijective).

Let U ′r1,...,rp

= w ∈ Ur1,...,rp|w is not a multiple of any element u

(i)gi (1 ≤

i ≤ d) and let U ′′r1,...,rp

= w ∈ Ur1,...,rp| there exists gj ∈ G and τ ∈ T

such that w = τu(1)gj and ordν(τu

(ν)gj ) > rν for some ν, 2 ≤ ν ≤ p. Then

Ur1,...,rp= U ′

r1,...,rp

⋃U ′′

r1,...,rpand U ′

r1,...,rp

⋂U ′′

r1,...,rp= ∅, whence

CardUr1,...,rp= CardU ′

r1,...,rp+ CardU ′′

r1,...,rp.

By Theorem 1.5.2, there exists a numerical polynomial ω(t1, . . . , tp) in p vari-ables t1, . . . tp such that ω(r1, . . . , rp) = CardU ′

r1,...,rpfor all sufficiently large

(r1, . . . , rp) ∈ Np.In order to express CardU ′′

r1,...,rpin terms of r1, . . . , rp, let us set aij =

ordiu(1)gj and bij = ordiu

(i)gj for i = 1, . . . , p; j = 1, . . . , d. Clearly, a1j = b1j and

aij ≤ bij for i = 1, . . . , p; j = 1, . . . , d. Furthermore, for any µ = 1, . . . , pand for any integers k1, . . . , kµ such that 2 ≤ k1 < · · · < kµ ≤ p, letVj;k1,...,kµ

(r1, . . . , rp) = τu(1)gj |ordiτ ≤ ri − aij for i = 1, . . . , p and ordντ >

rν − bνj if and only if ν is equal to one of the numbers k1, . . . , kµ.


Then CardVj;k1,...,kµ(r1, . . . , rp) = φj;k1,...,kµ

(r1, . . . , rp), where φj;k1,...,kµ

(t1, . . . , tp) is a numerical polynomial in p variables t1, . . . , tp defined by theformula

φj;k1,...,kµ(t1, . . . , tp)

=(

t1 + n1 − b1j

n1

). . .

(tk1−1 + nk1−1 − bk1−1,j

nk1−1

)×

[(tk1 + nk1 − ak1,j

nk1

)−

(tk1 + nk1 − bk1,j

nk1

)]×(

tk1+1 + nk1+1 − bk1+1,j

nk1+1

). . .

(tkµ−1 + nkµ−1 − bkµ−1,j

nkµ−1

)×

[(tkµ

+ nkµ− akµ,j

nkµ

)−

(tkµ

+ nkµ− bkµ,j

nkµ

)]. . .

(tp + np − bpj

np

).

(3.3.15)

(Statement (iii) of Theorem 1.5.2 shows that Card τ ∈T | ord1τ ≤ r1, . . . , ordpτ

≤ rp =p∏

i=1

(ri + ni

ni

)for any r1, . . . , rp ∈ N). Clearly, degti

φj;k1,...,kµ≤ ni for

i = 1, . . . , p.Now, for any j = 1, . . . , d, let Vj(r1, . . . , rp) = τu

(1)gj |ordiτ ≤ ri − aij

for i = 1, . . . , p and there exists ν ∈ N, 2 ≤ ν ≤ p, such that ordντ >rν − bνj. Then the combinatorial principle of inclusion and exclusion impliesthat CardVj(r1, . . . , rp) = φj(r1, . . . , rp), where φj(t1, . . . , tp) is a numericalpolynomial in p variables t1, . . . , tp defined by the formula

φj(t1, . . . , tp)

=p∑

k1=1

φj;k1(t1, . . . , tp) −∑

1≤k1<k2≤p

φj;k1,k2(t1, . . . , tp) + · · · + (−1)µ−1

×∑

1≤k1<···<kµ≤p

φj;k1,...,kµ(t1, . . . , tp) + · · · + (−1)p−1φj;2,...,p(t1, . . . , tp) .

It is easy to see that degtiφj(t1, . . . , tp) ≤ ni for i = 1, . . . , p.

Applying the principle of inclusion and exclusion once again we obtain that

CardU ′′r1...rp

= Card

d⋃j=1

Vj(r1, . . . , rp) =d∑

j=1

CardVj(r1, . . . , rp)

−∑

1≤j1<j2≤d

Card (Vj1(r1, . . . , rp)⋂

Vj2(r1, . . . , rp))

+ · · · + (−1)d−1Cardd⋂

ν=1

Vjν(r1, . . . , rp),

so it is sufficient to prove that for any s = 1, . . . , d and for any indicesj1, . . . , js, 1 ≤ j1 < · · · < js ≤ d, Card (Vj1(r1, . . . , rp)

⋂ · · ·⋂Vjs(r1, . . . , rp)) =


φj1,...,js(r1, . . . , rp), where φj1,...,js

(t1, . . . , tp) is a numerical polynomial inp variables t1, . . . , tp such that degti

φj1,...,js≤ ni for i = 1, . . . , p. It is

clear that the intersection Vj1(r1, . . . , rp)⋂ · · ·⋂Vjs

(r1, . . . , rp) is not empty(therefore, φj1,...,js

= 0) if and only if the leaders u(1)gj1

, . . . , u(1)gjs

containthe same element ei (1 ≤ i ≤ m). Let us consider such an intersectionVj1(r1, . . . , rp)

⋂. . .

⋂Vjs

(r1, . . . , rp), let v(j1, . . . , js) = lcm(u(1)j1

, . . . , u(1)js

),

and let v(j1, . . . , js) = γνu(1)gν (1 ≤ ν ≤ s; γν ∈ T ). Then Vj1(r1, . . . , rp)

⋂. . .⋂

Vjs(r1, . . . , rp) is the set of all terms u = τv(j1, . . . , js) such that ordiu ≤ ri

(that is, ordiτ ≤ ri − ordiv(j1, . . . , js)) for i = 1, . . . , p, and for any l = 1, . . . , s,there exists at least one index ν ∈ 2, . . . , p such that ordν(τγlu

(ν)gjl

) > rν (i.e.,

ordντ > rν −ordνv(j1, . . . , js)−ordνu(ν)gjl

+ordνu(1)gjl

). Denoting ordiv(j1, . . . , js)

by c(i)j1,...,js

(1 ≤ i ≤ p) and applying the principle of inclusion and exclusion

one more time, we obtain that Card

s⋂µ=1

Vjµ(r1, . . . , rp) is an alternating sum of

terms of the form CardW (j1, . . . , js; k11, k12, . . . , k1q1 , k21, . . . , ksqs; r1, . . . , rp)

where W (j1, . . . , js; k11, k12, . . . , k1q1 , k21, . . . , ksqs; r1, . . . , rp) = τ ∈ T |ordiτ ≤

ri − c(i)j1,...,js

for i = 1, . . . , p, and for any l = 1, . . . , s, ordkτ > rk − c(k)j1,...,js

+akjl

− bkjlif and only if k = kli for some i = 1, . . . , ql (q1, . . . , qs are some

positive integers from the set 1, . . . , p and kiµ|1 ≤ i ≤ s, 1 ≤ µ ≤ qs is afamily of integers such that 2 ≤ ki1 < ki2 < · · · < kiqi

≤ p for i = 1, . . . , s).Thus, it is sufficient to show that CardW (j1, . . . , js; k11, . . . , ksqs

; r1, . . . , rp)= ψj1,...,js

k11,...,ksqs(r1, . . . , rp) where ψj1,...,js

k11,...,ksqs(t1, . . . , tp) is a numerical polyno-

mial in p variables t1, . . . , tp such that degiψj1,...,js

k11,...,ksqs≤ ni (i = 1, . . . , p).

But this is almost evident: as in the process of evaluation of the value ofCardVj;k1,...,kq

(r1, . . . , rp) (when we used Theorem 1.5.2 (iii) to obtain formula(3.3.15)), we see that CardW (j1, . . . , js; k11, . . . , ksqs

; r1, . . . , rp) is a product

of terms of the form(

rν + nν − c(ν)j1,...,js

− Sν

nν

)(such a term corresponds to a

number ν ∈ 1, . . . , p which is different from all kiµ (1 ≤ i ≤ s, 1 ≤ µ ≤ qs);Sν is defined as maxbνjl

− aνjl|1 ≤ l ≤ s) and also terms of the form[(

rν + nν − c(ν)j1,...,js

nν

)−

(rν + nν − c

(ν)j1,...,js

− S′ν

nν

)](such a term appears in

the product if ν = kiµ for some i, µ; if ki1µ1 , . . . , kieµeare all elements of the

set kiµ|1 ≤ i ≤ s, 1 ≤ µ ≤ qs that are equal to ν (1 ≤ e ≤ s, 1 ≤ i1 < · · · <ie ≤ s), then S′

ν is defined as minbνjiλ− aνjiλ

|1 ≤ λ ≤ l). The correspondingnumerical polynomial ψj1,...,js

k11,...,ksqs(t1, . . . , tp) is a product of p “elementary” nu-

merical polynomials, each of which is equal to either(

tν + nν − c(ν)j1,...,js

− Sν

nν

)or

[(tν + nν − c

(ν)j1,...,js

nν

)−

(tν + nν − c

(ν)j1,...,js

− S′ν

mν

)](1 ≤ ν ≤ p).


Since the degree of such a product with respect to any variable ti (1 ≤ i ≤ p)does not exceed ni, this completes the proof of the theorem.

Definition 3.3.17 The polynomial φ(t1, . . . , tp), whose existence is establishedby Theorem 3.3.16, is called a (σ1, . . . , σp)-dimension (or simply dimension)polynomial of the σ-K-vector space M associated with the p-dimensional filtra-tion Mr1...rp

|(r1, . . . , rp) ∈ Zp.Example 3.3.18 With the notation of Theorem 3.3.16, let n = 2, σ1 =α1, σ2 = α2, and let a σ-K-module M be generated by one element x thatsatisfies the defining equation

p∑i=0

aiαi1x +

q∑j=1

bjαj2x = 0.

where p, q ≥ 1, ai, bj ∈ K (1 ≤ i ≤ p, 1 ≤ j ≤ q), ap = 0, bq = 0. In otherwords, M is a factor module of a free D-module E = De with a free generator

e by its D-submodule N = D⎛⎝ p∑

i=0

aiαi1 +

q∑j=1

bjαj2

⎞⎠ e.

It is easy to see that the set consisting of a single element g = (∑p

i=0 aiαi1 +∑q

j=1 bjαj2)e is a Grobner basis of N with respects to the orders <1, <2. In this

case, the proof of Theorem 3.3.15 shows that the (σ1, σ2)-dimension polynomial

of M associated with the natural bifiltration

⎛⎝Mrs =r∑

i=0

s∑j=0

Kαi1α

j2x

⎞⎠r,s∈N

is

as follows:

φ(t1, t2) =[(

t1 + 11

)(t2 + 1

1

)−

(t1 + 1 − p

1

)(t2 + 1

1

)]+

(t1 + 1 − p

1

)×

[(t2 + 1

1

)−

(t2 + 1 − q

1

)]= qt1 + pt2 + p + q − pq.

(With the notation of the proof, the polynomial in the first brackets gives

CardU ′rs while the polynomial

(t11

)[(t2 + 1

1

)−

(t2 − 1

1

)]gives CardU ′′

rs for

all sufficiently large (r, s) ∈ N2.)

Example 3.3.19 Using the notation of Theorem 3.3.16 once again, let n = 2,σ1 = α1, σ2 = α2, and let a vector σ-K-space M be generated by twoelements f1 and f2 with the system of defining equations

α21f1 + α2f2 = 0,

α1α2f1 + α2f2 = 0.

Let E be the free left D-module with free generators e1, e2 and let N bethe D-submodule of E generated by the elements g1 = α2

1e1 + α2e2, g2 =


α1α2e1+α2e2. Then M ∼= E/N and g1, g2 is a Grobner basis of N with respectto the order <2 (the 2-leaders of g1 and g2 are the terms α2e2 and α1α2e1,respectively). Since S1(g1, g2) = α2g1 − α1g2 = α2

2e2 − α1α2e2 is (<1, <2)-reduced with respect to g1, g2, the method of construction of a Grobner basisof N with respect to the orders <1, <2 justified in Theorem 3.3.13 (see alsoExercise 3.3.14) requires that we extend g1, g2 to the set g1, g2, g3 whereg3 = −S1(g1, g2) = α1α2g2 − α2

2e2. Since S1(gi, g3) = 0 (i = 1, 2), Theorem3.3.13 shows that G is a Grobner basis of N with respect to <1, <2.

Let φ(t1, t2) be the dimension polynomial of the D-module M which cor-

responds to the excellent bifiltration

⎛⎝Mr1r2 =r∑

i=0

s∑j=0

2∑i=1

Kαi1α

j2fi

⎞⎠r1,r2∈N

associated with the generators f1 and f2. With the notation of Theorem3.3.15, Vr1r2 = θe1 | θe1 = αi

1αj2e1 is not a multiple of u

(1)g1 = α2

1e1 oru

(1)g2 = α1α2e1

⋃τ ′e2 | τ ′e2 = αk1αl

2e2 is not a multiple of u(1)g3 = α1α2e2.

Applying Theorem 1.5.7 we obtain that CardVr1r2 = 2r1 + r1 + 2 for allsufficiently large r1, r2. The corresponding set Wr1r2 consists of the termsαr1

1 α2e2, αr11 α2

2e2 . . . , αr11 αr2

2 e2, αr1−11 α2e2, so CardWr1r2 = r2 + 1. Thus,

φ(t1, t2) = 2t1 + 2t2 + 3.

Let K be a difference field and let partition (3.3.1) of its basic set σ befixed. As we have seen, if M is a finitely generated vector σ-K-space, then everyfinite set of generators Σ of M over the ring of σ-operators D produces anexcellent p-dimensional filtration of M and therefore a dimension polynomialof M associated with this filtration. Generally speaking, different finite systemsof generators of M over D produce different (σ1, . . . , σp)-dimension polynomials(we leave the correspondent example to the reader as an exercise), however everydimension polynomial carries certain integers that do not depend on the systemof generators. These integers, that characterize the vector σ-K-space M , arecalled invariants of a dimension polynomial. In what follows, we describe someof the invariants.

For any permutation (j1, . . . , jp) of the set 1, . . . , p, we define the lexico-graphic order <j1,...,jp

on Np as follows: (r1, . . . , rp) <j1,...,jp(s1, . . . , sp) if and

only if either rj1 < sj1 or there exists k ∈ N, 1 ≤ k ≤ p− 1, such that rjν= sjν

for ν = 1, . . . , k and rjk+1 < sjk+1 . Now, if Σ ⊆ Np, then Σ′ will denote the sete ∈ Σ|e is a maximal element of Σ with respect to one of the p! lexicographicorders <j1,...,jp

.

Example 3.3.20 Let Σ = (3, 0, 2), (2, 1, 1), (0, 1, 4), (1, 0, 3), (1, 1, 6), (3, 1, 0),(1, 2, 0) ⊆ N3. Then Σ′ = (3, 0, 2), (3, 1, 0), (1, 1, 6), (1, 2, 0).

The following result provides some invariants of dimension polynomial asso-ciated with a multi-dimensional filtration.


Theorem 3.3.21 Let K be a difference field with a basic set σ = α1, . . . , αnwhose partition (3.3.1) is fixed. Let M be a finitely generated vector σ-K-space,Mr1...rp

|(r1, . . . , rp) ∈ Zp an excellent p-dimensional filtration of M , and

φ(t1, . . . , tp) =n1∑

i1=0

. . .

np∑ip=0

ai1...ip

(t1 + i1

i1

). . .

(tp + ip

ip

)

the dimension polynomial associated with this filtration. Let Σφ = (i1, . . . , ip) ∈Np | 0 ≤ ik ≤ nk (k = 1, . . . , p) and ai1...ip

= 0.Then d = deg φ, an1...np

, elements (k1, . . . , kp) ∈ Σ′φ, the corresponding

coefficients ak1...kp, and the coefficients of the terms of total degree d do not

depend on the choice of an excellent filtration.

PROOF. Let Mr1...rp|(r1, . . . , rp) ∈ Zp and M ′

r1...rp|(r1, . . . , rp) ∈ Zp

be two excellent p-dimensional filtrations of the same finitely generated σ-K-vector space M and let φ(t1, . . . , tp) and φ′(t1, . . . , tp) be dimension polynomialsassociated with these excellent filtrations, respectively. Then there exists anelement (s1, . . . , sp) ∈ Np such that Mr1...rp

⊆ M ′r1+s1,...,rp+sp

and M ′r1...rp

⊆Mr1+s1,...,rp+sp

for all sufficiently large (r1, . . . , rp) ∈ Zp. It follows that thereexist u1, . . . , up ∈ Z such that

φ(t1, . . . , tp) ≤ φ′(t1 + s1, . . . , tp + sp) (3.3.16)

andφ′(t1, . . . , tp) ≤ φ(t1 + s1, . . . , tp + sp) (3.3.17)

for all integer (and real) values of t1, . . . , tp such that t1 ≥ u1, . . . , tp ≥ up. Ifwe set ti = t (1 ≤ i ≤ p) in (3.3.16) and (3.3.17) and let t → ∞ we obtain thatφ(t1, . . . , tp) and φ′(t1, . . . , tp) have the same degree d and the same coefficientof the monomial tn1

1 . . . tnpp .

If (k1, . . . , kp) ∈ Σ′φ is the maximal element of Σφ with respect to the lexico-

graphic order ≤j1,...,jp, then we set tjp

= t, tjp−1 = 2tjp = 2t,. . . , tj1 = 2tj2 andlet t → ∞ in (3.3.16) and (3.3.17). We obtain that (k1, . . . , kp) is the maximalelement of Σφ′ with respect to ≤j1,...,jp

and the coefficients of tk11 . . . t

kpp in the

polynomials φ(t1, . . . , tp) and φ′(t1, . . . , tp) are equal. Finally, let us order theterms of the total degree d in φ and φ′ using the lexicographic order ≤p,p−1,...,1,set w1 = t, w2 = 2w1 = 2t, . . . , wp = 2wp−1 , T = 2wp , ti = wiT (1 ≤ i ≤ p) andlet t → ∞. Then the inequalities (3.3.16) and (3.3.17) immediately imply thatthe polynomials φ(t1, . . . , tp) and φ′(t1, . . . , tp) have the same coefficients of theterms of total degree d.

Exercise 3.3.22 With the notation of the last theorem, prove that an1...np= σ-

dimKM .

Exercise 3.3.23 Let K be a difference field with a basic set σ = α1, . . . , αn(n ≥ 2) and let a partition of σ into two disjoint subsets σ1 = α1, . . . , αmand σ2 = αm+1, . . . , αn be fixed (1 ≤ m < n). Let D, D′ and D′′ be rings of

3.4. INVERSIVE DIFFERENCE MODULES 185

σ-, σ1-, and σ2-operators over K, respectively, considered as filtered rings withstandard filtrations (Dr)r∈Z, (D′

r)r∈Z, and (D′′r )r∈Z. Furthermore, let M be a

finitely generated D-module with generators g1, . . . , gk and for any r ∈ Z, let

M ′r =

k∑i=1

D′′D′rgi.

Mimic the proofs of Theorems 3.2.3, 3.2.9, 3.2.11, and 3.2.12 to show thatthere exists a numerical polynomial λ(t) in one variable t with the followingproperties:

(i) λ(r) = σ2-dimKMr for all sufficiently large r ∈ Z(ii) deg λ(t) ≤ m, and the polynomial λ(t) can be written as λ(t) =

m∑i=0

ai

(t + i

i

)where a0, . . . , am ∈ Z.

(iii) am = σ-dimKM .

3.4 Inversive Difference Modules

Let R be an inversive difference ring with a basic set of automorphisms σ =α1, . . . , αn and let Γ (or Γσ, if one needs to specify the basic set) denotethe free commutative group generated by the set σ. As before, we set σ∗ =α1, . . . , αn, α−1

1 , . . . , α−1n and call R a σ∗-ring.

If γ = αk11 . . . αkn

n ∈ Γ, then the number∑n

i=1 |ki| is called the order of theelement γ, it is denoted by ord γ. For any r ∈ N, the set γ ∈ Γ|ord γ ≤ r isdenoted by Γ(r).


γ∈Γ aγγ, where aγ ∈ R for anyγ ∈ Γ and only finitely many elements aγ are different from 0, is called aninversive difference (or σ∗-) operator over R. Two σ∗-operators

∑γ∈Γ aγγ and∑

γ∈Γ bγγ are considered to be equal if and only if aγ = bγ for any γ ∈ Γ.

The set of all inversive difference operators over a σ∗-ring R can be naturallyequipped with a ring structure if one sets

∑γ∈Γ aγγ +

∑γ∈Γ bγγ =

∑γ∈Γ(aγ+

bγ)γ, a∑

γ∈Γ aγγ =∑

γ∈Γ(aaγ)γ, (∑

γ∈Γ aγγ)γ1 =∑

γ∈Γ aγ(γγ1), and γ1a =γ1(a)γ1 for any σ∗-operators

∑γ∈Γ aγγ,

∑γ∈Γ bγγ and for any a ∈ R, γ1 ∈ Γ,

and extends the multiplication by distributivity. The resulting ring is called thering of inversive difference (or σ∗-) operators over R and denoted by E . It is easyto see that the ring of difference (σ-) operators D introduced in the precedingsection is a subring of E .

If A =∑

γ∈Γ aγγ ∈ E , then the number ordA = maxord γ|aγ = 0 is calledthe order of the σ∗-operator A. For any r ∈ N, the set of all σ∗-operators whoseorder does not exceed r will be denoted by Er. Furthermore, we set Er = 0 ifr ∈ Z, r < 0.

It is easy to see that the ring E can be considered as a filtered ring withthe ascending filtration (Er)r∈Z called a standard filtration of the ring E . Below,while considering E as a filtered ring, we always mean this filtration.


Theorem 3.4.2 Let R be an inversive difference ring with a basic set of auto-morphisms σ = α1, . . . , αn and let E be the ring of σ∗-operators over R. Thenthe ring E is left Noetherian.

PROOF. Let I be a left ideal of E and let D denote the ring of σ-operatorsover R considered in Section 3.1 (as we have noticed, D is a subring of the ringE). By Lemma 3.2.7, the ring D is left Noetherian, so that its left ideal I

⋂Dhas a finite system of generators w1, . . . , wm.

Now, let u ∈ E . Then there exists an element γ ∈ Tσ ⊆ Γσ such that

γu ∈ I⋂D, so that γu =

m∑i=1

viwi for some v1, . . . , vm ∈ D. It follows that,

u =m∑

i=1

(γ−1vi)wi, so that w1, . . . , wm generate the ideal I. Thus, the ring E is

left Noetherian.

Definition 3.4.3 Let R be an inversive difference ring with a basic set σ =α1, . . . , αn and let E be the ring of inversive difference operators over R.Then a left E-module is said to be an inversive difference R-module (or a σ∗-R-module). In other words, an R-module M is called a σ∗-R-module if elementsof the set σ∗ act on M in such a way that the following conditions hold:

(i) α(x + y) = αx + αy ;(ii) α(βx) = β(αx) ;(iii) α(ax) = α(a)α(x) ;(iv) α(α−1x) = x

for any α, β ∈ σ∗; x, y ∈ M ; a ∈ R. If R is a σ∗-field, then a σ∗-R-module M issaid to be a vector σ∗-R-space (or an inversive difference vector space over R).

It is clear that any σ∗-R-module can be also treated as a difference module,that is, as a σ-R-module. Furthermore, if M and N are two σ∗-R-modules,then any difference (σ-) homomorphism f : M → N has the property thatf(αx) = αf(x) for any x ∈ M and α ∈ σ∗.

Let R be an inversive difference ring with a basic set σ = α1, . . . , αn, andlet M and N be two σ∗-R-modules. Then each of the R-modules HomR(M,N)and M

⊗R N can be equipped with a structure of a σ∗-R-module if for any

f ∈ HomR(M,N),∑k

i=1 xi

⊗yi ∈ M

⊗R N (x1, . . . , xk ∈ M ; y1, . . . , yk ∈

N), and α ∈ σ∗, one defines α(f) (also denoted as αf) by the formula(α(f))x = α(f(α−1x)) and sets α(

∑ki=1 xi

⊗yi) =

∑ki=1 αxi

⊗αyi. It is

easy to check that αf ∈ HomR(M,N) and the action of elements of σ∗

on HomR(M,N) satisfies conditions (i) - (iv) of Definition 3.4.3. Further-more, α(a

∑ki=1 xi

⊗yi) = α(

∑ki=1 axi

⊗yi) =

∑ki=1(α(a)αxi

⊗αyi) =

α(a)α(∑k

i=1 xi

⊗yi) for any

∑ki=1 xi

⊗yi ∈ M

⊗R N , a ∈ R, α ∈ σ∗, and

also α(α−1z) = z, α(z1 + z2) = αz1 + αz2, α(βz) = β(αz) for any α, β ∈ σ∗ andz, z1, z2 ∈ M

⊗R N . Thus, the action of elements of σ∗ on M

⊗R N satisfies


conditions (i) - (iv) of Definition 3.4.3 as well. In what follows, while consideringHomR(M,N) and M

⊗R N as σ∗-R-modules, we always mean the foregoing

inversive differential structures of these modules.

Lemma 3.4.4 Let R be an inversive difference (σ∗-) ring and let M , N , andP be three σ∗-R-modules. Then the canonical mapping

η : HomR(P⊗R

M,N) → HomR(P,HomR(M,N))

(defined by [(η(f))x](y) = f(x⊗

y) for any f ∈ HomR(P⊗

R M,N), x ∈ P ,y ∈ M) is a σ-isomorphism of σ∗-R-modules.

PROOF. The fact that η is an isomorphism of R-modules is well-known (see,for example [176, Theorem 3.4.3]). If f ∈ HomR(P

⊗R M,N), x ∈ P , y ∈ M ,

and α ∈ σ, then (η(α(f))(x))(y) = (α(f))(x⊗

y) = α(f(α−1(x⊗

y))) =α(f(α−1x

⊗α−1y)) = α(η(f)(α−1x))(α−1y) = (α(η(f)(α−1x)))(y) = ((αη)(f))

(x))(y). Thus, η is a σ-isomorphism. Let R be an inversive difference ring with a basic set σ and E the ring of

σ∗-operators over R. For any σ∗-R-module M , the set C(M) = x ∈ M |αx = xfor all α ∈ σ is called the set of constants of the module M , elements of this setare called constants. It is easy to see that C(M) is a subgroup of the additivegroup of M and the mapping C : M → C(M) is a functor from the categoryof σ∗-R-modules (i. e., the category of all left E-modules) to the category ofAbelian groups.

Lemma 3.4.5 Let R be an inversive difference (σ∗-) ring and E the ring ofσ∗-operators over R. Then

(i) C(HomR(M,N)) = HomE(M,N) for any two σ∗-R-modules M and N .(ii) The functors C and HomE(R, ·) are naturally isomorphic. (In this case

we write C HomE(R, ·) .)(iii) The functor C is left exact and for any positive integer p, its p-th right

derived functor is naturally isomorphic to the functor ExtpE(R, ·).(iv) If M and N are two σ∗-R-modules, then HomE(·⊗R M,N) HomE

(·,HomR(M,N)) and HomE(M⊗

R ·, N) HomE(M,HomR(·, N)).

PROOF. The first statement follows from the definition of the action ofelements of σ on HomR(M,N). Indeed, φ ∈ C(HomR(M,N)) if and only ifα(φ(α−1(x)) = φ(x) for every α ∈ σ, x ∈ M , that is equivalent to the inclusionφ ∈ HomE(M,N). Statement (ii) is a direct consequence of (i) and the obviousfact that the functors C(·) and C(HomR(R, ·) are naturally isomorphic.

Since the functor HomE(R, ·) is left exact, statement (ii) implies that C(·)is left exact as well. Now, the natural isomorphism of the functors C(·) andHomE(R, ·) implies the natural isomorphism of their p-th right derived functorsRpC and RpHomE(R, ·) = ExtpE(R, ·) for any p > 0.

By Lemma 3.4.4, HomR(·⊗R M,N) HomR(·,HomR(M,N)), whenceC(HomR(·⊗R M,N)) C(HomR(·,HomR(M,N))). Applying (i) we obtain


that HomE(·⊗R M,N) HomE(·,HomR(M,N)). The statement about theother pair of functors in (iv) can be proved in the same way.

Theorem 3.4.6 Let R be an inversive difference ring with a basic set σ, E thering of σ∗-operators over R, and M , N two σ∗-R-modules. Then for any positiveintegers p and q, there exists a spectral sequence converging to Extp+q

E (M,N)whose second term is equal to Ep,q

2 = (RpC)(ExtqR(M,N)).

PROOF. Because of the statement of Theorem 1.1.13, it is sufficient to provethe following fact:

Let N be an injective E-module and p a positive integer. Then (RpC)(HomR

(M,N)) = 0 for any E-module M .First, let us prove the last equality for an E-module M which is flat as

an R-module. In this case the functor HomE(·⊗R M,N) is exact, hence thefunctor HomE(·,HomR(M,N)) is also exact (by Lemma 3.4.5 (iv) these twofunctors are naturally isomorphic). It follows that HomR(M,N) is an injectiveE-module, hence ExtpE(R,HomR(M,N)) = 0 for all p > 0. Applying Lemma3.4.5 (iii) we obtain that (RpC)(HomR(M,N)) = 0 for all p > 0.

Now, let M be arbitrary E-module and let F : · · · → F1 → F0 → M → 0 bea flat (e. g., free) resolution of M as an R-module. (Each Fi is a flat R-moduleand the mappings are homomorphisms of R-modules.)

By Lemma 3.4.5 (iv), HomE(E ⊗R ·, N) HomE(E ,HomR(·, N)), hence

HomE(E ⊗R ·, N) HomR(·, N). Since the E-module N is injective, the

functor HomE(E ⊗ ·, N) is exact, hence HomR(·, N) is also exact. Applyingfunctor C to the injective resolution 0 → HomR(M,N) → HomR(F,N)of the E-module HomR(M,N) we obtain that (RpC)(HomR(M,N)) =Hp(C(HomR(F,N))) is isomorphic to Hp(HomE(F,N)) for every p > 0. By thefirst part of the proof, Hp(HomE(F,N)) = 0, hence (RpC)(HomR(M,N)) = 0for any E-module M and for any p > 0. This completes the proof.

The last theorem finds its applications in the analysis of systems of lineardifference equations considered in the rest of this section.

Inversive difference modules and systemsof linear difference equations

Let R be an inversive difference ring with a basis set σ = α1, . . . , αn,σ∗ = α1, . . . , αn, α−1

1 , . . . , α−1n , Γ the free commutative group generated by

σ, and E the ring of σ∗-operators over R. For any two σ∗-R-modules M andN , let B(M,N) denote the set of all additive mappings from M to N withthe following property. For every β ∈ B(M,N), there exists γβ ∈ Γ such thatβ(ax) = γβ(a)β(x) for any a ∈ R, x ∈ M . (The mapping β → γβ is notsupposed to be injective or surjective.) Furthermore, let P(M,N) denote theset of all formal sums

∑β∈B(M,N) aββ, where aβ ∈ R for any β ∈ B(M,N) and

only finitely many elements aβ are different from 0.It is easy to see that P(M,N) becomes a σ∗-R-module if one defines

α(∑

β∈B(M,N) aββ) =∑

β∈B(M,N) α(aβ)(αβ) for every α ∈ σ∗. (Clearly,αβ ∈ B(M,N) if β ∈ B(M,N); in this case γαβ = αγβ .) In what follows,


we also treat HomR(M,N) and E ⊗R M as left E- (that is, σ∗-R-) modules.

The corresponding structure of the first module is defined as in the begin-ning of this section, and the E-module structure on the second one is natural:ω(ω1

⊗x) = (ωω1)

⊗x for every ω, ω1 ∈ E , x ∈ M .

Lemma 3.4.7 Let M be a σ∗-R-module and M∗ = HomR(M,R). Then theE-modules P(M,R) and E ⊗

R M∗ are isomorphic.

PROOF. Consider the mapping φ : E ⊗R M∗ → P(M,R) such that

(φ(∑k

i=1 ai(ωi

⊗e∗i )))(e) =

∑ki=1 aiωi(e∗i (e)) (ai ∈ R, ωi ∈ E , e∗i ∈ M∗ for

i = 1, . . . , k and e ∈ M). It is easy to see that φ is a σ-homomorphism.To show that φ is bijective, one just needs to verify that the mappingψ : P(M,R) → E ⊗

R M∗ defined by ψ(∑s

i=1 aiβi) =∑s

i=1 ai(γβi

⊗γ−1

βiβi)

(ai ∈ R, βi ∈ B(M,R) for i = 1, . . . , s) is inverse of φ. Let P = (ωij)1≤i≤s,1≤j≤m be an s × m-matrix over E , and let f1, . . . , fs be

elements of the σ∗-ring R. We are going to consider the problem of solvabilityof a system of linear equations

Pu = g (3.4.1)

with respect to unknown elements u1, . . . , um of the ring R (u and g denotethe column of the unknowns (u1, . . . , um)T and the column (g1, . . . , gs)T , re-spectively). In what follows we treat the R-modules E = Rm and F = Rs asσ∗-R-modules such that α((a1, . . . , ak)T ) = (α(a1), . . . , α(ak))T for any α ∈ σ∗

(k = m or k = s, a1, . . . , ak ∈ R). The ring of s × m-matrices Es×m (with en-tries from E) will be also treated as a σ∗-R-module where α(ωij)1≤i≤s, 1≤j≤m =(α(ωij))1≤i≤s, 1≤j≤m (α ∈ σ∗ (ωij)1≤i≤s, 1≤j≤m ∈ Es×m).

Lemma 3.4.8 With the above notation, the E-modules P(E,F ) and Es×m areisomorphic.

PROOF. Let Pij denote the matrix from Es×m whose only nonzero en-try is 1 at the intersection of the ith row and jth column. Since matrices Pij

(1 ≤ i ≤ s, 1 ≤ j ≤ m) generate the E-module Es×m, it is sufficient to definean isomorphism φ : Es×m → P(E,F ) on these matrices. We define φ(Pij) by itsaction on elements of E as follows. If e = (c1, . . . , cm)T ∈ E, then (φ(Pij))(e) =(0, . . . , cj , . . . , 0)T (the ith coordinate is cj , and all other coordinates are zeros).The inverse mapping ψ : P(E,F ) → Es×m acts on generators β ∈ B(E,F )of the E-module P(E,F ) as follows: ψ(β) = (γβ(aij)γβ)1≤i≤s,1≤j≤m where el-ements aij ∈ R are defined by the relationships γ−1

β β(ek) =∑s

j=1 ajkfj forthe R-homomorphism γ−1

β β. (e1, . . . , em and f1, . . . , fs are standard bases ofE = Rm and F = Rs, respectively.) It is easy to check that φψ and ψφ areidentical mappings of Es×m and P(E,F ), respectively.

In the rest of this section we use the notation introduced before Lemma3.4.8. Furthermore, we consider P(E,F ), E∗ = HomR(E,R), F

⊗R P(E,R),

and F⊗

R(E ⊗R E∗) as σ∗-R-modules with respect to the action of σ∗ defined

as at the beginning of the section and treat E ⊗E∗ as a left E-module with


the natural structure (ω′(ω⊗

e∗) = ω′ω⊗

e∗ for any e∗ ∈ E∗, ω ∈ E). Inparticular, if f

⊗(ω

⊗e∗) is a generator of F

⊗R(E ⊗

R E∗) and α ∈ σ∗, thenα(f

⊗(ω

⊗e∗)) = α(f)

⊗(αω

⊗e∗).

Let us consider the diagram

P(E,F ) η

F⊗

R P(E,R))ξ

F⊗

R(E ⊗R E∗)

λ

ν

δµ (3.4.2)

where all six mappings are difference homomorphisms defined at the generatorsas follows:

δ(f⊗

(ω⊗

e∗)) = f⊗

ω(e∗(·)), µ(f⊗

β) = f⊗

(γβ

⊗γ−1

β β),

ν(f⊗

(ω⊗

e∗)) = ω(e∗(·))f, λ(β) =m∑

i=1

β(ei)⊗

(γβ

⊗e∗i ),

η(β) =m∑

i=1

β(ei)⊗

γβe∗i , ξ(f⊗

β) = β(·)f

for any f ∈ F, ω ∈ E , e∗ ∈ E∗, β ∈ B(E,R), β ∈ B(E,F ) ( (ei)1≤i≤m denotesthe standard basis of E over R, and (e∗i )1≤i≤m denotes the dual basis of E∗).

Lemma 3.4.9 All mappings in diagram (3.4.2) are difference isomorphisms,η = ξ−1, µ = δ−1 λ = ν−1, and the diagram is commutative.

PROOF. We shall prove that µ = δ−1 and νµ = ξ. (The other requi-red relationships can be proved in a similar way.) Let f ∈ F , e∗ ∈ E∗,∑

γ∈Γ aγγ ∈ E , and β ∈ B(E,R). Then (µδ)(f⊗

(∑

γ∈Γ aγγ⊗

e∗)) =µ(f

⊗∑γ∈Γ aγγ(e∗(·)))=

∑γ∈Γ aγ(f

⊗(γ

⊗γ−1γe∗))= f

⊗(∑

γ∈Γ aγγ⊗

e∗)and (δµ)(f

⊗β) = δ(f

⊗(γβ

⊗γ−1

β β)) = f⊗

γβγ−1β β = f

⊗β, so µ =

δ−1. Furthermore, for any generator f⊗

β of the E-module F⊗

R P(E,R)(f ∈ F, β ∈ B(E,R)), we have (νµ)(f

⊗β) = ν(f

⊗(γβ

⊗γ−1

β β)) =γβ(γ−1

β β(·))f = β(·)f = ξ(f⊗

β) that proves the equality νµ = ξ.

Let P ∈ P(E,F ) and let P : P(F,R) → P(E,R) be the homomorphism ofE-modules such that P (β) = βP for any β ∈ B(E,R). Let φF : E ⊗

R F ∗ →P(F,R) and ψE : P(E,R) → E ⊗

R E∗ be difference isomorphisms defined inthe proof of Lemma 3.4.7. Let P ∗ = ψEP φF : E ⊗

R F ∗ → E ⊗R E∗, N =


Ker P ∗, and M = Coker P ∗. Applying functor HomE(·, R) to the standardexact sequence of left E-modules

0 → Ni−→ E

⊗R

F ∗ P∗−−→ E

⊗R

E∗ j−→ M → 0 (3.4.3)

(i and j are the natural injection and projection, respectively), we obtain theexact sequence of σ∗-R-modules

0 → HomE(M,R)j∗−→ HomE(E

⊗R

E∗, R) P∗∗−−→ HomE(E

⊗R

F ∗, R)

i∗−→ HomE(N,R) (3.4.4)

Now let us consider the homomorphism of σ-modules θ : P(E,F ) →HomE(P(F,R),P(E,R)) such that (θ(P ))(P1) = P1P for every P ∈ P(E,F ),P1 ∈ P(F,R) and the exact sequence of σ∗-R-modules

P(E,F ) λ−→ F⊗R

(E⊗R

E∗) ε−→ HomR(F ∗, E⊗R

E∗)

ρ−→ HomE(E⊗R

F ∗, E⊗R

E∗) π−→ HomE(P(F,R),P(E,R))

where λ is the same as in diagram (3.4.2) and the other σ-homomorphismsare defined as follows: (ε(f

⊗(ω

⊗e∗))(f∗) = f∗(f)

⊗e∗, (ρ(h))(ω

⊗f∗) =

ωh(f∗), and π(χ) = φEχψF for any f ∈ F, ω ∈ E , e∗ ∈ E∗, h ∈ HomR(F ∗,E ⊗

R E∗), and χ ∈ HomE(E ⊗R F ∗, E ⊗

R E∗). (φE and ψF denote the σ-homomorphisms defined in the proof of Lemma 3.4.7.) It is easy to check thatλ, ε, ρ, and π are isomorphisms of σ∗-R-modules. For example, ε−1 is defined byε−1(ζ) =

∑si=1 fi

⊗ζ(f∗

i ) for every ζ ∈ HomR(F ∗, E ⊗R E∗) ((fi)1≤i≤s is the

standard basis of F and (f∗i )1≤i≤s is the dual basis of the σ∗-R-module F ∗).

Lemma 3.4.10 With the above notation, θ = πρελ.

PROOF. Clearly, it is sufficient to verify the equality at an element β ∈B(E,F ). If ω ∈ E and f∗ ∈ F ∗, then

(ρελ(β))(ω⊗

f∗) = ρε

(m∑

i=1

β(ei)⊗

(γβ

⊗e∗i )

)(ω

⊗f∗)

=m∑

i=1

ωf∗(β(ei))γβ

⊗e∗i .

Now, for any β ∈ B(F,R), we have (πρελ(β))(β) = φE(ρελ(β))ψF (β) =φE(ρελ(β))(γβ

⊗γ−1

β β) = φE

∑mi=1 β(β(ei))γβ

⊗e∗i . To complete the proof it

is sufficient to notice that for every e =∑m

k=1 ckek ∈ E ( e1, . . . , em is the


standard basis of E and c1, . . . , cm ∈ R), φE

(m∑

i=1

β(β(ei))γβ

⊗e∗i

)(e) =

m∑i=1

ββ(ciei) = ββ

(m∑

i=1

ciei

)= ββ(e) = (θ(β))(β)(e) whence θ = πρελ.

Let us associate with every mapping P ∈ P(E,F ) a set ComP consistingof all f ∈ F such that for every pair (G = Rt, P1 ∈ P(F,G)) with the conditionP1P = 0, one has P1f = 0. Clearly, the image ImP of the mapping P is a subsetof ComP . The following example shows that the inclusion can be proper.

Example 3.4.11 Let R = Q(x) be the field of rational fractions over Q treatedas an inversive difference field with one translation α such that (αf)(x) = f(2x)for every f(x) ∈ R. Let E denote the ring of inversive difference operatorsover R, E = R2, F = R2, and P the element of P(E,F ) defined by the ma-

trix(

α − 1 00 1

). (By Lemma 3.4.8, every element of P(E,F ) can be defined

by a 2 × 2-matrix over E .) Note that ImP = (

α − 1 00 1

)(u1

u2

)|u1, u2 ∈

R = (

(α − 1)u1

u2

)|u1, u2 ∈ R is a proper E-submodule of F . It follows

from the fact that 1 cannot be written as (α − 1)u1 with u1 ∈ R: if u1 =anxn + · · · + a1x + a0

bmxm + · · · + b1x + b0(all coefficients ai, bj belong to R, an = 0, and bm = 0),

then (α − 1)u1 = h1(x)/h2(x) where h1(x) = (2n − 2m)anbmxm+n + · · · +2(a1b0 − a0b1)x and h2(x) = 2mb2

mx2m + · · ·+ 3b1b0x + b20. It is easy to see that

h1(x) = h2(x).

On the other hand, ComP = F . Indeed, v =(

v1

v2

)∈ ComP if and only if

for every s = 1, 2, . . . and for every matrix W = (ωij)1≤i≤s,1≤j≤2, the equality

W

(α − 1 0

0 1

)= 0 implies Wv = 0. (In the last two equalities 0 in the right-

hand sides denotes the zero s × 2- and s × 1- matrices, respectively.) Since the

ring E does not have zero divisors, the equality W

(α − 1 0

0 1

)= 0 implies that

W is a zero matrix, so Wv = 0 for all v ∈ F , whence ComP = F .

The following theorem gives a connection between ImP and ComP underthe above conventions. The theorem allows to reduce the description of the imageof the operator P to the description of ComP using the spectral sequence fromTheorem 3.4.6.

Theorem 3.4.12 With the above notation, for every P ∈ P(E,F ), thereexist isomorphisms of σ∗-R-modules ϑE : E → HomE(E ⊗

R E∗, R) andϑF : F → HomE(E ⊗

R F ∗, R) such that the following diagram is commutative.


HomE(E ⊗R E∗, R) P ∗∗

HomE(E ⊗R F ∗, R)

E PF

ϑE ϑF (3.4.5)

Furthermore, we have the following properties of the exact sequence (3.4.4).

(i) Im j∗ ∼= Ker P ;(ii) Ker i∗ ∼= ComP ;(iii) ComP/ImP ∼= Ext1E(M,R).

PROOF. The difference homomorphisms φE : E ⊗R E∗ → P(E,R) and

φF : E ⊗R F ∗ → P(F,R) defined in the proof of Lemma 3.4.7 induce ho-

momorphisms of E-modules φE : HomE(E ⊗R E∗, R) → HomE(P(E,R) and

φF : HomE(E ⊗R F ∗, R) → HomE(P(F,R) where (φE(g))(β) = g(γβ

⊗γ−1

β β)for every g ∈ HomE(E ⊗

R E∗, R), β ∈ B(E,R), and φF acts in a similarway. Now one can define the mappings χE : E → HomE(P(E,R), R) andχF : F → HomE(P(F,R), R) by setting (χE(e))(β) = β(e) and (χF (f))(β1) =β1(f) for any e ∈ E, f ∈ F, β ∈ P(E,R), and β1 ∈ P(F,R). It is easyto check that χE and χF are difference isomorphisms and the diagram

HomE(P(E,R), R) P ∗HomE(P(F,R), R)

E PF

χE χF (3.4.6)

is commutative (the mapping P ∗ is obtain by applying functor HomE(·, R) tothe mapping P : P(F,R) → P(E,R) considered above). The inverse differenceisomorphism of χE is the mapping λE = χ−1

E : HomE(P(E,R), R) → E such

that λE(g) =m∑

i=1

g(e∗i )ei for any g ∈ HomE(P(E,R), R), and the inverse map-

ping of χF is defined similarly. (As before, (ei)1≤i≤m is the standard basis ofthe R-module E and (e∗i )1≤i≤m ⊆ HomR(E,R) ⊆ P(E,R) is the correspondingdual basis.) Indeed, for any e ∈ E,

(λEχE)(e) =m∑

i=1

(χE(e))(e∗i )ei =m∑

i=1

e∗i (e)ei = e,

hence λEχE is the identity automorphism of E. Also, a routine computationshows that the mapping χEλE leaves fixed every element of HomE(P(E,R), R)(thus, λE = χ−1

E ) and P ∗χE = χF P .


The commutativity of diagram (3.4.6) and Lemma 3.4.7 imply the commuta-tivity of diagram (3.4.5) with ϑE = HomE(φE , R)χE and ϑF = HomE(φF , R)χF

(φE : E ⊗R E∗ → P(E,R) and φF : E ⊗

R F ∗ → P(F,R) are difference isomor-phisms defined in the proof of Lemma 3.4.7). Therefore, Ker P ∼= Ker P ∗∗ =Im j∗ = HomE(M,R) (see the exact sequence (3.4.3) where M = Coker P ∗).This proves statement (i).

To prove (ii) assume first that ζ ∈ Ker i∗ and PP ′ = 0 for every P ′ ∈P(F,G) (G = Rt for some positive integer t). Then P ∗P ′∗ = (P ′P )∗ = 0hence ImP ′∗ ⊆ Ker P ∗ = Im i and one can consider the well-defined mapping = i−1P ′∗ : E ⊗

R G∗ → N (as in sequence (3.4.3), N = Ker i). Now, ifζ = ϑF (z) ∈ Ker i∗ (z ∈ F ), then P ′∗∗(ζ) = (i)∗(ζ) = ∗i∗(ζ) = 0 hence(ϑGP ′)(ζ) = 0. It follows that P ′(z) = 0, so that z ∈ ComP . Thus, Ker i∗ ⊆ϑF (ComP ).

Conversely, let z ∈ ComP and ζ = ϑF (z). Let us fix some x ∈ N andconsider the homomorphism of E-modules δ : E → N such that δ(1) = x. Thecomposition of the σ∗-R-isomorphisms

HomE(E , E⊗R

F ∗) → E⊗R

F ∗ φF−−→ P(F,R),

where the first mapping is the natural isomorphism, sends the element iδ ∈HomE(E , E ⊗

R F ∗) to some element P ′ ∈ P(F,R). Denoting the natural iso-morphism of E-modules E → E ⊗

R R∗ by τ , we obtain that iδ = P ′∗τ . Indeed,let x = iδ(1) =

∑dk=1 ai(ωk

⊗f∗

k ) where ak ∈ R, ωk =∑dk

l=1 bklγkl ∈ E(γkl ∈ Γ), and f∗

k ∈ F ∗ (1 ≤ k ≤ d). Then

(P ′∗τ)(1) = P ′∗(1⊗

1∗) = ψF P ′φR(1⊗

1∗) = ψF P ′(1∗(·))

= ψF

(d∑

k=1

akωk(f∗k (·))

)= ψF

(d∑

k=1

ak

dk∑l=1

bklγkl(f∗l (·))

)

=d∑

k=1

ak

dk∑l=1

bkl(γkl

⊗γ−1

kl γklf∗l =

d∑k=1

ak(ωk

⊗f∗

k = x = iδ(1)

(ψF is the same as in Lemma 3.4.7), whence iδ = P ′∗τ.Since Im i = Ker P ∗, P ∗P ′∗τ = P ∗iδ = 0 whence P ∗P ′∗ = 0. Applying ∗

we obtain that P ′P = 0. Furthermore, since z ∈ ComP , we have P ′(z) = 0and P ′∗∗(ζ) = ζP ′∗ = 0. Therefore, ζ(i(x)) = ζ(iδ(1)) = ζ(P ′∗τ(1)) = 0, thatis, (i∗(ζ))(x) = 0. Since x is an arbitrary element of N , i∗(ζ)) = 0, that is, ζ ∈Ker i∗. Thus, ϑF (ComP ) is contained in Ker i∗ whence ϑF (ComP ) = Ker i∗.

In order to prove the last statement of the theorem, let us break the ex-act sequence (3.4.3) into two short exact sequences of E-modules, 0 → N

i−→E ⊗

R F ∗ q−→ L → 0 and 0 → Lε−→ E ⊗

R E∗ j−→ M → 0 where L = ImP ∗, q is aprojection, and ε is the embedding. Applying functor HomE(·, R) to these twosequences we obtain the exact sequences

3.5. σ∗-DIMENSION POLYNOMIALS AND THEIR INVARIANTS 195

0 → HomE(L,R)q∗−→ HomE(E ⊗

R F ∗, R) i∗−→ HomE(N,R) and

0 → HomE(M,R)j∗−→ HomE(E ⊗

R E∗, R) ε∗−→ HomE(L,R) →

Ext1E(M,R) → Ext1E(E ⊗R E∗, R) = 0.

(The mapping q∗ identifies HomE(L,R) with the module Ker i∗ = ϑF (ComP )and ϑ−1

F q∗ε∗ϑE = P . Thus,Ext1E(M,R) ∼= HomE(L,R) / ε∗(HomE(E ⊗

R E∗, R)) =HomE(L,R) / q∗ϑE(E)(q∗)−1ϑF (ComP ) / (q∗)−1ϑF P (E) ∼= ComP / ImP .

3.5 σ∗-Dimension Polynomialsand their Invariants

Let R be an inversive difference ring with a basic set σ and let E be the ringof σ∗-operators over R considered as a filtered ring with the filtration (Er)r∈Z

introduced in the preceding section.

Definition 3.5.1 Let M be a σ∗-R-module. An ascending chain (Mr)r∈Z ofR-submodules of M is called a filtration of M if ErMs ⊆ Mr+s for all r, s ∈ Z,Mr = 0 for all sufficiently small r ∈ Z, and

⋃r∈Z Mr = M . A filtration (Mr)r∈Z

of the the σ∗-R-module M is called excellent if all R-modules Mr (r ∈ Z) arefinitely generated and there exists r0 ∈ Z such that Mr = Er−r0Mr0 for anyr ∈ Z, r ≥ r0.

Theorem 3.5.2 Let R be an Artinian σ∗-ring with a basic set σ = α1, . . . , αnand let (Mr)r∈Z be an excellent filtration of a σ∗-R-module M . Then there existsa numerical polynomial χ(t) in one variable t such that

(i) χ(r) = lR(Mr) for all sufficiently large r ∈ Z ;(ii) deg χ(t) ≤ n and the polynomial χ(t) can be represented in the form

χ(t) =n∑

i=0

2iai

(t + i

i

)(3.5.1)

where a0, . . . , an ∈ Z.

PROOF. Let gr E denote the graded ring associated with the filtration(Er)r∈Z, that is, gr E =

⊕s∈Z grsE where gr E = Es/Es−1 for all s ∈ Z. Let

x1, . . . , x2n be the images in gr E of the elements α1, . . . , αn, α−11 , . . . , α−1

n , re-spectively (so that xi = αi + E0 ∈ gr1E = E1/E0 and xn+i = α−1

i + E0 ∈ gr1Efor i = 1, . . . , n). It is easy to see that the elements x1, . . . , x2n generate thering gr E over R, xixj = xjxi (1 ≤ i, j ≤ 2n,) and xixn+i = 0 for i = 1, . . . , n.Furthermore, xia = α(a)xi and xn+ia = α−1(a)xn+i (1 ≤ i ≤ n).

In the rest of the proof the ring gr E will be also denoted by Rx1, . . . , x2n.A homogeneous component grsE (s ∈ N) of this graded ring is an R-module


generated by the set of all monomials xk1i1

. . . xknin

such that k1, . . . , kn ∈ N,∑nν=1 kν = s, and iµ − iν = n for any µ, ν ∈ 1, . . . , n.Let gr M =

⊕s∈Z grsM be the graded gr E-module associated with the

excellent filtration (Mr)r∈Z. (In what follows, the module gr M will be alsodenoted by M and its homogeneous components grsM = Ms/Ms−1 (s ∈ Z)will be denoted by M (s)). First of all, repeating the beginning of the proof ofTheorem 3.2.3, we obtain that gr M is a finitely generated gr E-module. SincelR(Mr) =

∑s≤r

lR(M (s)), the theorem will be proved if one can prove the existence

of a numerical polynomial f(t) in one variable t such that

(a) f(s) = lR(M (s)) for all sufficiently large s ∈ Z;(b) deg f(t) ≤ n − 1 and the polynomial f(t) can be represented as f(t) =

n−1∑i=0

2i+1bi

(t + i

i

)where b0, . . . , bn−1 ∈ Z.

(Indeed, if such a polynomial f(t) exists, then Corollary 1.4.7 implies theexistence of the polynomial χ(t) with the desired properties.) We are going toprove the existence of the polynomial f(t) with the properties (a) and (b) byinduction on n = Card σ.

If n = 0, then gr M is a finitely generated module over the Artinian ring R.In this case, M (s) = 0 for all sufficiently large s ∈ Z, so that the polynomialf(t) = 0 satisfies conditions (a) and (b).

Now, suppose that n > 0. Let us consider the exact sequence of finitelygenerated Rx1, . . . , x2n-modules

0 → Ker θn → M θn−→ xnM → 0 (3.5.2)

where θn is the mapping of multiplication by xn, that is, θn(y) = xnyfor any y ∈ M. It is easy to see that θn is an additive mapping andθn(ay) = αn(a)θn(y) for any y ∈ M, a ∈ R. By Proposition 3.1.4 (withK = (Ker θn)(s),M = M (s), N = (xnM)(s), and δ = θn), we obtain thatlR((Ker θn)(s))+ lR((xnM)(s)) = lR(M (s)) for every s ∈ Z. (In accordance withthe above notation, we denote the s-th homogeneous component of a gradedmodule P by P (s).)

Since Ker θn and xnM are annihilated by the elements xn and x2n, re-spectively, one can consider Ker θn as a graded Rx1, . . . , xn−1, xn+1, . . . , x2n-module and treat xnM as a graded Rx1, . . . , x2n−1-module. (By Rx1, . . . ,xn−1, xn+1, . . . , x2n and Rx1, . . . , x2n−1 we denote the graded subrings of thering Rx1, . . . , x2n whose elements can be written in the form that does notcontain xn and x2n, respectively. The homogeneous components of each of thesesubrings are the intersections of the corresponding homogeneous components ofRx1, . . . , x2n with the subring.)

Let σ = α1, . . . , αn−1 and let E denote the ring of σ∗-operators over R(when R is treated as an inversive difference ring with the basic set σ). Thengr E = Rx1, . . . , xn−1, xn+1, . . . , x2n−1. By the induction hypothesis, for anyfinitely generated Rx1, . . . , xn−1, xn+1, . . . , x2n−1-module N =

⊕p∈Z N (p),


there exists a numerical polynomial fN (t) in one variable t such that fN (t) =lR(N (p)) for all sufficiently large p ∈ Z. Furthermore, deg fN (t) ≤ n − 2 and

the polynomial fN (t) can be written as fN (t) =n−2∑i=0

2i+1ci

(t + i

i

)for some

c0, . . . , cn−2 ∈ Z. (We assume that n ≥ 2. If n = 1, then E = R and fN (t) = 0).If L =

⊕q∈Z L(q) is a finitely generated graded module over the ring

Rx1, . . . , xn−1, xn+1, . . . , x2n, then the first and the last terms of the exactsequence of R-modules

0 → (Ker θ2n)(q) → L(q) θ2n−−→ L(q+1) → L(q+1)/x2nL(q) → 0 (3.5.3)

(θ2n is the mapping of multiplication by x2n) are homogeneous components offinitely generated graded Rx1, . . . , xn−1, xn+1, . . . , x2n−1-modules. Breakingthe sequence (3.5.3) into two short exact sequences and applying Proposition3.1.4 (with δ = θ2n) we obtain that

lR(L(q+1)) − lR(L(q)) = lR(L(q+1)/x2nL(q)) − lR((Ker θ2n)(q)).

By the induction hypothesis, there exists a numerical polynomial gL(t) =n−2∑i=0

2i+1c′i

(t + i

i

)(c′0, . . . , c

′n−2 ∈ Z) such that gL(s) = lR(L(q+1)) − lR(L(q))

for all sufficiently large q ∈ Z, say, for all q ≥ q0 for some integer q0. Since

lR(L(q)) = lR(L(q0)) +q∑

s=q0+1

[lR(L(s)) − lR(L(s−1))

],

Corollary 1.4.7 implies that there exists a numerical polynomial hL(t) such thathL(q) = lR(L(q)) for all sufficiently large q ∈ Z and the polynomial hL(t) can

be written as hL(t) =n−1∑i=0

2ic′′i

(t + i

i

)where c′′0 , . . . , c′′n−1 ∈ Z and c′′i = c′i−1 for

i = 1, . . . , n − 1. Clearly, a similar statement is true for any finitely generatedgraded module K =

⊕q∈Z K(q) over the ring Rx1, . . . , x2n−1.

Applying the above reasoning to the graded modules of the exact sequence

(3.5.2) we obtain that there exist numerical polynomials f1(t) =n−1∑i=0

2ib′i

(t + i

i

)and f2(t) =

n−1∑i=0

2ib′′i

(t + i

i

)(b′i, b

′′i ∈ Z for i = 0, . . . , n − 1) such that

f1(s) = lR((Ker θn)(s)) and f2(s) = lR((xnM)(s)) for all sufficiently large s ∈ Z.Furthermore, the exact sequence (3.5.2) shows that lR(M (s)) = lR(Ker θn)(s))+lR((xnM)(s)) for all sufficiently large s ∈ Z.

Now, let us show that lR((xiM)(s)) = lR((xn+iM)(s)) for any s ∈ Z, i =1, . . . , n. Indeed, lR((xn+iM)(s)) = lR

((α−1

i Ms + Ms)/Ms

)= lR(α−1

i Ms +Ms) − lR(Ms). Applying Proposition 3.1.4 (where δ is the bijective mapping of


multiplication by αi) we obtain that lR(α−1i Ms +Ms) = lR(αi(α−1

i Ms +Ms)) =lR(αiMs + Ms) whence lR((xn+iM)(s)) = lR(αiMs + Ms/Ms) = lR((xiM)(s)).

Thus, lR(M (s)) = lR(Ker θn)(s)) + lR((xnM)(s)) = lR(Ker θn)(s)) +lR((x2nM)(s)) = f1(s) + f2(s) for all sufficiently large s ∈ Z.

Let us consider the exact sequence of graded Rx1, . . . , xn−1, xn+1, . . . , x2n-modules

0 → x2nM → Ker θn → Ker θn/x2nM → 0

whose last term is a graded Rx1, . . . , xn−1, xn+1, . . . , x2n−1-module. Applyingthe induction hypothesis to this term we obtain that there exists a numerical

polynomial f3(t) =n−2∑i=0

2i+1di

(t + i

i

)(d0, . . . , dn−2 ∈ Z) such that f3(s) =

lR

((Ker θn/x2nM)(s)

)for all sufficiently large s ∈ Z.

Since f2(s) = lR((x2nM)(s)) = lR((Ker θn)(s)) − lR

((Ker θn/x2nM)(s)

)for any s ∈ Z, f2(t) = f1(t) − f3(t). Setting f(t) = f1(t) + f2(t) we ob-tain that f(s) = lR((x2nM)(s)) for all sufficiently large s ∈ Z and f(t) =

2f1(t) − f3(t) = 2

[n−1∑i=0

2ib′i

(t + i

i

)]−

n−2∑i=0

2i+1di

(t + i

i

)=

n−1∑i=0

2i+1bi

(t + i

i

)where bn−1 = b′n−1, and bi = b′i − di for i = 0, . . . , n − 2.

Thus, the polynomial f(t) satisfies conditions (a) and (b). This completesthe proof of the theorem.

Definition 3.5.3 Let R be an Artinian inversive difference ring with a basic setσ = α1, . . . , αn and let (Mr)r∈Z be an excellent filtration of a σ∗-R-module M .Then the polynomial χ(t) whose existence is established by Theorem 3.5.2 iscalled the σ∗-dimension or characteristic polynomial of the module M associatedwith the excellent filtration (Mr)r∈Z.

Example 3.5.4 Let R be an inversive difference field with a basic set σ =α1, . . . , αn and let E be the ring of σ∗-operators over R. Then E can be treatedas a σ∗-R-module equipped with the excellent filtration (Er)r∈Z, the standardfiltration of E . If χE(t) is the corresponding dimension polynomial then χE(r) =lR(Er) = dimR(Er) = Card γ = αk1

1 . . . αknn ∈ Γσ|ord γ =

∑ni=1 |ki| ≤ r for

all sufficiently large r ∈ Z. Proposition 1.4.2 gives three expressions for the lastnumber that imply the following three expressions for χE(t).

χE(t) =n∑

i=0

2i

(n

i

)(t

i

)=

n∑i=0

(n

i

)(t + i

n

)=

n∑i=0

(−1)n−i2i

(n

i

)(t + i

i

).

(3.5.4)

Let R be an inversive difference ring with a basic set σ, E the ring of σ∗-operators over R, and Fm a free left E-module of rank m (m ∈ N) with freegenerators f1, . . . , fm. Then for every l ∈ Z, one can consider an excellent fil-tration ((F l

m)r)r∈Z of the module Fm such that (F lm)r =

∑mi=1 Er−lfi for every

r ∈ Z. We obtain a filtered σ∗-R-module that will be denoted by F lm. A finite


direct sum of such filtered σ∗-R-modules will be called a free filtered σ∗-R-module. Example 3.5.4 shows that the dimension polynomial χ(t) of the filteredσ∗-R-module F l

m can be found using one of the following formulas:

χ(t) = mχE(t − l) = m

n∑i=0

2i

(n

i

)(t − l

i

)= m

n∑i=0

(n

i

)(t + i − l

n

)

= mn∑

i=0

(−1)n−i2i

(n

i

)(t + i − l

i

). (3.5.5)

Let R be an inversive difference ring with a basic set σ = α1, . . . , αn, E thering of σ∗-operators over R, M a filtered σ∗-R-module with a filtration (Mr)r∈Z,and R[x] the ring of polynomials in one indeterminate x over R. Let E ′ denote thesubring

∑r∈N

Er

⊗R

Rxr of the ring E ⊗R R[x] and M ′ denote the left E ′-module∑

r∈N Mr

⊗R Rxr. (The ring structure on E ⊗

R R[x] is induced by the ringstructure of E [x] via the natural isomorphism of E-modules E ⊗

R R[x] ∼= E [x]).The proof of the following lemma is similar to the proof of Lemma 3.2.6.

Lemma 3.5.5 With the above notation, let all components of the filtration(Mr)r∈Z be finitely generated R-modules. Then the following conditions areequivalent:

(i) The filtration (Mr)r∈Z is excellent;(ii) M ′ is a finitely generated E ′-module.

Lemma 3.5.6 Let R be a Noetherian inversive difference ring with a basic setσ = α1, . . . , αn. Then the ring E ′ is left Noetherian.

PROOF. Let E be the ring of σ-operators over R when R is treated as a dif-ference ring with the basic set σ = α1, . . . , αn, αn+1, . . . , α2n whose elementsαn+1, . . . , α2n act on the ring R as follows: αn+i(a) = α−1

i (a) for any a ∈ R

(i = 1, . . . , n). Clearly, if J is the two-sided ideal of the ring E generated by theset αiαn+i − 1|1 ≤ i ≤ n, then E is isomorphic to the ring E/J .

It is easy to see that if α is an element of the set σ∗, then there exists a uniqueautomorphism β of the ring R[x] such that β(x) = x and β(a) = α(a) for anya ∈ R. Let β1, . . . , β2n be the automorphisms of R[x] obtained as such extensionsof α1, . . . , αn, α−1

1 , . . . , α−1n , respectively. Furthermore, let S denote the ring

of skew polynomials R[x][z1, . . . , z2n; d1, . . . , d2n;β1, . . . , β2n] constructed in theproof of Lemma 3.2.7 (di(f) = xβi(f) − xf for any f ∈ R[x], i = 1, . . . , 2n).Then the arguments of the proof of Lemma 3.2.7 and the above remark show thatE ′ is isomorphic to the factor ring of S by the two-sided ideal generated by allelements of the form (zi +x)(zn+i +x)−x2 = zizn+i +xzi +xzn+i (i = 1, . . . , n).By Theorem 1.7.20, the ring of skew polynomials S is left Noetherian, whencethe ring E ′ is also Noetherian.


Let R be an inversive difference ring with a basic set σ and let M and N befiltered σ∗-R-modules with filtrations (Mr)r∈Z and (Nr)r∈Z, respectively. Thena σ-homomorphism f : M → N is said to be a σ-homomorphism of filteredσ∗-R-modules if f(Mr) ⊆ Nr for any r ∈ Z.

The proof of the following result is similar to the proof of Theorem 3.2.8.

Theorem 3.5.7 Let R be a Noetherian inversive difference ring with a basic setσ and let ρ : N −→ M be an injective homomorphism of filtered σ∗-R-modules.Furthermore, suppose that the filtration of the module M is excellent. Then thefiltration of N is also excellent.

Let K be an inversive difference field with a basic set σ = α1, . . . , αn,E the ring of σ∗-operators over K, and M a finitely generated σ∗-K-modulewith generators z1, . . . , zm. Then one can easily see that the ascending chain ofvector K-spaces Mr =

∑mi=1 Erzi (r ∈ Z) is an excellent filtration of the mod-

ule M . This filtration is called a standard filtration of M associated with thesystem of generators z1, . . . , zm. Let χ(t) be the corresponding characteristicpolynomial of M and let d = deg χ(t). By Theorem 3.5.2, d ≤ n and the polyno-

mial χ(t) has a unique representation in the form χ(t) =d∑

i=0

2iai

(t + i

i

)where

a0, . . . , ad ∈ Z and ad = 0.Let (M ′

r)r∈Z be another excellent filtration of the σ∗-K-module M . Thenthere exists some non-negative integer p such that ErM

′p = M ′

r+p for any r ∈ N.Since

⋃r∈Z Mr =

⋃r∈Z M ′

r = M , there exists some q ∈ N such that Mp ⊆ M ′p+q

and M ′p ⊆ Mp+q, hence Mr ⊆ M ′

r+q and M ′r ⊆ Mr+q for all r ∈ Z, r ≥ p.

Thus, if χ1(t) is the characteristic polynomials of the σ∗-K-module M asso-ciated with the excellent filtration (M ′

r)r∈Z, then χ(r) ≤ χ1(r + q) and χ1(r) ≤χ(r + q) for all sufficiently large r ∈ Z. It follows that deg χ1(t) = deg χ(t) = d

and χ1(t) can be written as χ1(t) =d∑

i=0

2ibi

(t + i

i

)where b0, . . . , bd ∈ Z and

bd = ad. Using the finite differences we can write ad = bd =∆dχ(t)

2d=

∆dχ1(t)2d

.

Furthermore, we have ∆nχ(t) = ∆nχ1(t) (if d < n, then both sides of the lastequality are equal to zero). We have arrived at the following result.

Theorem 3.5.8 Let K be an inversive difference field with a basic set σ =α1, . . . , αn, let M be a σ∗-K-module, and let χ(t) be a characteristic poly-

nomial associated with an excellent filtration of M . Then the integers∆nχ(t)

2n,

d = deg χ(t), and∆dχ(t)

2ddo not depend on the choice of the excellent filtration

of M .

Definition 3.5.9 Let K be an inversive difference field with a basic setσ = α1, . . . , αn, M a finitely generated σ∗-K-module, and χ(t) a char-acteristic polynomial associated with an excellent filtration of M . Then the


numbers∆nχ(t)

2n, d = deg χ(t), and

∆dχ(t)2d

are called the inversive difference

(or σ∗-) dimension, inversive difference (or σ∗-) type, and typical inversivedifference (or typical σ∗-) dimension of the module M , respectively. Thesecharacteristics of the σ∗-K-module M are denoted by iδ(M) (or σ∗-dimKM),it(M) (or σ∗-typeKM) and tiδ(M) (or σ∗-tdimKM), respectively.

Definition 3.5.10 Let K be an inversive difference field with a basic set σ =α1, . . . , αn, Γ the free commutative group generated by σ, E the ring of σ∗-operators over K, and M a σ∗-K-module. Elements z1, . . . , zm ∈ M are said tobe σ∗-linearly independent over K if the set γzi|1 ≤ i ≤ m, γ ∈ Γ is linearlyindependent over the field K.

The following two theorems can be proved precisely in the same way as thesimilar statements on difference vector spaces (see Theorems 3.2.11 and 3.2.12).

Theorem 3.5.11 Let K be an inversive difference field with a basic set σ =α1, . . . , αn and let

0 −→ Ni−→ M

j−→ P −→ 0

be an exact sequence of finitely generated σ∗-K-modules. Then iδ(N)+ iδ(P ) =iδ(M).

Theorem 3.5.12 Let K be an inversive difference field with a basic set σ =α1, . . . , αn, E the ring of σ∗-operators over K, and M a finitely generatedσ∗-K-module. Then iδ(M) is equal to the maximal number of elements of Mσ∗-linearly independent over K.

Exercise 3.5.13 Let K be a σ∗-field with a basic set σ = α1, α2, E thering of σ∗-operators over K, and M a finitely generated σ∗-K-module with twogenerators z1 and z2 connecting by the defining relation α1z1+α2z2 = 0. (Thus,one can treat M as a factor module of a free left E-module F2 with two freegenerators f1 and f2 by its E-submodule E(α1f1 + α2f2). ) Let (Mr)r∈Z bethe standard filtration of the σ∗-K-module M associated with the system ofgenerators z1, z2. Find the corresponding σ∗-dimension polynomial χ(t) andthe invariants iδ(M), it(M), and tiδ(M).

Transformations of the basic set

Let R be an inversive difference ring with a basic set σ = α1, . . . , αn andlet Γ be the free commutative group generated by the set σ. It is easy to see thatif σ1 = τ1, . . . , τn is another system of free generators of the group Γ, thenthere exists a matrix K = (kij)1≤i,j≤n ∈ GL(n,Z) such that αi = τki1

1 . . . τkinn

for i = 1, . . . , n. Since σ1 is a set of pairwise commuting automorphisms of Rthe ring R can be treated as an inversive difference ring with the basic set σ1.Clearly, the ring E of σ∗-operators over R is the same as the ring of σ∗

1-operatorsover R.

In accordance with the notation of section 2.4, we define the orders of anelement γ = αi1

1 . . . αinn = τ j1

1 . . . τ jnn ∈ Γ with respect to the sets σ and σ1


as numbers ordσγ =∑n

ν=1 |iν | and ordσ1γ =∑n

ν=1 |jν |, respectively. If u =∑γ∈Γ aγγ ∈ E , then the orders of u with respect to σ and σ1 are defined as

usual: ordσu = maxordσγ|aγ = 0 and ordσ1u = maxordσ1γ|aγ = 0.In the rest of this section, an inversive difference ring R with a basic set

σ = α1, . . . , αn will be also denoted by (R, σ). If we treat R as an inversivedifference ring with another basic set σ1, it will be denoted by (R, σ1). Further-more, the ring of σ∗-operators over an inversive difference ring (R, σ) will bedenoted by Eσ or by R〈α1, . . . , αn〉, and the free commutative group generatedby the set σ will be denoted by Γσ. If K is a σ∗-field and M a σ∗-K-module,then the inversive difference dimension, the inversive difference type, and thetypical inversive difference dimension of M will be denoted by iδσ(M), itσ(M),and tiδσ(M), respectively.

Definition 3.5.14 Let (R, σ) and (R, σ1) be inversive difference rings with thebasic sets σ = α1, . . . , αn and σ1 = τ1, . . . , τn, respectively. The sets σ andσ1 are said to be equivalent if there exists a matrix K = (kij)1≤i,j≤n ∈ GL(n,Z)such that αi = τki1

1 . . . τkinn (1 ≤ i ≤ n). In this case we write σ ∼ σ1 and say that

the transformation of the set σ into the set σ1 is an admissible transformationof σ. The matrix K is called the matrix of this transformation.

As we have noticed, if (R, σ) and (R, σ1) are inversive difference rings suchthat σ ∼ σ1, then Eσ = Eσ1 . In this case, Theorem 3.5.12 shows that iδσ(M) =iδσ1(M) for any finitely generated σ∗-(or, equivalently, σ∗

1-)R-module M .

Theorem 3.5.15 Let K be an inversive difference field with a basic set σ =α1, . . . , αn and let M be a finitely generated σ∗-K-module with iδσ(M) = 0.Then there exists a set σ1 = τ1, . . . , τn of pairwise commuting automorphismsof K with the following properties.

(i) The sets σ and σ1 are equivalent.(ii) Let σ2 = τ1, . . . , τn−1. Then M is a finitely generated σ∗

2-K-module.

PROOF. Let E = Eσ be the ring of σ∗-operators over K. Let us, first,consider the case when M is a cyclic left E-module with a generator z: M = Ez.Since iδσ(M) = 0 the element z is σ∗-linearly independent over K (see Theorem3.5.12), that is, there exists a non-zero σ∗-operator U ∈ E such that Uz = 0.Clearly, one can assume that U /∈ K (otherwise, M = Ez = E(U−1Uz) = 0 andour statement becomes trivial).

Let U =m∑

i=1

aiγi, where 0 = ai ∈ K and γi = αt1i1 . . . αtni

n (1 ≤ i ≤ m)

are distinct elements of the group Γσ (tki ∈ Z for k = 1, . . . , n, i = 1, . . . , m).Without loss of generality, we can assume that all exponents tki (1 ≤ k ≤ n, 1 ≤i ≤ m) are nonnegative. (If the σ∗-operator U does not satisfy this condition,then for every k = 1, . . . , n one can choose the minimal number jk ∈ N such thatjk + tki ≥ 0 for all i = 1, . . . , m and consider the σ∗-operator U1 = αj1

1 . . . αjnn U =

m∑i=1

αj11 . . . αjn

n (aiαj1+t1i

1 . . . αjn+tnin ) instead of U .) Furthermore, in what follows


we assume that the elements γ1, . . . , γm do not have a common factor in Tσ

different from 1. (Indeed, if there exists 1 = γ ∈ Tσ such that γi = γγ′i for

i = 1, . . . , m, then (γ−1U)z = 0, so one can replace U by the σ∗-operator

U ′ = γ−1U =m∑

i=1

γ−1(ai)γ′i such that U ′z = 0 and γ′

1, . . . , γ′m lie in Tσ and do

not have a common factor different from 1.) Finally, without loss of generality, weassume that t1i > 0 for some i. Indeed, since all exponents tki are non-negativeand U /∈ K, there exists a strictly positive exponent tki. If k = 1, we can performan admissible transformation of the basic set σ into the set σ′ = β1, . . . , βnsuch that β1 = αk, βk = α1 and βj = αj if 1 < j ≤ n, j = k. Clearly, σ′ ∼ σ and

the σ∗-operator U can be written as U =m∑

i=1

biβs1i1 . . . βsni

n where 0 = bi ∈ K,

sli ∈ N, s1i = tki > 0 (1 ≤ l ≤ n, 1 ≤ i ≤ m) and (s1i, . . . , sni) = (s1j , . . . , snj)if i = j.

Summarizing our assumptions, we can say that Uz = 0 for some σ∗-operator

U =m∑

i=1

aiαt1i1 . . . αtni

n ∈ E \ K such that 0 = ai ∈ K, tki ∈ N (1 ≤ i ≤ m, 1 ≤

k ≤ n), t1i > 0 for some i, elements γ1 = αt111 . . . αtn1

n , . . . , γm = αt1m1 . . . αtnm

n

do not have a non-trivial common factor in Tσ, and (t1i, . . . , tni) = (t1j , . . . , tnj)if i = j.

Let Λ denote the set of all elements of the form aαk11 . . . αkn

n with non-zerocoefficients a ∈ K (k1, . . . , kn ∈ Z). Then one can consider a partial ordering of the set Λ such that aαk1

1 . . . αknn bαl1

1 . . . αlnn if and only if (k1, . . . , kn) is

greater than (l1, . . . , ln) with respect to the lexicographic order on Zn. It is easyto see that if Σ is a finite subset of Λ such that λ1 = aλ2 for any λ1, λ2 ∈ Σ,0 = a ∈ K, then Σ is well-ordered with respect to . Therefore, we can writethe σ∗-operator U as

U = b1αk111 . . . αk1n

n + · · · + bmαkm11 . . . αkmn

n , (3.5.6)

where 0 = bi ∈ K, kij ∈ N (1 ≤ i ≤ m, 1 ≤ j ≤ n) and

b1αk111 . . . αk1n

n · · · bmαkm11 . . . αkmn

n .

Let dj be the maximal exponent of the element αj in the representation (3.5.6),let d(U) =

∑nj=1 dj and let p be a positive integer such that p > 2d(U). Let us

consider the admissible transformation of the basic set σ into another basic setσ1 = τ1, . . . , τn such that

α1 = τ1τp2 τp2

3 . . . τpn−1

n ,

α2 = τ2τp3 . . . τpn−2

n ,

. . . (3.5.7)αn = τn.


(The transformation is admissible, since the determinant of its matrix isequal to 1.)

Expressing our σ∗-operator U in terms of τ1, . . . , τn, we obtain that

U = b1τl111 . . . τ l1n

n + · · · + bmτ lm11 . . . τ lmn

n (3.5.8)

where

li1 = ki1,

li2 = ki1p + ki2,

. . . (3.5.9)lin = ki1p

n−1 + ki2pn−2 + · · · + kin

(i = 1, . . . , m).

Since ki1 > 0 for some i (1 ≤ i ≤ m), k11 > 0 whence l1n > 0. Let us showthat

l1n > l2n ≥ l3n ≥ · · · ≥ lmn .

Indeed, since b1αk111 . . . αk1n

n b2αk211 . . . αk2n

n , there exists an index j, 1 ≤ j ≤ nsuch that k1ν = k2ν for all ν < j and k1j > k2j . Therefore,

l1n − l2n =n∑

ν=1

(k1ν − k2ν)pn−ν > pn−j +n∑

ν=j+1

(k1ν − k2ν)pn−ν

≥ pn−j − d(U)n∑

ν=j+1

pn−ν > pn−j − p

2· pn−j − 1

p − 1> 0

whence l1n > l2n. Similarly, for any i = 2, . . . , m−1, one can obtain the inequal-ity lin ≥ li+1 (the equality holds if lin = li+1,n = 0, that is kij = ki+1,j = 0 forall j = 1, . . . , n).

Writing the relationship Uz =(b1τ

l111 . . . τ l1n

n + · · · + bmτ lm11 . . . τ lmn

n

)z = 0

in the form

b1τl111 . . . τ l1n

n z = −b2τl211 . . . τ l2n

n z − · · · − bmτ lm11 . . . τ lmn

n z ,

we obtain that

τ l1nn z =

[−τ−l11

1 . . . τ−l1,n−1n−1 (b−1

1 )τ−l111 . . . τ

−l1,n−1n−1 (b2)τ l21−l11

1 . . . τl2,n−1−l1,n−1n−1

]× τ l2n

n z + . . . +[−τ−l11

1 . . . τ−l1,n−1n−1 (b−1

1 )τ−l111 . . . τ

−l1,n−1n−1 (bm)τ lm1−l11

1

× . . . τlm,n−1−l1,n−1n−1

]τ lmnn z. (3.5.10)

The last equality shows that the element τ l1nn z can be written as a lin-

ear combination of the elements z, τnz, . . . , τ l1n−1n z with coefficients from


K〈τ1, . . . , τn−1〉. Multiplying both sides of equality (3.5.10) by τn from theleft, we obtain that

τ l1n+1n z =

[−τn(b′2)τ

l21−l111 . . . τ

l2,n−1−l1,n−1n−1

]τ l2n+1n z + . . .

+[−τn(b′m)τ lm1−l11

1 . . . τlm,n−1−l1,n−1n−1

]τ lmn+1n z ′

where b′i = τ−l111 . . . τ

−l1,n−1n−1 (b−1

1 )τ−l111 . . . τ

−l1,n−1n−1 (bi) (1 ≤ i ≤ m).

Thus, the element τ l1n+1n z can be also written as a linear combination of the

elements z, τnz, . . . , τ l1n−1n z with coefficients from K〈τ1, . . . , τn−1〉. Let

τ l1n+1n z = Voz + V1(τnz) + · · · + Vl1n−1(τ l1n−1

n z) , (3.5.11)

where V0, . . . , Vl1n−1 ∈ K〈τ1, . . . , τn−1〉. Multiplying both sides of (3.5.11) byτn from the left we obtain that the element τ l1n+2

n z can be expressed as alinear combination of the elements z, τnz, . . . , τ l1n−1

n z with coefficients fromK〈τ1, . . . , τn−1〉. Continuing this process we obtain that every element τs

nz (s ≥l1n) can be expressed as such a linear combination.

Now, let us consider the σ∗-operator U written in the form (3.5.6). As above,for any j = 1, . . . , n, let dj = maxkij |1 ≤ i ≤ m and let

V = α−d11 . . . α−dn

n U =m∑

i=1

biαki1−d11 . . . αkin−dn

n . (3.5.12)

Then V z = 0 and all exponents of the elements α1, . . . , αn in the last represen-tation of the σ∗-operator V are non-positive. One can also note that the elementα1 really appears in the representation (3.5.12) (otherwise ki1 = d1 = k11 > 0for i = 1, . . . , m, so the monomials biα

ki11 . . . αkin

n would have the common factorαk11

1 that contradicts our assumption).Let us write the σ∗-operator V as

V = c1αr111 . . . αr1n

n + · · · + cmαrm11 . . . αrmn

n (3.5.13)

where 0 = ci ∈ K, rij ≤ 0(1 ≤ i ≤ m, 1 ≤ j ≤ n), and

cmαrm11 . . . αrmn

n · · · c1αr111 . . . αr1n

n .

It is easy to see that the m-tuple (c1, . . . , cm) can be obtained by a permutationof the coordinates of the m-tuple (b1, . . . , bm). Furthermore, if for any j =1, . . . , n, we set d′j = max|rij ||1 ≤ i ≤ m and d(V ) =

∑nj=1 d′j , then d(V ) =

d(U).Let us consider transformation (3.5.7) of the basic set σ (where the integer p

satisfies the condition p > 2d(U) = 2d(V )). Writing the σ∗-operator V in termsof τ1, . . . , τn we arrive at the expression

V = c1τs111 . . . τs1n

n + · · · + cmτsm11 . . . τsmn

n (3.5.14)


where sin = rin, si2 = ri1p + ri2, . . . sin = ri1pn−1 + ri2p

n−2 + · · · + rin. Fur-thermore, the same arguments that were used in the proof of the fact thatl1n > l2n ≥ . . . lmn for the σ∗-operator U written in the form (3.5.8) show thats1n < s2n ≤ · · · ≤ smn.

Writing the equality V z = 0 as

c1τs111 . . . τs1n

n z = −c2τs211 . . . τs2n

n z + · · · − cmτsm11 . . . τsmn

n z

we obtain that

τs1nn z =[−τ−s11

1 . . . τ−s1,n−1n−1 (c−1

1 )τ−s111 . . . τ

−s1,n−1n−1 (c2)τs21−s11

1 . . . τs2,n−1−s1,n−1n−1 ]

× τs2nn z + . . . + [−τ−s11

1 . . . +τ−s1,n−1n−1 (c−1

1 )τ−s111 . . . τ

−s1,n−1n−1 (cm)τsm1−s11

1

. . . τsm,n−1−s1,n−1n−1 ]τ smn

n z (3.5.15)

where s1n < s2n ≤ · · · ≤ smn ≤ 0. Repeatedly multiplying both sides of (3.5.15)by τ−1

n and applying the same equality (3.5.15), we obtain that for any r ∈ Z,r ≤ s1n, the element τ r

nz can be written as a linear combination of the elementsz, τ−1

n z, . . . , τs1n+1n z with coefficients in K〈τ1, . . . , τn−1〉. Thus,

M = Ez =∞∑

i=−∞K〈τ1, . . . , τn−1〉τ i

nz =l1n−1∑

i=s1n+1

K〈τ1, . . . , τn−1〉τ inz .

It follows that the finite set τknz|s1n + 1 ≤ k ≤ lin − 1 generates M as an

inversive difference module with the basic set σ2 = τ1, . . . , τn−1. Since the setσ1 is obtained by the admissible transformation (3.5.7) of the set σ, σ1 ∼ σ.This completes the proof of the theorem fo r the case of cyclic E-modules.

Now, let M be a σ∗-K-module generated by m elements z1, . . . , zm (that

is M =m∑

i=1

Ezi) where m > 1. Then for every i = 1, . . . , m, one can apply

Theorem 3.5.11 to the canonical exact sequence of σ∗-K-modules 0 → Ezi →M → M/Ezi → 0 and obtain that 0 ≤ iδσ(Ezi) = iδσ(M) − iδσ (M/Ezi) =−iδσ (M/Ezi) ≤ 0, so that iδσ(Ezi) = 0 for i = 1, . . . , m. Now, the first partof the proof shows that for every i = 1, . . . , m, there exist a σ∗-operator Ui =ni∑

j=1

aijαqij11 . . . αqijn

n ∈ E such that all qijk (1 ≤ i ≤ m, 1 ≤ j ≤ ni, 1 ≤ k ≤ n)

are non-negative, the monomials αqi111 . . . αqi1n

n , . . . , αqini1

1 . . . αqininn do not have

non-trivial common factors in Tσ, and U1z1 = · · · = Umzm = 0. Let us choosean integer p ∈ Z such that p > 2maxd(Ui)|1 ≤ i ≤ m (as above, d(Ui) =∑m

k=1 maxqijk|1 ≤ j ≤ ni for i = 1, . . . , m) and consider the correspondingadmissible transformation (3.5.7) of the basic set σ into another basic set ofautomorphisms σ1 = τ1, . . . , τn. By the above observations, if one sets σ2 =τ1, . . . , τn−1, then every σ∗-K-module Ezi (1 ≤ i ≤ m) is finitely generated asa σ∗

2-K-module. Thus, the σ∗-K-module M is a finitely generated σ∗2-K-module

as well. This completes the proof of the theorem. The following result can be considered as a generalization of Theorem 3.5.15.


Theorem 3.5.16 Let K be an inversive difference field with a basic set σ =α1, . . . , αn, M a finitely generated σ∗-K-module, and d = itσ(M). Then thereexists a set σ′ = β1, . . . , βn of pairwise commuting automorphisms of K withthe following properties.

(i) The sets σ and σ′ are equivalent (so that σ′ can be obtained from σ by anadmissible transformation).

(ii) Let σ′′ = β1, . . . , βd. Then M is a finitely generated σ′′∗-K-module andiδσ′′(M) > 0.

PROOF. Suppose that a set σ0 = τ1, . . . , τn of automorphisms of thefield K is obtained from σ by an admissible transformation such that αi =τki11 . . . τkin

n and τi = αli11 . . . αlin

n (1 ≤ i ≤ n and kij , lij ∈ Z for i = 1, . . . , n; j =1, . . . , n). Let q = max∑n

j=1 |k1j |, . . . ,∑n

j=1 |knj |,∑n

j=1 |l1j |, . . . ,∑n

j=1 |lnj |and let E denote the ring of σ∗- (and σ∗

0-) operators over K (so that E = Eσ =Eσ0). Furthermore, let (Er)r∈Z and (E0

r )r∈Z be the standard filtrations of thering E associated with the basic sets σ and σ0, respectively. (For any r ∈ N,Er = U ∈ E|ordσU ≤ r and E0

r = U ∈ E|ordσ0U ≤ r where ordσU andordσ0U are the orders of an element U ∈ E when U is treated as a σ∗-operatorand a σ∗

0-operator, respectively. If r < 0, then Er = E0r = 0.) In what follows,

E denotes the ring of σ∗-operators over K equipped with the filtration (Er)r∈Z,while the same ring equipped with the filtration (E0

r )r∈Z will be denoted by E0.Let z1, . . . , zm be any finite system of generators of a σ∗-K-module M , so

that M =m∑

i=1

Ezi. Let χσ(t) and χσ0(t) be the σ∗-dimension polynomials of M

associated with the excellent filtrations (Er)r∈Z and (E0r )r∈Z, respectively. Then

deg χσ(t) = iδσ(M) = d and deg χσ0(t) = iδσ0(M).It is easy to see that Er ⊆ E0

rq ⊆ Erq2 for any r ∈ N whence

dimK

(m∑

i=1

Erzi

)≤ dimK

(m∑

i=1

E0rqzi

)≤ dimK

(m∑

i=1

Erq2zi

).

Therefore, χσ(r) ≤ χσ0(rq) ≤ χσ(rq2) for all sufficiently large r ∈ Z henceitσ0(M) = itσ(M) = d.

Suppose that iδσ(M) = 0 (otherwise, the statement of our theorem be-comes obvious). By Theorem 3.5.15, there exists a set σ1 = θ1, . . . , θn ofpairwise commuting automorphisms of the field K such that σ1 ∼ σ and Mis a finitely generated σ∗

2-K-module where σ2 = θ1, . . . , θn−1. Therefore,

M =m∑

i=1

p∑j=−p

Eσ2(θjnzi) for some positive integer p, so that there exist elements

wijl, vijl ∈ Eσ2 (1 ≤ i ≤ m, −p ≤ j ≤ p, 1 ≤ l ≤ m) such that

θp+1n zl =

m∑i=1

p∑j=−p

wijl(θjnzi), θ−p−1

n zl =m∑

i=1

p∑j=−p

vijl(θjnzi) . (3.5.16)


for l = 1, . . . , m. Let s = maxordσ2wijl, ordσ2vijl|1 ≤ i ≤ m, −p ≤ j ≤p, 1 ≤ l ≤ m and let F denote the ring of σ∗

2-operators over the σ∗2-field K.

Furthermore, let (Fr)r∈Z denote the standard filtration of the ring F (that isFr = 0 for any r < 0 and if r ∈ N, then Fr is a vector K-space generated bythe set of all σ∗

2-operators w such that ordσ2w ≤ r).

It is easy to see that

⎛⎝ m∑i=1

p∑j=−p

Fr(θjnzi)

⎞⎠r∈Z

is an excellent filtration of the

σ∗2-K-module M and

m∑i=1

p∑j=−p

Fr(θjnzi) ⊆

m∑i=1

E0r+pzi ⊆

m∑i=1

p∑j=−p

F(r+p)s(θjnzi) .

(The first inclusion follows from the fact that Frθjn ⊆ E0

r+p for j = −p,−p +1, . . . , p, and the second inclusion is a direct consequence of equalities (3.5.16)).Therefore,

χσ2(r) ≤ χσ1(r + p) ≤ χσ2((r + p)s) (3.5.17)

for all sufficiently large r ∈ Z (χσ2(t) denotes the σ∗2-dimension polynomial of

M associated with the filtration

⎛⎝ m∑i=1

p∑j=−p

Fr(θjnzi)

⎞⎠r∈Z

.

It is easy to see that inequalities (3.5.17) and the equivalence σ1 ∼ σ implythe equalities itσ2(M) = itσ1(M) = d. Furthermore, if iδσ2(M) > 0 (that is,d = n − 1), then the sets σ′ = σ1 and σ′′ = σ2 satisfy conditions (i) and (ii)of the theorem. If iδσ2(M) = 0, then we can repeat the above reasonings andarrive at the set σ3 = λ1, . . . , λn−1 of pairwise commuting automorphismsof K such that σ3 ∼ σ2 and M is a finitely generated σ∗

4-module, where σ4 =λ1, . . . , λn−2. Furthermore, itσ3(M) = itσ2(M) = d.

If iδσ4(M) > 0 (that is, d = n − 2), then we set σ′ = β1, . . . , βn whereβi = λi for i = 1, . . . , n − 1 and βn = θn. Clearly, this set and the set σ′′ =σ4 satisfy conditions (i) and (ii) of our theorem. In this case, the admissibletransformation of σ into σ′ is the composition of the admissible transformationsof σ into σ1 and σ1 into σ′ (the last transformation is obtained by adjoining therelationship βn = θn to the admissible transformation of σ2 into σ3).

If iδσ4(M) = 0 (that is, d < n−2), we can repeat the the above procedure andso on. After a finite number of steps we obtain a set σ′ = β1, . . . , βd, . . . , βnof pairwise commuting automorphisms of the field K such that σ′ ∼ σ, M isa finitely generated σ′′∗-K-module, where σ′′ = β1, . . . , βd, and iδσ′′(M) > 0(that is itσ′′(M) = d). This completes the proof of the theorem.

Exercise 3.5.17 Let K be an inversive difference field with a basic set σ =α1, α2, let E be the ring of σ∗-operators over K, and let M be a finitelygenerated σ∗-K-module with two generators z1, z2 and two defining relationsα1z1 + α2z2 = 0 and α2z1 + α1z2 = 0. (Thus, one can treat, M as a factormodule of a free left E-module F2 with two free generators f1 and f2 by its


E-submodule E(α1f1 + α2f2) + E(α2f1 + α1f2). ) Show that iδσ(M) = 0 andfind an admissible transformation of the set σ into some new set σ1 = τ1, τ2of commuting automorphisms of K such that M is a finitely generated σ∗

2-K-module where σ2 = τ1.

We conclude this section with results on multivariable dimension polynomialsof inversive difference vector spaces associated with partitions of basic sets ofautomorphisms.

Let K be an inversive difference field with a basic set σ = α1, . . . , αn, Γthe free commutative group of all power products αk1

1 . . . αknn (k1, . . . , kn ∈ Z)

and E the ring of σ∗-operators over K. Let us fix a partition of the set σ into adisjoint union of its subsets:

σ = σ1∪· · ·∪σp (3.5.18)

where p ∈ N, and σ1 = α1, . . . , αn1, σ2 = αn1+1, . . . , αn1+n2, . . . , σp =αn1+···+np−1+1, . . . , αn (ni ≥ 1 for i = 1, . . . , p ;n1 + · · · + np = n).

If γ = αk11 . . . αkn

n ∈ Γ (k1, . . . , kn ∈ Z), then for any i = 1, . . . , p, the numberordiγ =

∑n1+···+ni

ν=n1+···+ni−1+1 |kν | (1 ≤ i ≤ p) is called the order of γ with respectto σi (we assume that n0 = 0, so the indices in the sum for ord1γ change from 1to n1). As before, the order of the element γ is defined as ord γ =

∑nν=1 |ki| =∑p

i=1 ordiγ.

We shall consider p orders <1, . . . , <p on the group Γ defined as follows:γ = αk1

1 . . . αknn <i γ′ = αl1

1 . . . αlnn if and only if the vector (ordiγ, ord γ,

ord1γ,. . . , ordi−1γ, ordi+1γ, . . . , ordpγ, kn1+···+ni−1+1, . . . , kn1+···+ni, k1, . . . ,

kn1+···+ni−1 , kn1+···+ni+1, . . . , kn) is less than the vector (ordiγ′, ord γ′, ord1

γ′, . . . , ordi−1γ′, ordi+1γ

′, . . . , ordpγ′, ln1+···+ni−1+1, . . . , ln1+···+ni

, l1, . . . ,ln1+···+ni−1 , ln1+···+ni+1, . . . , ln) with respect to the lexicographic order onZn+p+1.

It is easy to see that Γ is well-ordered with respect to each of the orders<1, . . . , <p.

If r1, . . . , rp are non-negative integers, we set Γ(r1, . . . , rp) = γ ∈ γ | ordiγ ≤ri (i = 1, . . . , p). The vector K-subspace of E generated by the set Γ(r1, . . . , rp)will be denoted by Er1,...,rp

.Setting Er1,...,rp

= 0 for any (r1, . . . , rp) ∈ Zp \ Np, we obtain a familyEr1,...,rp

|(r1, . . . , rp) ∈ Zp of vector K-subspaces of E which is called thestandard p-dimensional filtration of the ring E . Clearly, Er1,...,rp

⊆ Es1,...,spif

(r1, . . . , rp) ≤P (s1, . . . , sp) (as before, ≤P denotes the product order on Zp)and Ei1,...,ip

Ej1,...,jp= Ei1+j1,...,ip+jp

for any (i1, . . . , ip), (j1, . . . , jp) ∈ Np.

Definition 3.5.18 Let M be a vector σ∗-K-space (that is, a left E-module). Afamily Mr1,...,rp

|(r1, . . . , rp) ∈ Zp is said to be a p-dimensional filtration of Mif the following four conditions hold.

(i) Mr1,...,rp⊆ Ms1,...,sp

for any p-tuples (r1, . . . , rp), (s1, . . . , sp) ∈ Zp suchthat (r1, . . . , rp) ≤P (s1, . . . , sp).


(ii)⋃

(r1,...,rp)∈Zp

Mr1,...,rp= M .

(iii) There exists a p-tuple (r(0)1 , . . . , r

(0)p ) ∈ Zp such that Mr1,...,rp

= 0 if ri <

r(0)i for at least one index i (1 ≤ i ≤ p).(iv) Er1,...,rp

Ms1,...,sp⊆ Mr1+s1,...,rp+sp

for any p-tuples (r1, . . . , rp), (s1, . . . , sp)∈ Zp.

If every vector K-space Mr1,...,rpis finite-dimensional and there exists an

element (h1, . . . , hp) ∈ Zp such that Er1,...,rpMh1,...,hp

= Mr1+h1,...,rp+hpfor any

(r1, . . . , rp) ∈ Np, the p-dimensional filtration Mr1,...,rp|(r1, . . . , rp) ∈ Zp is

called excellent.

It is easy to see that if z1, . . . , zk is a finite system of generators of a vector σ∗-

K-space M , then k∑

i=1

Er1,...,rpzi|(r1, . . . , rp) ∈ Zp is an excellent p-dimensional

filtration of M .Let K be an inversive difference field with a basic set σ = α1, . . . , αn, Γ

the free commutative group generated by σ, and E the ring of σ∗-operators overK. Furthermore, we assume that partition (3.5.18) of the set σ is fixed.

In what follows, a free E-module is also called a free σ∗-K-module or afree vector σ∗-K-space. If such a module F has a finite family f1, . . . , fm offree generators, it is called a finitely generated free vector σ∗-K-space. In thiscase the elements of the form γfν (γ ∈ Γ, 1 ≤ ν ≤ m) are called terms while theelements of the group Γ are called monomials. The set of all terms is denotedby Γf ; it is easy to see that this set generates F as a vector space over the fieldK. By the order of a term γfν with respect to σi (1 ≤ i ≤ p) we mean the orderof the monomial γ with respect to σi.

We shall consider p orderings of the set Γf that correspond to the orderingsof the group Γ introduced above. These orderings are denoted by the samesymbols <1, . . . , <p and defined as follows: if γfµ, γ′fν ∈ Γf , then γfµ <i γ′fν

if and only if γ <i γ′ in Γ or γ = γ′ and µ < ν. As in Section 1.5, we considerthe representation

Zn =⋃

1≤j≤2n

Z(n)j (3.5.19)

of the set Zn where Z(n)1 , . . . ,Z(n)

2n are all distinct Cartesian products of n factorseach of which is either Z− = a ∈ Z | a ≤ 0 or N (we assume that Z1 = Nn).Then the group Γ and the set of terms Γf can be represented as the unions

Γ =2n⋃

j=1

Γj and Γf =2n⋃

j=1

Γjf

where Γj = γ = αk1111 . . . α

kpnppnp |(k11, . . . , kpnp

) ∈ Z(n)j and Γjf = γfi | γ ∈

Γj , 1 ≤ i ≤ m. (Such a representation of Γ was also considered in Section 2.4where we studied autoreduced sets of inversive difference polynomials.)


Two elements γ, γ′ ∈ Γ are said to be similar if they belong to the same setΓj (1 ≤ j ≤ 2n). In this case we write γ γ′ or γ j γ′. Note that is not atransitive relation on Γ.

Let F be a finitely generated free vector σ∗-K-space and let f1, . . . , fm befree generators of F over the ring of σ∗-operators E . An element γ ∈ Γ anda term γ′fi ∈ Γf (γ′ ∈ Γ, 1 ≤ i ≤ m) are called similar if γ j γ′ for somej = 1, . . . , 2n. It is written as γ γ′fi or γ j γ′fi. Furthermore, we say thattwo terms γfi, γ

′fk ∈ Γf (1 ≤ i, k ≤ m) are similar and write γfi γ′fk orγfi j γ′fk, if γ j γ′ for some j = 1, . . . , 2n. It is easy to see that if γ u forsome term u ∈ Γf (or γ γ′ for some γ′ ∈ Γ), then ordν(γu) = ordνγ + ordνu(respectively, ordν(γγ′u) = ordνγ + ordνγ′) for ν = 1, . . . , p.

Definition 3.5.19 Let γ1, γ2 ∈ Γ. We say that γ1 is a transform of γ2 andwrite γ2 | γ1, if γ1 and γ2 belong to the same set Γj (1 ≤ j ≤ 2n) and thereexists γ ∈ Γj such that γ1 = γγ2. (In this case we write γ =

γ1

γ2.) Furthermore,

we say that a term u = γ1fi is a transform of a term v = γ2fk and write v |u,if i = k and γ1 is a transform of γ2. (If u = γv, we write γ =

u

v.)

In what follows, if two terms u = γ1fi and v = γ2fi belong to the sameset Γjf (1 ≤ j ≤ 2n) and γ1 = γγ2 for some γ ∈ Γj (so that v|u), then themonomial γ will be sometimes written as a fraction

u

v.

Since the set Γf is a basis of F over the field K, any nonzero element h ∈ Fhas a unique representation in the form

h = a1γ1fi1 + · · · + alγfil(3.5.20)

where γν ∈ Γ, aν ∈ K, aν = 0 (1 ≤ ν ≤ l), 1 ≤ i1, . . . , il ≤ m, and γνfiν= γµfiµ

whenever ν = µ (1 ≤ ν, µ ≤ l).

Definition 3.5.20 Let h be a nonzero element of the E-module F written in theform (3.5.20). Then for every k ∈ 1, . . . , p, the greatest with respect to <k termγνfiν

(1 ≤ ν ≤ l) is called the k-leader of h. It is denoted by u(k)h . The coefficient

of u(k)h in (3.5.20) is called the k-leading coefficient of h and denoted by lck(h).

Remark 3.5.21 Let an element h ∈ F be written in the form (3.5.20). Thenfor any j, 1 ≤ j ≤ 2n, there is a unique term vj in h (vj = γνfiν

for some ν,1 ≤ ν ≤ l) such that u

(1)γh = γvj for every γ ∈ Γj .

Indeed, suppose that there are two terms, vj and wj in h such that γ1vj =u

(1)γ1h and γ2wj = u

(1)γ2h for some elements γ1, γ2 ∈ Γj . Then γ2γ1vj is the 1-leader

of the element γ2γ1h and γ1γ2wj is also the 1-leader of this element. It followsthat γ2γ1vj = γ1γ2wj whence vj = wj .

The term vj with the above property is denoted by ltj(h).

Exercise 3.5.22 Let h be a nonzero element of F written in the form (3.5.20)and let 0 = ω =

∑si=1 biγ

′i ∈ E (0 = bi ∈ K for i = 1, . . . , s, γ′

i = γ′j whenever


i = j). Prove that for every k = 1, . . . , p, there exist unique elements γνfiνand

γ′i (1 ≤ ν ≤ l, 1 ≤ i ≤ s) such that u

(1)ωh = γ′

iγνfiν.

Definition 3.5.23 Let f, g ∈ F and let k, i1, . . . , il be distinct elements inthe set 1, . . . , p. Then the element f is said to be (<k, <i1 , · · · <il

)-reducedwith respect to g if f does not contain any transform γu

(k)g such that ordiν

γ +ordiν

u(iν)g ≤ ordiν

u(iν)f (ν = 1, . . . , l).

An element f ∈ F is said to be (<k, <i1 , · · · <il)-reduced with respect to a

set Σ ⊆ F , if f is (<k, <i1 , · · · <il)-reduced with respect to every element of Σ.

In what follows we introduce a technique of characteristic sets of a free inver-sive difference vector space in the spirit of the theory developed in section 2.4.Our considerations, which involve several term orderings <1, . . . , <p consideredabove, will lead to an analog of Theorem 3.3.16 for finitely generated inversivedifference modules and new invariants of such modules.

Definition 3.5.24 A subset Σ of a free E-module F is called (<1, . . . , <p)-autoreduced if either it is empty or every element of Σ is reduced with respect toany other element of this set. A (<1, . . . , <p)-autoreduced set Σ is called normalif lc1(g) = 1 for every element g ∈ Σ.

Our first goal is to show that every (<1, . . . , <p)-autoreduced set is finite. Inorder to prove this fact we need the following result.

Lemma 3.5.25 Let F be a free E-module with a basis f1, . . . , fm consideredabove and let S be an infinite sequence of terms of the set Γf . Then there existsan index j (1 ≤ j ≤ m) and an infinite subsequence γ1fj , . . . , γ2fj , . . . of thesequence S such that γν |γν+1 for all ν = 1, 2, . . . .

PROOF. Since the sequence S is infinite, there exists k ∈ 1, . . . , 2n suchthat Γkf contains an infinite subsequence of S. Applying Lemma 1.5.1 we ob-tain that this subsequence contains an infinite subsequence with the requiredproperty.

Theorem 3.5.26 Every (<1, . . . , <p)-autoreduced subset of a finitely generatedfree E-module F is finite.

PROOF. Suppose that Σ is an infinite (<1, . . . , <p)-autoreduced subset ofF . Then Σ contains an infinite subset Σ1 such that any two elements of Σ1 havedistinct 1-leaders. Indeed, if it is not so, then there exists an infinite set Σ′ ⊆ Σsuch that all elements of Σ′ have the same 1-leader u. By Lemma 1.5.1, theinfinite set (ord2u

(2)h , . . . , ordpu

(p)h ) ∈ Np−1|h ∈ Σ′ contains a nondecreasing

infinite sequence (ord2u(2)h1

, . . . , ordpu(p)h1

) ≤P (ord2u(2)h2

, . . . , ordpu(p)h2

) ≤P . . .

where h1, h2, · · · ∈ Σ′ (<P denotes the product order on Np−1). We obtain thatwhenever i > j, hi is not (<1, . . . , <p)-reduced with respect to hj , contrary tothe fact that Σ is a (<1, . . . , <p)-autoreduced set.


Thus, we can assume that all 1-leaders of our infinite (<1, . . . , <p)-autoreduced set Σ are distinct. Applying Lemma 3.5.25 we obtain that thereexists an infinite sequence g1, g2, . . . of elements of Σ whose leaders belong tothe same set Γqf (1 ≤ q ≤ 2n) and u

(1)gi |u(1)

gi+1 for i = 1, 2, . . . .

Let kij = ordju(1)gi and lij = ordju

(j)gi (i = 1, 2, . . . , 2 ≤ j ≤ p). Obvi-

ously, lij ≥ kij (i = 1, 2, . . . ; j = 2, . . . , p), so that (li2 − ki2, . . . , lip − kip)|i =1, 2, . . . ⊆ Np−1. By Lemma 1.5.1, there exists an infinite sequence of indicesi1 < i2 < . . . such that (li12 − ki12, . . . , li1p − ki1p) ≤P (li22 − ki22, . . . , li2p −ki2p) ≤P . . . .

Let γ be an element of Γq such that u(1)gi2

= γu(1)gi1

. Then for any j = 2, . . . , p,we have ordj γ+ordju

(1)gi1

= ki2j−ki1j +li1j ≤ ki2j +li2j−ki2j = li2j = ordju(j)gi2

,so that gi2 contains a term γu

(1)gi1

= u(1)gi2

such that γ ∈ Γq and ordj(γu(j)gi1

) ≤ordju

(j)gi2

for j = 2, . . . , p. Thus, gi2 is not (<1, . . . , <p)-reduced with respectto gi1 that contradicts the fact that Σ is a (<1, . . . , <p)-autoreduced set. Thiscompletes the proof of the theorem.

Theorem 3.5.27 Let Σ = g1, . . . , gr be a (<1, . . . , <p)-autoreduced subset ofa free E-module F with free generators f1, . . . , fm and let f ∈ F . Then there

exist elements g ∈ F and λ1, . . . , λr ∈ E such that f − g =r∑

i=1

λigi and g is

(<1, . . . , <p)-reduced with respect to the set Σ.

PROOF. If f is (<1, . . . , <p)-reduced with respect to Σ, the statement isobvious (one can set g = f).

Suppose that f is not (<1, . . . , <p)-reduced with respect to Σ. In what fol-lows, a term wh, that appears in an element h ∈ F , will be called a Σ-leader ofh if wh is the greatest with respect to the order <1 term among all transformsγu

(1)gj (γ ∈ Γ, γ u

(1)gj , 1 ≤ j ≤ r) which appear in h and satisfy the condition

ordνγ + ordνu(ν)gj ≤ ordνu

(ν)h for ν = 2, . . . , p.

Let wf be the Σ-leader of the element f and let cf be the coefficient of wf

in f . Then wf = γu(1)gj for some gj (1 ≤ j ≤ r) and for some γ ∈ Γ such that

γ u(1)gj and ordνγ + ordνu

(ν)gj ≤ ordνu

(ν)f for ν = 2, . . . , p. (Without loss of

generality we may assume that j corresponds to the maximum (with respect tothe order <1) 1-leader u

(1)gj in the set of all 1-leaders of elements of Σ.)

Let f ′ = f − cf (γ(lc1(gj)))−1

γgj . Obviously, f ′ does not contain wf andordν(γu

(ν)f ′ ) ≤ ordνu

(ν)f for ν = 2, . . . , p. Furthermore, f ′ cannot contain any

transform γ′u(1)gi (γ′ ∈ Γ, γ u

(1)gi , 1 ≤ i ≤ r) which is greater than wf with

respect to <1 and satisfies the condition ordνγ′ + ordνu(ν)gi ≤ ordνu

(ν)f ′ for ν =

2, . . . , p. Indeed, the last inequality implies that ordνγ′ + ordνu(ν)gi ≤ ordνu

(ν)f

for ν = 2, . . . , p, so that the term γ′u(1)gi cannot appear in f . This term cannot

appear in γgj either, since u(1)γgj = γu

(1)gj = wf <1 γ′u(1)

gi .


Thus, γ′u(1)gi cannot appear in f ′ = f − cf (γ(lc1(gj)))

−1γgj , whence the Σ-

leader of f ′ is strictly less with respect to the order <1 than the Σ-leader of f .Applying the same procedure to the element f ′ and continuing in the same way,we obtain an element g ∈ F such that f − g is a linear combination of elementsg1, . . . , gr with coefficients in E and g is (<1, . . . , <p)-reduced with respect toΣ. This completes the proof.

The process of reduction described in the proof of the last theorem can berealized by the following algorithm.

Algorithm 3.5.28 (f, r, g1, . . . , gr; g, λ1, . . . , λr)Input: f ∈ F , a positive integer r, Σ = g1, . . . , gr ⊆ F where gi = 0

for i = 1, . . . , rOutput: Element g ∈ F and elements λ1, . . . , λr ∈ E such that

g = f − (λ1g1 + · · · + λrgr) and g is reduced with respect to ΣBeginλ1 := 0, . . . , λr := 0, g := fWhile there exist i, 1 ≤ i ≤ r, and a term w, which appears in g with a

nonzero coefficient c(w), such that u(1)gi |w and

ordk(w

u(1)gi

u(ν)gi ) ≤ ordνu

(ν)g for ν = 2, . . . , p do

z:= the greatest (with respect to <1) of the terms w that satisfythe above conditions.j:= the smallest number i for which u

(1)gi is the greatest (with

respect to <1) 1-leader of an element of gi ∈ Σ such thatu

(1)gi |z and ordν(

z

u(1)gi

u(ν)gi

) ≤ ordνu(ν)g for ν = 2, . . . , p

γ :=z

ugj(1)

λj := λj + c(ω)(γ(lc1(gj))−1 γg := g − c(ω)(γ(lc1(gj))−1 γgj

End

In what follows we keep our notation: E is the ring of σ∗-operators over aninversive difference field with a basic set σ = α1, . . . , αn, (3.5.18) is a fixedpartition of σ, and F is a free left E-module with free generators f1, . . . , fm.

Definition 3.5.29 Let f and g be two elements of the module F . We say thatthe element f has lower rank than g and write rk(f) < rk(g) if either u

(1)f <1

u(1)g or there exists some ν, 2 ≤ ν ≤ p, such that u

(µ)f = u

(µ)g for µ = 1, . . . , ν−1

and u(ν)f <ν u

(ν)g . If u

(i)f = u

(i)g for i = 1, . . . , p, we say that f and g have the

same rank and write rk(f) = rk(g).

In what follows, while considering (<1, . . . , <p)-autoreduced subsets of F ,we always assume that their elements are arranged in order of increasing rank.


Definition 3.5.30 Let Σ = h1, . . . , hr and Σ′ = h′1, . . . , h

′s be two

(<1, . . . , <p)-autoreduced subsets of the free vector σ-K-space F . We say that the(<1, . . . , <p)-autoreduced set Σ has lower rank than Σ′ and write rk(Σ) < rk(Σ′)if one of the following two cases holds:

(1) There exists k ∈ N such that k ≤ minr, s, rk(hi) = rk(h′i) for i =

1, . . . , k − 1 and rk(hk) < rk(h′k).

(2) r > s and rk(hi) = rk(h′i) for i = 1, . . . , s.

If r = s and rk(hi) = rk(h′i) for i = 1, . . . , r, then Σ is said to have the

same rank as Σ′. In this case we write rk(Σ) = rk(Σ′).

Theorem 3.5.31 In every nonempty set of (<1, . . . , <p)-autoreduced subsetsof the free vector σ-K-space F there exists a (<1, . . . , <p)-autoreduced subset oflowest rank.

PROOF. Let Φ be any nonempty set of (<1, . . . , <p)-autoreduced subsetsof F . Define by induction an infinite descending chain of subsets of Φ as follows:Φ0 = Φ, Φ1 = Σ ∈ Φ0|Σ contains at least one element and the first element ofΣ is of lowest possible rank, . . . , Φj = Σ ∈ Φj−1|Σ contains at least j elementsand the j-th element of Σ is of lowest possible rank, . . . . It is clear that if aset Φj is nonempty, then j-th elements of (<1, . . . , <p)-autoreduced sets in Φj

have the same 1-leader u(1)j , the same 2-leader u

(2)j , etc. If Φj are nonempty for

all j = 1, 2, . . . , then the set fj |fj is the j-th element of some (<1, . . . , <p)-autoreduced set in Φj would be an infinite (<1, . . . , <p)-autoreduced set, andthis would contradict Theorem 3.5.26. Therefore, there is the smallest j suchthat Φj is empty.(Since, Φ0 = Φ is nonempty, j > 0.) It is clear that everyelement of Φj−1 is an autoreduced subset in Φ of lowest rank.

Since every E-submodule of F contains at least one (<1, . . . , <p)-autoreducedsubset (e. g., the empty set), we can consider such a subset of lowest rank.

Definition 3.5.32 Let N be an E-submodule of the free E-module F . Then a(<1, . . . , <p)-autoreduced subset of N of lowest rank is called a (<1, . . . , <p)-characteristic set of the module N .

Theorem 3.5.33 Let N be an E-submodule of F and let Σ = g1, . . . , gr be a(<1, . . . , <p)-characteristic set of N . Then an element f ∈ N is (<1, . . . , <p)-reduced with respect to Σ if and only if f = 0.

PROOF. Suppose that f is a nonzero element of N (<1, . . . , <p)-reducedwith respect to Σ. If rk(f) < rk(g1), then the (<1, . . . , <p)-autoreduced set fhas lower rank than Σ. If rk(g1) < rk(f) (f and g1 cannot have the same rank,since f is (<1, . . . , <p)-reduced with respect to Σ), then f and the elementsg ∈ Σ, which have lower rank than f , form a (<1, . . . , <p)-autoreduced set thathas lower rank than Σ. In both cases we arrive at the contradiction with thefact that Σ is a (<1, . . . , <p)-characteristic set of N .


Theorem 3.5.34 Let N be a E-submodule of the free E-module F and letΣ = g1, . . . , gr be a (<1, . . . , <p)-characteristic set of N . Then the elementsg1, . . . , gr generate the E-module N .

PROOF. Let f be any element of N . By Theorem 3.5.27, there exist elementsλ1, . . . , λr ∈ E and an element g ∈ F such that g is (<1, . . . , <p)-reduced with

respect to Σ and f − g =r∑

i=1

λigi. It follows that g ∈ N , and Theorem 3.5.33

shows that g = 0. Thus, f =r∑

i=1

λigi.

Theorem 3.5.35 Let Σ1 = g1, . . . , gr and Σ2 = h1, . . . , hs be two normal(<1, . . . , <p)-characteristic sets of some E-submodule N of the free E-module F .Then r = s and gi = hi for all i = 1, . . . , r.

PROOF. Since Σ1 and Σ2 are two (<1, . . . , <p)-autoreduced sets of thesame (lowest possible) rank, r = s and u

(ν)gi = u

(ν)hi

for i = 1, . . . , r; ν = 1, . . . , p.Suppose that there exists i, 1 ≤ i ≤ r, such that gi = hi. Setting ti = gi − hi

we obtain that u(1)ti

<1 u(1)gi (since the coefficients of u

(1)gi in gi and hi are equal

to 1), u(ν)ti

≤ν u(ν)gi (2 ≤ ν ≤ p), and ti is (<1, . . . , <p)-reduced with respect

to any element gj (1 ≤ j ≤ r). Indeed, suppose that ti contains a transformγu

(1)gj of some 1-leader u

(1)gj such that ordνγ + ordνu

(ν)gj ≤ ordνti

(ν) for ν =2, . . . , p (obviously, ti is (<1, . . . , <p)-reduced with respect to gi, so we canassume that j = i). Then at least one of the elements gi, hi must containγu

(1)gj and ordνγ + ordνu

(ν)gj ≤ ordνti

(ν) ≤ ordνu(ν)gi = ordνu

(ν)hi

for ν = 2, . . . , pthat contradicts the fact that the sets Σ1 and Σ2 are (<1, . . . , <p)-autoreduced.Applying Theorem 3.5.33 we obtain that ti = 0 whence gi = hi.

Theorem 3.5.36 Let Σ be a (<1, . . . , <p)-autoreduced set in F and let N bea E-submodule of F containing Σ. If N does not contain nonzero elements (<1

, . . . , <p)-reduced with respect to Σ, then Σ is a (<1, . . . , <p)-characteristic setof N .

PROOF. Let Σ = g1, . . . , gr and let Σ′ = h1, . . . , hs be a (<1, . . . , <p)-characteristic set of N . Since no nonzero element of N is (<1, . . . , <p)-reducedwith respect to Σ, h1 is not (<1, . . . , <p)-reduced with respect to some gk (1 ≤k ≤ r). Therefore, h1 contains a transform of u

(1)gk hence u

(1)g1 ≤1 u

(1)gk ≤1 u

(1)h1

.

Since Σ′ is a (<1, . . . , <p)-autoreduced set of lowest rank in N , u(1)h = u

(1)g1 and

also u(j)h1

= u(j)g1 for j = 2, . . . , p.

The element h2 is not (<1, . . . , <p)-reduced with respect to Σ, so h2 containsa transform γu

(1)gl of the 1-leader of some gl (1 ≤ l ≤ r) such that ordνγ +

ordνu(ν)gl ≤ ordνu

(ν)h2

for ν = 2, . . . , p (γ ∈ Γ, γ u(1)gl ). It is easy to see that

l = 1 (otherwise, h2 would be reduced with respect to h1). Therefore, h2 contains


a transform of some u(1)gl with l ≥ 2 hence u

(1)g2 ≤1 u

(1)gl ≤1 u

(1)h2

. Because of the

minimality of rank of Σ′ one has u(j)h2

= u(j)g2 for j = 1, . . . , p.

Continuing in the same way we obtain that s ≤ r and rk(hi) = rk(gi) fori = 1, . . . , s. Since Σ′ is a (<1, . . . , <p)-characteristic set of the module N , r = sand Σ is also a (<1, . . . , <p)-characteristic set of N .

The following result can be proved in the same way as Theorem 3.3.15 (onejust needs to use of Theorem 3.5.33 instead of Proposition 3.3.10). We leave thecorresponding adjustment of the proof of Theorem 3.3.15 to the reader as anexercise.

Theorem 3.5.37 Let K be an inversive difference field with a basic set σ,whose partition (3.5.18) into p disjoint subsets is fixed, and let E be the ringof σ∗-operators over K. Let M be a finitely generated E-module with a sys-tem of generators h1, . . . , hm, F a free E-module with a basis f1, . . . , fm,and π : F −→ M the natural E-epimorphism of F onto M (π(fi) = hi

for i = 1, . . . , m). Furthermore, let N = Ker π and let Σ = g1, . . . , gd bea (<1, . . . , <p)-characteristic set of N . Finally, for any r1, . . . , rp ∈ N, let

Mr1...rp=

p∑i=1

Er1...rpfi, and let Vr1...rp

= u ∈ Γf |ordiu ≤ ri for i = 1, . . . , p,

and u is not a transform of any u(1)gi (1 ≤ i ≤ d) ,

Wr1...rp= u ∈ Γf \ Vr1...rp

|ordiu ≤ ri for i = 1, . . . , p and whenever u =γu

(1)g is a transform of some u

(1)g (g ∈ G, γ ∈ Γ), there exists i ∈ 2, . . . , p

such that ordiγ + ordiu(i)g > ri,

Ur1...rp= Vr1...rp

⋃Wr1...rp

.Then for any (r1, . . . , rp) ∈ Np, the set π(Ur1...rp

) is a basis of the vectorK-space Mr1...rp

.

By Theorem 1.5.14, there is a numerical polynomial φV (t1, . . . , tp) in p vari-ables t1, . . . , tp such that φV (r1, . . . , rp) = CardVr1...rp

for all sufficiently large(r1, . . . , rp) ∈ Zp (we use the notation of the last theorem), The total degreeof the polynomial φV does not exceed n and degti

φV ≤ ni for i = 1, . . . , p. Fur-thermore, repeating the arguments of the proof of Theorem 3.3.16, we obtainthere is a certain linear combination φW (t1, . . . , tp) of polynomials of the form(

t1 + n1 + c1

n1

). . .

(tp + np + cp

np

)(c1, . . . , cp ∈ Z) with integer coefficients

such that φW (r1, . . . , rp) = CardWr1...rpfor all sufficiently large (r1, . . . , rp) ∈

Zp. We obtain the following statement which is an analog of the combination ofTheorems 3.3.16 and 3.3.21. (As in section 3.3, for any set Σ ⊆ Np, Σ′ denotesthe set of all maximal elements of Σ with respect to one of the lexicographicorders <j1,...,jp

where j1, . . . , jp is a permutation of 1, . . . , p.)

Theorem 3.5.38 Let K be an inversive difference field with a basic set σ =α1, . . . , αn and let E be the ring of σ∗-operators over K equipped with the


standard p-dimensional filtration corresponding to partition (3.3.18) of the set σ.Furthermore, let ni = Card σi (i = 1, . . . , p) and let Mr1...rp

|(r1, . . . , rp) ∈ Zpbe an excellent p-dimensional filtration of a vector σ∗-K-space M . Then thereexists a polynomial ψ(t1, . . . , tp) ∈ Q[t1, . . . , tp] such that

(i) ψ(r1, . . . , rp) = dimKMr1...rpfor all sufficiently large (r1, . . . , rp) ∈ Zp;

(ii) degtiψ ≤ ni (1 ≤ i ≤ p), so that deg ψ ≤ n and the polynomial

ψ(t1, . . . , tp) can be represented as

ψ(t1, . . . , tp) =n1∑

i1=0

. . .

np∑ip=0

ai1...ip

(t1 + i1

i1

). . .

(tp + ip

ip

)


(iii) The total degree d of the polynomial ψ, the coefficient an1...np, p-tuples

(j1, . . . , jp) ∈ Σ′, the corresponding coefficients aj1...jp, and the coefficients of

the terms of total degree d do not depend on the choice of the excellent filtration.Furthermore, 2n | an1...np

andan1...np

2nis equal to σ∗-dimKM , that is, to the

maximal number of elements of M linearly independent over E.

Definition 3.5.39 The polynomial ψ(t1, . . . , tp), whose existence is establishedby Theorem 3.5.38, is called a dimension (or (σ1, . . . , σp)∗-dimension) polyno-mial of the σ∗-K-vector space M associated with the p-dimensional filtrationMr1...rp

|(r1, . . . , rp) ∈ Zp.

The (σ1, . . . , σp)∗-dimension polynomial ψ(t1, . . . , tp) of an excellently fil-tered E-module M can be computed by constructing a (<1, . . . ,<p)-characteristicset of the corresponding E-submodule N = Ker π of F (we use the notationof Theorem 3.5.37). Such a construction can be fulfilled by analogy with themethod of building of characteristic sets considered in Section 2.4.

Exercise 3.5.40 Introduce an analog of the concept of a coherent autoreducedset (see Definition 2.4.10) for the (<1, . . . , <p)-reduction in a free σ∗-K-vectorspace. Formulate and prove the corresponding analogs of Theorem 2.4.11 andthe two-step method of building characteristic sets considered after Corollary2.4.12.

Another method of computation of (σ1, . . . , σp)∗-dimension polynomials ofinversive difference vector spaces is based on the technique of Grobner baseswith respect to several orderings in free difference vector spaces (this techniqueis developed in Section 3.3).

Let K be an inversive difference field with a basic set σ = α1, . . . , αnand let E be the ring of σ∗-operators over K equipped with the standardp-dimensional filtration corresponding to a fixed partition (3.3.18) of theset σ: σ = σ1 ∪ · · · ∪ σp where p ∈ N, and σ1 = α1, . . . , αn1, σ2 =αn1+1, . . . , αn1+n2, . . . , σp = αn1+···+np−1+1, . . . , αn (ni ≥ 1 for i =1, . . . , p ;n1 + · · · + np = n).


Let M be a finitely generated E-module with a system of generatorsh1, . . . , hm and let Mr1...rp

|(r1, . . . , rp) ∈ Zp be an excellent p-dimensional

filtration of M associated with these generators (Mr1...rp=

m∑i=1

Er1...rphi

for all (r1, . . . , rp) ∈ Zp ). Furthermore, let F be a free E-module with a basisf1, . . . , fm, π : F −→ M the natural E-epimorphism of F onto M (π(fi) = hi fori = 1, . . . , m), and g1, . . . , gd a set of generators of the E-module N = Ker π.

Let gj =m∑

k=1

ωjkfk where ωjk ∈ E (1 ≤ j ≤ d, 1 ≤ k ≤ m).

Let us denote the automorphisms α−11 , . . . , α−1

n of the field K by symbolsαn+1, . . . , α2n, respectively, and let us consider K as a difference field with abasic set σ = α1, . . . , α2n. Let T denote the free commutative semigroupgenerated by the elements α1, . . . , α2n and let D denote the ring of σ-operatorsover K. Furthermore, let us fix the following partition of the set σ:

σ = σ1 ∪ · · · ∪ σp (3.5.21)

where σ1 = α1, . . . , αn1 , αn+1, . . . , αn+n1, . . . , σp = αn1+···+np−1+1, . . . ,αn, αn+n1+···+np−1+1, . . . , α2n. In what follows we shall consider D as a fil-tered ring with the standard p-dimensional filtration Dr1...rp

| (r1 . . . rp) ∈ Zpcorresponding to partition (3.5.21).

Let E be a free D-module with m free generators e1, . . . , em and let N be a

D-submodule of E generated by d+mn elements gj =m∑

k=1

ωjkek (1 ≤ j ≤ d, 1 ≤k ≤ m), ziν = (αiαn+i − 1)eν (1 ≤ i ≤ n, 1 ≤ ν ≤ m) where ωjk is a σ-operatorin D obtained by replacing every α−1

i (1 ≤ i ≤ n) in ωjk with αn+i (1 ≤ j ≤ d,1 ≤ k ≤ m). Then M = E/N is a D-module generated by the cosets ek = ek+N

(1 ≤ k ≤ m), and the vector K-spaces Mr1...rp=

m∑k=1

Dr1...rpek form an excellent

p-dimensional filtration of M (in the sense of Definition 3.3.1). Furthermore, itis easy to see that dimKMr1...rp

= dimKMr1...rpfor every (r1 . . . rp) ∈ Zp.

Thus, the (σ1, . . . , σp)∗-dimension polynomial ψ(t1, . . . , tp) associated withthe filtration Mr1...rp

| (r1 . . . rp) ∈ Zp of the E-module M coincides withthe (σ1, . . . , σp)-dimension polynomial φ(t1, . . . , tp) associated with the filtra-tion Mr1...rp

| (r1 . . . rp) ∈ Zp of the D-module M . It follows that one cancompute the (σ1, . . . , σp)∗-dimension polynomial of the module M using thetechnique of Grobner bases with respect to the orderings <1, . . . , <p developedin Section 3.3.

Example 3.5.41 Let K be an inversive difference field whose basic set σ con-sists of two automorphisms α1 and α2. Let E be the ring of σ∗-operators overK and let M be an E-module with two generators h1 and h2 and one definingrelation α1h1 − α2h2 = 0. In other words, M can be treated as a factor mod-ule of a free E-module F with free generators f1 and f2 by its E-submoduleN = E(α1f1 − α2f2).


Let us fix a partitionσ = σ1 ∪ σ2, (3.5.22)

where σ1 = α1 and σ2 = α2, and compute the (σ1, σ2)∗-dimension poly-nomial ψ(t1, t2) of the module M associated with the excellent 2-dimensionalfiltration Mrs = Ersh1 + Ersh2 | (r, s) ∈ Z2.

Following the above scheme, we denote the automorphisms α−1 and α−2 ofthe field K by α3 and α4, respectively, consider K as a difference field with abasic set σ = α1 . . . , α4, and set a partition

σ = σ1 ∪ σ2, (3.5.23)

where σ1 = α1, α3 and σ2 = α2, α4. Furthermore, we denote the set ofσ-operators over K by D, consider a free D-module E with two free generatorse1, e2 and a D-submodule N of E generated by the elements g1 = α1e1 − α2e2,g2 = α1α3e1 − e1, g3 = α2α4e1 − e1, g4 = α1α3e2 − e2, and g4 = α2α4e2 − e2.Let G0 = g1, g2, g3, g4, g5.

Clearly, u(1)g1 = α1e1, u

(2)g1 = α2e2, u

(1)g2 = u

(2)g2 = α1α3e1, u

(1)g3 = u

(2)g3 =

α2α4e1, u(1)g4 = u

(2)g4 = α1α3e2, u

(1)g5 = u

(2)g5 = α2α4e2.

Applying the Grobner basis method developed in Section 3.3. (and usingthe notation of Definition 3.3.11 and Theorem 3.3.13) we find that S2(g1, g2) =S2(g1, g3) = S2(g2, g4) = S2(g2, g5) = S2(g3, g4) = S2(g3, g5) = 0 (the 2-leadersof the elements of each of these pairs have different free generators of E, so theirleast common multiple is 0),

S2(g2, g3) = α2α4g2 − α1α3g3 = α1α3e1 − α2α4e1g3−−→<2

α1α3e1 − e1g2−−→<2

0

and similarly S2(g4, g5)G0−−→<2

0. Furthermore,

S2(g1, g4) = α1α3g1 + α2g4 = α21α3e1 − α2e2

g2−−→<2

−α2e2 + α1e1g1−−→<2

0.

The only 2-d S-polynomial of elements of G0 which does not <2-reduce to 0with respect to G0 is S2(g1, g5) = α4g1 + g5 = α1α4e1 − e2. We denote thiselement by g6 and set G1 = G0 ∪ g6.

Obviously, S2(g1, g6) = S2(g4, g6) = S2(g5, g6) = 0 (u(2)g6 = u

(1)g6 = α1α4e1

while the 2-leaders of g1, g4, and g5 contain e2). Furthermore,

S2(g3, g6) = α1g3 − α2g6 = −α1e1 + α2e2g1−−→<2

0,

but S2(g2, g6) = α4g2 −α3g6 = −α4e1 −α3e2 does not reduce to 0 with respectto G1. We set g7 = −S2(g2, g6) = α4e1 + α3e2 and G2 = G1 ∪ g7.

Now we have u(1)g7 = α3e2 and u

(2)g7 = α4e1 whence

S2(g1, g7) = S2(g4, g7) = S2(g5, g7) = 0,

S2(g2, g7) = α4g2−α1α3g7 = −α4e1+α1α23e2

g4−−→<2

−α4e1+α3e2g4−−→<2

0,


S2(g3, g7) = g3 − α2g7 = α2α3e2 − e1g1−−→<2

α1α3e1 − e1g2−−→<2

0,

S2(g6, g7) = g6 − α1g7 = α1α3e2 − e2g4−−→<2

0.

Thus, G2 = g1, . . . , g7 is a Grobner basis of N with respect to <2.Considering the 1-leaders of elements of G, we see that S1(g1, g4) = S1(g1, g5)

= S1(g1, g7) = S1(g2, g4) = S1(g2, g5) = S1(g2, g7) = S1(g3, g4) = S1(g3, g5) =S1(g3, g7) = S1(g4, g6) = S1(g5, g6) = S1(g6, g7) = 0 (the 1-leaders of elements ofeach of these pairs include different free generators of E) and S1(g2, g3)

G2−−−−→<1,<2

0,

S1(g2, g6)G2−−−−→

<1,<20, S1(g3, g6)

G2−−−−→<1,<2

0, S1(g4, g5)G2−−−−→

<1,<20 (the 1-leaders of the

elements g2, . . . , g6 are the same as their 2-leaders, so the last four reductionscan be obtained as above, when we considered the corresponding reductionswith respect to <2). Furthermore,S1(g1, g3) = α2α4g1 − α1g3 = α1e1 − α2

2α4e2g1−−−−→

<1,<2α2e2 − α2

2α4e2g5−−−−→

<1,<20,

S1(g1, g6) = α4g1 − g6 = −α2α4e2 + e2g5−−−−→

<1,<20,

S1(g2, g6) = α4g2 − α3g6 = α3e2 − α4e1g7−−−−→

<1,<20,

S1(g3, g6) = α1g3 − α2g6 = α2e2 − α1e1g1−−−−→

<1,<20,

S1(g4, g7) = g4 − α1g7 = α1α4e1 − e2g6−−−−→

<1,<20,

S1(g5, g7) = α3g5 − α2α4g7 = α2α24e1 − α3e2

g3−−−−→<1,<2

α4e1 − α3e2g7−−−−→

<1,<20,

S1(g1, g2) = α3g1 − g2 = −α2α23e2 + e1.

The last element does not (<1, <2)-reduces to 0, so we set g8 = α2α23e2 − e1

and G = G2 ∪ g8. Now one can easily see that S1(g1, g8) = S1(g2, g8) =S1(g3, g8) = S1(g6, g8) = 0, S1(g4, g8) = α2g4 − α1g8 = α1e1 − α2e2

g1−−−−→<1,<2

0,

S1(g5, g8) = α3g5 − α4g8 = α4e1 − α3e2g7−−−−→

<1,<20, and S1(g7, g8) = α2g7 − g8 =

−α2α4e1 + e2g3−−−−→

<1,<20.

Thus, G = g1, . . . , g8 is a Grobner basis of N with respect to (<1, <2).If for any term u = τei = αa

1αb2α

c3α

d4ei ∈ T (a, b, c, d ∈ N, i = 1, 2) we con-

sider the corresponding 4-tuple (a, b, c, d) of exponents of τ , then the set of such4-tuples associated with the 1-leaders of elements of G containing e1 is as follows:

A1 = (1, 0, 0, 0), (1, 0, 1, 0), (0, 1, 0, 1), (1, 0, 0, 1)

(the elements of A1 correspond to u(1)g1 , u

(1)g2 , u

(1)g3 , and u

(1)g6 ).

A similar set A2 associated with the 1-leaders of elements of G containinge2 is of the form

A2 = (1, 0, 1, 0), (0, 1, 0, 1), (0, 0, 1, 0), (0, 1, 1, 0).Using the notation of the proof of Theorem 3.3.16, we obtain that the

numerical polynomial ω(t1, t2), whose values describe the number of elements


of the set U ′rs for all sufficiently large r, s ∈ N, is the sum of the dimen-

sion polynomials ωA1(t1, t2) and ωA2(t1, t2) of the sets A1 and A2, respec-tively (see Definition 1.5.3). Applying formula (1.5.3) one can easily find thatωA1(t1, t2) = ωA2(t1, t2) = 2t1t2 + t1 + 2t2 + 1. Therefore,

ω(t1, t2) = 4t1t2 + 2t1 + 4t2 + 2.

With the notation of Theorem 3.3.16, the set U ′′rs is the union of two

sets U(1)rs and U

(2)rs such that U

(1)rs = u = αa

1αb2α

c3α

d4e1 ∈ T |u = τu

(1)gi

is a transform of some 1-leader u(1)gi containing e1 and ord2(τu

(2)gi ) > s =

u = αa1αb

2αc3α

d4e1 ∈ T | 1 ≤ a ≤ r, c = 0, d = 0, and b = s, U

(2)rs =

u = αa1αb

2αc3α

d4e2 ∈ T |u = τu

(1)gi is a transform of some 1-leader u

(1)gj

containing e2 and ord2(τu(2)gj ) > s = u = αa

1αb2α

c3α

d4e1 ∈ T | 1 ≤ c ≤ r,

a = 0, b = 0, and d = s.It follows that CardU ′′

rs = 2r, so the size of U ′′rs is described by the

polynomial φ(t1, t2) = 2t1. As it is shown in the proof of Theorem 3.3.16,the (σ1, σ2)-polynomial the D-module N is ω(t1, t2) + φ(t1, t2). Therefore, the(σ1, σ2)∗-dimension polynomial ψ(t1, t2) of the module M associated with theexcellent 2-dimensional filtration Mrs = Ersh1 + Ersh2 | (r, s) ∈ Z2 is asfollows:

ψ(t1, t2) = 4t1t2 + 4t1 + 4t2 + 2.

Furthermore, as it follows from Theorem 3.5.38, σ∗-dimKM =422

= 1.

As we have seen, the computation of dimension polynomials associated withfinitely generated inversive difference vector spaces is quite a long process.However, one can always reduce the problem to the computation of dimen-sion polynomials of difference modules and use computer algebra systems forthe realization of the difference analog of the Buchberger Algorithm. Anotherpossible approach is to develop a Grobner basis method for inversive differencevector spaces using the concept of reduction implied by in Definition 3.5.23.

Exercise 3.5.42 With the notation of Definitions 3.5.19, 3.5.20, and 3.5.23, letus consider p−1 new symbols z1, . . . , zp−1 and the free commutative semigroupΛ of all power products λ = γzl1

1 . . . zlp−1p−1 with γ ∈ Γ; l1, . . . , lp−1 ∈ N. Let

Λf = λfj |λ ∈ Λ, 1 ≤ j ≤ m = Λ × f1, . . . , fm. Furthermore, for anyelement f ∈ F , let di(f) = ordiu

(i)f − ordiu

(1)f (2 ≤ i ≤ p) and let ρ : F → Λf

be defined by ρ(f) = zd2(f)1 . . . z

dp(f)p−1 u

(1)f .

Let N be a E-submodule of F . A finite set G = g1, . . . , gt ⊆ N will becalled a Grobner basis of N with respect to the orders <1, . . . , <p if for anyf ∈ N , there exists gi ∈ G such that ρ(gi) | ρ(f) in Λf .

As in section 3.3, for any f, g, h ∈ F , with g = 0, we say that the element f

(<k, <i1 , · · · <il)-reduces to h modulo g in one step and write f

g−−−−−−−−−→<k,<i1 ,···<il

h


iff u(k)g |w for some term w in f with a coefficient a, w = γu

(k)g (γ ∈ Γ),

h = f − a(γ(lck(g)))−1γg and ordiνγ + ordiν

u(iν)g ≤ ordiν

u(iν)f (1 ≤ ν ≤ l).

If f, h ∈ F and G ⊆ F , then we say that the element f (<k, <i1 , · · · <il)-

reduces to h modulo G and write fG−−−−−−−−→

<k;<i1 ,···<il

h iff there exist two se-

quences g(1), g(2), . . . g(q) ∈ G and h1, . . . , hq−1 ∈ F such that fg(1)

−−−−−−−−−→<k,<i1 ,···<il

h1g(2)

−−−−−−−−−→<k,<i1 ,···<il

. . .g(q−1)

−−−−−−−−−→<k,<i1 ,···<il

hq−1g(q)

−−−−−−−−−→<k,<i1 ,···<il

h.

Let us define the least common multiple of two terms u = γu1fi =αk1

1 . . . αknn fi and v = γ′fj = αl1

1 . . . αlnn in Γf , as follows: lcm(u, v) = 0 if

either i = j or i = j and the n-tuples (k1, . . . , kn) and (l1, . . . , ln) do not belongto the same ortant of Zn.

If i = j and the n-tuples of exponents of u and v belong to the same ortantZ(n)

j , then lcm(u, v) = αε1 max|k1|,|l1|1 . . . α

εn max|kn|,|ln|n fi where (ε1, . . . , εn) is

an n-tuple with entries 1 and −1 that belongs to Z(n)j .

Now, for any nonzero elements f, g ∈ F , we can define the rth S-polynomial

of f and g as the element Sr(f, g)=

(lcm(u(r)

f , u(r)g )

u(r)f

(lcr(f))

)−1lcm(u(r)

f , u(r)g )

u(r)f

f

−(

lcm(u(r)f , u

(r)g )

u(r)g

(lcr(g))

)−1lcm(u(r)

f , u(r)g )

u(r)g

g.

Prove the following two statements that lead to an algorithm of computationof Grobner bases of an inversive difference vector spaces.

1. Let G = g1, . . . , gt be a Grobner basis of an E-submodule N of F withrespect to the orders <1, . . . , <p. Then

(i) f ∈ N if and only if fG−−−−−−−−→

<1,<2,···<p

0 .

(ii) If f ∈ N and f is (<1, <2, · · · <p)-reduced with respect to G, then f = 0.2. Let G = g1, . . . , gt be a Grobner basis of an E-submodule N of F

with respect to each of the following sequences of orders: <p; <p−1, <p; . . . ;<r+1, . . . , <p (1 ≤ r ≤ p − 1). Furthermore, suppose that

Sr(gi, gj)G−−−−−−−−−→

<r,<r+1,···<p

0 for any gi, gj ∈ G.

Then G is a Grobner basis of N with respect to <r, <r+1, . . . , <p.Formulate and prove analogs of Theorems 3.3.15 and 3.3.16 using the concept

of a Grobner basis of an E-submodule N of F introduced above.

We conclude this section with a brief discussion of one important general-ization of the results on dimension polynomials of inversive difference modules.Namely, we are going to extend these results to the case of modules with theaction of a finitely generated commutative group. The results obtained in thisdirection is useful in the study of algebraic “G-equations” (that is, algebraicequations with respect to the indeterminates and their images under the action


of elements of a group) and also in the group ring theory. We shall see someapplications of these results in Chapter 7.

Let A be a commutative ring and let elements of a finitely generated com-mutative group G act on A as automorphisms of this ring. Then A is said tobe a G-ring. If J is a subring (an ideal) of A such that g(J) ⊆ J for everyg ∈ G, then J is called a G-subring (respectively, a G-ideal) of the G-ring A.By a prime G-ideal we mean a G-ideal which is prime in the usual sense. If A0

is a G-subring of A and S ⊆ A, then A0SG (or simply A0S if the group Gis fixed) denotes the smallest G-subring of A containing A0 and S (this is theintersection of all G-subrings of A containing A0 and S). In this case we say thatA = A0SG is the G-ring extension of A0 generated by S, and S is called theset of G-generators of A over A0. Clearly, A0SG = A0[g(η) | g ∈ G, η ∈ S].If S = η1, . . . , ηs, we write A = A0η1, . . . , ηsG (or A = A0η1, . . . , ηs if Gis fixed) and say that A0 is a finitely generated G-ring extension of A0 with theset of G-generators η1, . . . , ηs.

Let A and B be G-rings. A ring homomorphism φ : A → B is called aG-homomorphism if φ(g(a)) = g(φ(a)) for any a ∈ A, g ∈ G. The concepts ofG-epimorphism, G-monomorphism, G-isomorphism, etc. are introduced in theusual way. Clearly, if φ : A → B is a G-homomorphism, then Ker A is a G-ideal of A and the natural isomorphism A/Ker φ ∼= φ(B) is a G-isomorphism.Also, if J is a G-ideal of A, then the natural ring epimorphism A → A/J is aG-epimorphism.

If a G-ring is a field, it is called a G-field. It is easy to see that if a G-ring A isan integral domain, then the corresponding quotient field Q(A) can be treated

as a G-field (called a quotient G-field of A) such that g(a

b) =

g(a)g(b)

for every

a ∈ A, 0 = b ∈ A, g ∈ G.If K is a subfield of a G-field L such that g(K) ⊆ K for every g ∈ G, then K

is said to be a G-subfield of L, and L is called a G-overfield or G-field extensionof K (we also say that L/K is a G-field extension). If S ⊆ L, then the smallestG-subfield of L containing K and S (that is, the intersection of all G-subfieldsof L containing K and S) is denoted by K〈S〉G (or K〈S〉 if the group G is fixed)and called the G-field extension of K generated by S; the set S is called theset of G-generators of K〈S〉 over K. Obviously, K〈S〉 coincides with the fieldK(g(η) | g ∈ G, η ∈ S). If S is finite, S = η1, . . . , ηs, and L = K〈S〉, wewrite L = K〈η1, . . . , ηs〉G (or L = K〈η1, . . . , ηs〉) and say that L is a finitelygenerated G-field extension of K (or that the G-field extension L/K is finitelygenerated).

Exercise 3.5.43 Let A be a G-ring and S a multiplicative subset of A suchthat g(S) ⊆ S for every g ∈ G (hence g(S) = S for all g ∈ G). Prove that the

ring of quotients S−1A can be treated as a G-ring such that g(a

s) =

g(a)g(s)

for

all g ∈ G, a ∈ A, s ∈ S. (Show that this action of G on S−1A is well-definedand S−1A is a G-ring with respect to this action of G.) This ring is called theG-ring of quotients of A over S.


Let A be a G-ring and let a primary decomposition of G into a direct productof cyclic subgroups be fixed:

G = α1∞ × . . . αn∞ × β1q1 × · · · × βmqm. (3.5.24)

Then every element g ∈ G has a unique representation of the form

g = αk11 . . . αkn

n βl11 . . . βlm

m (3.5.25)

where ki, lj ∈ Z and 0 ≤ lj ≤ qj − 1 (1 ≤ i ≤ n, 1 ≤ j ≤ m). The numberord g =

∑ni=1 |ki|+

∑mj=1 lj is called the order of the element g. It is easy to see

that ord g ≥ 0, ord g = 0 if and only if g = 1 (the identity of the group G), andord(g1g2) ≤ ord g1 + ord g2 for any g1, g2 ∈ G. Furthermore, for any r ∈ N, weset G(r) = g ∈ G | ord g ≤ r.

An expression of the form∑

g∈G agg, where ag ∈ A and only finitely many ag

are distinct from zero, is called a G-operator over A. Two G-operators∑

g∈G aggand

∑g∈G bgg are considered to be equal if and only if ag = bg for all g ∈ G.

The set of all G operators over A has a natural structure of a left A-module.This A-module becomes a ring if for any elements g1, g2 ∈ G, treated asG-operators, we define their product g1g2 as it is defined in G, set ga = g(a)g forany g ∈ G, a ∈ A and expand these rules to the product of any two G-operatorsby distributivity. The ring we obtain is called the ring of G-operators over Aor the ring of G-A-operators; it is denoted by FA or simply by F . The ringA and the group G are naturally considered as a subring and a subset of F ,respectively (we use the same symbol 1 for the unity of A, the identity of G,and the unity of F). Note that F is a twisted group ring of the group G over Aif one uses the ring theory terminology.

The order of a G-operator P =∑

g∈G agg is defined as the numberordP = maxord g | ag = 0. Clearly, for any two G operators P and Q, onehas ord(PQ) ≤ ordP + ordQ and ord(P + Q) ≤ maxordP, ordQ. SettingFr = P ∈ F | ordP ≤ r for any r ∈ N and Fr = 0 for any r ∈ Z, r < 0,we obtain an ascending filtration (Fr)r∈Z of the ring F called the standardfiltration of F associated with decomposition (3.5.24). Clearly, FrFs = Fr+s

for every r, s ∈ N. In what follows, while considering F as a filtered ring wealways mean this filtration of F .

Definition 3.5.44 With the above notation, a left F-module is called a G-A-module or a G-module over the G-ring A. Thus, a G-A-module is a left A-moduleM such that the elements of group G act on M as endomorphisms of the additivegroup of M satisfying the following conditions:

(i) The identity of G acts as the identity automorphism of M ;(ii) g1(g2(x)) = g2(g1(x)) = (g1g2)(x) for any g1, g2 ∈ G, x ∈ M ;(iii) g(ax) = g(a)g(x) for any g ∈ G, a ∈ A, x ∈ M .

If M and N are two G-A-modules, then a homomorphism of A-modules φ :M → N is called a G-homomorphism (or a G-A-homomorphism or a homomor-phism of G-A-modules) if φ(g(x)) = g(φ(x)) for any g ∈ G, x ∈ M . The notions


of G-epimorphism, G-monomorphism, G-isomorphism, etc. of G-A-modules aredefined in the usual way. We say that a G-A-module is finitely generated if it isfinitely generated as a left F-module.

By a filtration of a G-A-module M we mean a discrete and exhaustive as-cending filtration of M as module over the filtered ring F . If M has such afiltration, it is called a filtered G-A-module.

A filtration (Mr)r∈Z of a G-A-module M is called finite if every A-moduleMr is finitely generated. This filtration is called good if there exists r0 ∈ Zsuch that FsMr = Mr+s for all integers r ≥ r0 and s ≥ 0. A finite and goodfiltration is called excellent. One can easily see that if A is a G-field and M is a

finitely generated G-A-module, M =s∑

i=1

Fxi, then (s∑

i=1

Frxi)r∈Z is an excellent

filtration of M .

Exercise 3.5.45 With the above notation, prove that if the ringA is Noetherian,then the ring F is left and right Noetherian.

Exercise 3.5.46 Let M and N be two G-A-modules. Introduce the structure ofa G-A-module on each of the A-modules M

⊗A N and HomA(M,N) (mimic the

construction considered before Lemma 3.4.4) and prove the analogs of Lemmas3.4.4 and 3.4.5. Then prove an analog of Theorem 3.4.6 in this case.

Theorem 3.5.47 Let A be an Artinian G-ring and let (Mr)r∈Z be an excellentfiltration of a G-A-module M (associated with the fixed decomposition (3.5.24)of the group G). Then there exists a numerical polynomial θ(t) in one variablet such that

(i) θ(r) = lA(Mr) for all sufficiently large r ∈ Z ;(ii) deg χ(t) ≤ n (= rank G) and the polynomial θ(t) can be represented in

the form

θ(t) =n∑

i=0

ai

(t + i

i

)(3.5.26)

where a0, . . . , an ∈ Z and 2n|an.

PROOF. Let grF denote the graded ring associated with the filtered ringof G-operators F over A. Then the ring grF is generated over A by the pair-wise commuting elements x1, . . . , x2n, y1, . . . , ym which are the canonical imagesin grF of the elements α1, . . . , αn, α−1

1 , . . . , α−1n , β1, . . . , βm, respectively. Fur-

thermore, xia = αi(a)xi, xn+ia = α−1i (a)xn+i (1 ≤ i ≤ n) and yja = βj(a)yj

(1 ≤ j ≤ m) for any a ∈ A. A homogeneous component grsF (s ∈ N) of thegraded ring grF is an A-module generated by all monomials

xk1i1

. . . xknin

yl11 . . . ylm

m

such that

(a) ki, lj ∈ N, lj < qj (1 ≤ i ≤ n, 1 ≤ j ≤ m);


(b)n∑

i=1

ki +m∑

j=1

li = s;

(c) 1 ≤ i1, . . . , in ≤ 2n and iµ − iν = n or 0 whenever µ = ν.

In what follows we denote the ring grF by Rn, while R′n and R′′

n will de-note the the graded subrings of Rn whose elements can be written as linearcombinations over A of the monomials free of xn and x2n, respectively.

Let gr M =⊕

s∈Z grsM be the graded grF-module associated with the ex-cellent filtration (Mr)r∈Z. In what follows, the module grM will be also denotedby M and its homogeneous components grsM = Ms/Ms−1 (s ∈ Z) will be de-noted by M (s). Repeating the arguments of the proof of Theorem 3.2.3 (alsoused in the proof of Theorem 3.5.2), we obtain that gr M is a finitely generatedgrF-module. Furthermore, as in the proof of Theorem 3.5.2, one can see thatin order to prove our statement we need to prove the existence of a numericalpolynomial h(t) in one variable t with the following properties:

(a) h(s) = lA(M (s)) for all sufficiently large s ∈ Z;(b) deg h(t) ≤ n − 1 and the polynomial f(t) can be represented as h(t) =

n−1∑i=0

bi

(t + i

i

)where b0, . . . , bn−1 ∈ Z and 2n|bn−1.

We shall prove the existence of the polynomial h(t) by induction on n. Ifn = 0, then M = gr M is a finitely generated A-module, hence one can takeh(t) = 0. Suppose that n > 0 and let us consider the exact sequence of finitelygenerated Rx1, . . . , x2n-modules

0 → Ker πn → M πn−−→ xnM → 0

where πn is the mapping of multiplication by xn. Clearly πn is an additivemapping and πn(ay) = αn(a)πn(y) for any y ∈ M,a ∈ A. It follows fromProposition 3.1.4 (with K = (Ker πn)(s),M = M (s), N = (xnM)(s), and δ =πn) that

lA((Ker πn)(s)) + lA((xnM)(s)) = lA(M (s)) (3.5.27)

for every s ∈ Z (we denote the sth homogeneous component of a graded moduleP by P (s).)

Since Ker πn and xnM are annihilated by the elements xn and x2n, respec-tively, we can treat Ker πn as a graded R′

n-module and xnM as a graded R′′n-

module. By the inductive hypothesis, for any finitely generated Rn−1-moduleN =

⊕p∈Z N (p), there exists a numerical polynomial φN (t) such that φN (p) =

lA(N (p)) for all sufficiently large p ∈ Z, deg φN ≤ n − 2, and the polynomial

φN (t) can be written as φN (t) =n−2∑i=0

ci

(t + i

i

)where c0, . . . , cn−2 ∈ Z and

2n−1|cn−2.


If L =⊕

q∈Z L(q) is a finitely generated graded R′n-module, then the first

and the last terms of the exact sequence of R-modules

0 → (Ker π2n)(q) → L(q) π2n−−→ L(q+1) → L(q+1)/x2nL(q) → 0

(π2n is the mapping of multiplication by x2n) are homogeneous components of fi-nitely generated graded Rn−1-modules. By the inductive hypothesis, there exists

a numerical polynomial ψL(t) of the form ψL(t)=n−2∑i=0

di

(t + i

i

)(d0, . . . , dn−2∈Z

and 2n−1|dn−2) such that ψL(q) = lA((L(q+1)) − lA(L(q)) for all sufficientlylarge q ∈ Z, say, for all q ≥ q0 for some integer q0. Since

lA(L(q)) = lA(L(q0)) +q∑

s=q0+1

[lA(L(s)) − lA(L(s−1))

],

it follows from Corollary 1.4.7 that there exists a numerical polynomial χL(t)such that χL(q) = lA(L(q)) for all sufficiently large q ∈ Z and the polynomial

χL(t) can be written as χL(t) =n−1∑i=0

d′i

(t + i

i

)where d′0, . . . , d

′n−1 ∈ Z and

2n−1|d′n−1. Clearly, a similar statement is true for any finitely generated gradedmodule K =

⊕q∈Z K(q) over the ring R′′

n.

Thus, there exist numerical polynomials h1(t) =n−1∑i=0

a′i

(t + i

i

)and h2(t) =

n−1∑i=0

a′′i

(t + i

i

)(a′

i, a′′i ∈ Z for i = 0, . . . , n − 1 and 2n−1|a′

n−1, 2n−1|a′′n−1) such

that h1(s) = lA((Ker πn)(s)) and h2(s) = lA((xnM)(s)). Now we can use thearguments of the corresponding part of the proof of Theorem 3.5.2 to obtain thatlA((xiM)(s)) = lA((xn+iM)(s)) for any s ∈ Z, i = 1, . . . , n. The last equality,together with equality (3.5.27), implies that lA(M (s)) = h1(s) + h2(s) for allsufficiently large s ∈ Z.

The last term of the sequence of graded R′n-modules

0 → x2nM → Ker πn → Ker πn/x2nM

is a finitely generated graded Rn−1-module. By the inductive hypothesis, thereexists a numerical polynomial h3(t) of degree at most n − 2 such that h3(s) =lA

((Ker πn/x2nM)(s)

)for all sufficiently large s ∈ Z. It follows that a′

n−1 =

a′′n−1, hence the polynomial h(t) = h1(t) + h2(t) =

n−1∑i=0

bi

(t + i

i

), where bn−1 =

2a′n−1 is divisible by 2n, satisfies conditions (a) and (b). This completes the

proof of the theorem.

Definition 3.5.48 Let A be an Artinian G-ring and let (Mr)r∈Z be an excel-lent filtration of a G-A-module M . Then the polynomial θ(t) whose existence


is established by Theorem 3.5.47 is called the G-dimension polynomial of Massociated with the excellent filtration (Mr)r∈Z.

Example 3.5.49 Let A be a G-field and let decomposition (3.5.24) of the groupG be fixed. Let θF (t) denote the G-dimension polynomial of the correspondingring of G-operators associated with the excellent filtration (Fr)r∈Z. Then

θF (r) = dimAFr = Cardαk11 . . . αkn

n βl11 . . . βlm

m |ki, lj ∈ Z, 0 ≤ lj < qj

(1 ≤ i ≤ n, 1 ≤ j ≤ m),n∑

i=1

|ki| +m∑

j=1

lj ≤ r

for all sufficiently large r ∈ Z. Let Q =∑m

j=1(qj − 1) and, for any s ∈ N,let φ1(s) denote the number of distinct elements βl1

1 . . . βlmm with 0 ≤ lj < qj

(j = 1, . . . , m) and∑m

j=1 lj = s, and let φ1(s) denote the number of distinctelements αk1

1 . . . αknn with k1, . . . , kn ∈ Z and

∑ni=1 |ki| ≤ s. Clearly,

θF (r) =Q∑

i=0

φ1(i)φ2(s − i)

for all sufficiently large s ∈ N, hence we can find θF (t) if we can find formulas forφ1(s) and φ2(s). Applying formulas (1.4.17) and (1.4.18) we obtain the followingexpressions for φ1(s) and φ2(s):

φ2(s) =n∑

i=0

(−1)n−i2i

(n

i

)(s + i

i

),

and

φ1(s) =(

m + s − 1m − 1

)=

m∑i=1

(−1)k∑

1≤j1<···<jk≤mqj1+···+qjk

≤s

(m + s − qj1 − · · · − qjk

− 1m − 1

).

Therefore,

θF (t) =Q∑

i=0

[(m + i − 1

m − 1

)

+m∑

k=1

(−1)k∑

1≤j1<···<jk≤mqj1+···+qjk

≤i

(m + t − qj1 − · · · − qjk

− 1m − 1

)]

×[ n∑

l=0

(−1)n−l2l

(n

l

)(t − i + l

l

)]. (3.5.28)

Exercise 3.5.50 Formulate and proof analogs of Lemmas 3.5.5 and 3.5.6 forG-modules. Then prove an analog of Theorem 3.5.7 for this case.


Let K be a G-field and let (Fr)r∈Z be the standard filtration of the ringof G-K-operators associated with decomposition (3.5.24). Let M be a finitely

generated G-K-module with generators z1, . . . , zl and let (Mr =l∑

i=1

Frzi)r∈Z

be the corresponding excellent filtration of M . Then the field K can be alsotreated as an inversive difference field with the basic set σ = α1, . . . , αn andM can be considered as a finitely generated σ∗-K-module (i. e., module over thering of σ∗-operators E over K) with the set of generators hzi |h ∈ H, 1 ≤ i ≤ lwhere H denotes the (finite) periodical part of the group G (H = g ∈ G | gk = 1

for some k ∈ N). Clearly, (M ′r =

l∑i=1

∑h∈H

Erhzi)r∈Z is an excellent filtration

of the σ∗-K-module M , so we can consider the σ∗-dimension polynomial χ(t)

associated with this filtration. Let θ(t) =n∑

i=0

ai

(t + i

i

)be the G-dimension

polynomial of M associated with the filtration (Mr)r∈Z. Since ord h ≤ Q =∑mj=1(qj − 1) for every h ∈ H, one has Mr ⊆ M ′

r ⊆ Mr+Q for every r ∈ Z.Therefore, θ(r) ≤ χ(r) ≤ θ(r + Q) for all sufficiently large r ∈ Z. It followsthat if d = deg θ(t), then d = σ∗-typeKM ,

an

2n= σ∗-dimKM , the coefficient ad

is divisible by 2d, andad

2d= σ∗-tdimKM . Thus, the degree d of θ(t) and the

numbersan

2nand

ad

2d, do not depend on the choice of an excellent filtration of

M associated with decomposition (3.5.24). Let us show that these numbers donot depend on the decomposition of G either. Indeed, let

G = γ1∞ × . . . γn∞ × δ1q1 × · · · × δmqm

be another primary decomposition of G and let

αi = γki11 . . . γkin

n δli11 . . . δlim

m (1 ≤ i ≤ n),

γj = αuj11 . . . αujn

n βvj11 . . . βvjm

m (1 ≤ i ≤ n)

where kiµ, liν , uiλ, vjp ∈ Z, 0 ≤ liν < qν , 0 ≤ vjp < qp (1 ≤ µ, λ ≤ n, 1 ≤ν, p ≤ m). Setting q = q1 . . . qm we obtain that αq

i = (γq1)ki1 . . . (γq

n)kin and γqj =

(αq1)

uj1 . . . (αqn)ujn (1 ≤ i, j ≤ n). Let σ1 = αq

1, . . . , αqn and σ2 = γq

1 , . . . , γqn.

Then the ring of σ∗1-operators over K (when K is treated as a σ∗

1-field) coincideswith the ring of σ∗

2-operators over K (when K is considered as a σ∗2-field). Let

us denote this ring by E ′ and consider the standard filtration (E ′r)r∈Z of E ′ as a

ring of σ∗1-operators.

It is easy to see that the elements hαk11 . . . αkn

n zj (h ∈ H, 0 ≤ k1, . . . , kn < q,1 ≤ j ≤ l) generate M over E . Let

Mr =l∑

j=1

∑h∈H

∑0≤k1,...,kn<q

E ′rhαk1

1 . . . αknn zj


(r ∈ Z). Then (Mr)r∈Z is an excellent filtration of the σ∗-K-module M . Sincefor any element g = αqd1+e1

1 . . . αqdn+enn (0 ≤ ei < q for i = 1, . . . , n), one has

ord g =n∑

i=1

|qdi + ei| ≥ ord h+ q(n∑

i=1

|di|−n), the inequality ord g ≤ qr (r ∈ N)

implies thatn∑

i=1

|di| ≤ r + n. It follows that Mqr ⊆ Mr+n for all sufficiently

large r ∈ Z. Since we also have Mr ⊆ Mqr+qn+Q, the degree of the G-dimensionpolynomial θ(t) is equal to the degree d of a σ∗

1-dimension polynomial of M , and

the numberan

2nand

ad

2dare equal to the integers

σ∗1-dimKM

qn=

σ∗2-dimKM

qnand

σ∗1-tdimKM

qd=

σ∗2-tdimKM

qd, respectively. We obtain the following theorem.

Theorem 3.5.51 Let K be a G-field, let M be a finitely G-K-module, and let

θ(t) =n∑

i=0

ai

(t + i

i

)be the G-dimension polynomial of M associated with a

decomposition of the form (3.5.24) and an excellent filtration of M . Then theintegers d = deg θ(t), a =

an

2n, and a(d) =

ad

2dare independent of the choice of

such decomposition and filtration. If σ is a set of free generators of some freecommutative subgroup H of G whose rank is equal to the rank of G, then d =σ∗-typeKM , a = σ∗-dimKM , and a(d) = σ∗-tdimKM .

Definition 3.5.52 With the notation of the last theorem, the integers d, a, anda(d) are called the G-dimension, G-type, and typical G-dimension of the G-K-module M , respectively. They are denoted, respectively, by δG(M), tG(M), andtδG(M).

The following two propositions can be easily proved using the arguments ofthe proofs of Theorems 3.2.11 and 3.2.12.

Proposition 3.5.53 Let K be a G-field and let

0 −→ Ni−→ M

j−→ P −→ 0

be an exact sequence of finitely generated G-K-modules. Then δG(N)+δ(GP ) =δG(M).

Proposition 3.5.54 Let K be a G-field, F the ring of G-operators over K, andM a finitely generated G-K-module. Let G0 be any free commutative subgroupof G such that rank G0 = rank G. Then δG(M) is equal to the maximal numberof elements x1, . . . , xp ∈ M such that the set g(xi) | g ∈ G0, 1 ≤ i ≤ p islinearly independent over K.


3.6 Dimension of General Difference Modules

In this section we generalize the results on difference and inversive differencemodules obtained in the preceding sections. We also describe the main invariantsof characteristic polynomials of difference and inversive difference modules in theframe of the theory of Krull dimension.

Let R be a difference ring whose basic set is a union of a set σ = α1, . . . , αmof injective endomorphisms of the ring R and a set ε = β1, . . . , βn of auto-morphisms of R. (As usual, any two elements of the basic set σ ∪ ε commutewith each other.)

Let T (or Tσ) and Γ (or Γε) denote the free commutative semigroup generatedby the set σ and the free commutative group generated by the set ε, respectively.Furthermore, let Θ denote the commutative semigroup of all power products ofthe form

θ = αk11 . . . αkm

m βl11 . . . βln

n (3.6.1)

where k1, . . . , km ∈ N and l1, . . . , ln ∈ Z. This semigroup contains boththe semigroup T and the group Γ. In what follows, the subset β1, . . . , βn,β−1

1 , . . . , β−1n of Θ will be denoted by ε∗, and the ring R will be called a

σ-ε∗-ring. (As before, assigning ∗ to the set ε means that we treat R as aninversive difference ring with respect to the basic set ε. At the same time, theendomorphisms αi are not supposed to be bijective, so we consider only powerproducts (3.6.1) with positive exponents ki, even though some of the αi couldbe automorphisms of R.)

By the orders of the element θ of the form (3.6.1) relative to the sets σ and

ε we mean the numbers ordσθ =m∑

i=1

ki and ordεθ =n∑

j=1

|lj |, respectively; the

number ord θ = ordσθ + ordεθ is called the order of θ. Furthermore, for everyr ∈ N, Θ(r) will denote the set of all elements θ ∈ Θ such that ord θ ≤ r.


θ∈Θ aθθ, where aθ ∈ R for anyθ ∈ Θ and only finitely many elements aθ are different from zero, is called a σ-ε∗-operator over R. Two σ-ε∗-operators

∑θ∈Θ aθθ and

∑θ∈Θ bθθ are considered

to be equal if and only if aθ = bθ for all θ ∈ Θ.

The set of all σ-ε∗-operators over a σ-ε∗-ring R will be denoted by F . Asin the case of difference or inversive difference operators, the set F can beequipped with a ring structure if one sets

∑θ∈Θ aθθ +

∑θ∈Θ bθθ =

∑θ∈Θ(aθ +

bθ)θ, a∑

θ∈Θ aθθ =∑

θ∈Θ(aaθ)θ,(∑

θ∈Θ aθθ)δ =

∑θ∈Θ aθ(θδ), δa = δ(a)δ for

any elements∑

θ∈Θ aθθ,∑

θ∈Θ bθθ ∈ F , a ∈ R, δ ∈ σ ∪ ε∗ and extends thisoperations to the whole set F by distributivity. The resulting ring is called thering of σ-ε∗-operators over R. (Note that the ring of σ-operators D and the ringof ε∗-operators E introduced in Sections 3.1 and 3.4, respectively, are subringsof the ring F .)

By the order of a σ-ε∗-operator u =∑

θ∈Θ aθθ ∈ F we mean the non-negative integer ord u = maxord θ|aθ = 0. If r ∈ N, then the set of all σ-ε∗-operators whose orders do not exceed r will be denoted by Fr. Setting Fr = 0 for

3.6. DIMENSION OF GENERAL DIFFERENCE MODULES 233

any negative integer r, we obtain a discrete and exhaustive ascending filtration(Fr)r∈Z of the ring F . This filtration is called the standard filtration of the ringof σ-ε∗-operators F . In what follows, while considering F as a filtered ring wealways mean that F is equipped with the standard filtration.

Definition 3.6.2 Let R be a σ-ε∗-ring and F the ring of σ-ε∗-operators over R.Then a left F-module is called a σ-ε∗-R-module. In other words, an R-module Mis called a σ-ε∗-R-module if the elements of the set σ∪ε∗ act on M in such a waythat δ(x + y) = δx + δy, δ1(δ2x) = δ2(δ1x), δ(ax) = δ(a)δx, and β(β−1x) = xfor any x, y ∈ M,a ∈ R, δ, δ1, δ2 ∈ σ ∪ ε∗, and β ∈ ε∗.

If R is a field then a σ-ε∗-R-module is also called a vector σ-ε∗-space overR or a vector σ-ε∗-R-space

It is easy to see that if R is a σ-ε∗-ring, then any σ-ε∗-R-module is at the sametime a σ-R-module and an ε∗-R-module in the sense of definitions 3.1.2 and 3.4.3,respectively. In accordance with the definition of the difference homomorphism(see Definition 3.1.3), if M and N are two σ-ε∗-R-modules, then a mappingf : M → N is called a σ-ε∗-homomorphism if f is a homomorphism ofR-modules such that f(δx) = δf(x) for any x ∈ M, δ ∈ σ ∪ ε. (It is easy to seethat in this case f commutes with the elements β−1

1 , . . . , β−1n as well). An in-

jective (respectively, surjective or bijective) σ-ε∗-homomorphism is said to be aσ-ε∗-monomorphism (respectively, a σ-ε∗-epimorphism or a σ-ε∗-isomorphism).

Definition 3.6.3 Let R be a σ-ε∗-ring and F the ring of σ-ε∗-operatorsover R equipped with the standard filtration (Fr)r∈Z. An ascending chainof R-submodules (Mr)r∈Z of a σ-ε∗-R-module M is called a filtration of M ifFsMr ⊆ Mr+s for all r, s ∈ Z,

⋃r∈Z

Mr = M , and Mr = 0 for all sufficiently

small r ∈ Z. (Thus, by a filtration of a σ-ε∗-R-module M we mean an exhaustiveand discrete filtration of M as a left F-module.)

A filtration (Mr)r∈Z of a σ-ε∗-R-module M is called excellent if every itscomponent Mr (r ∈ Z) is a finitely generated R-module and there exists r0 ∈ Zsuch that FsMr = Mr+s for any r ∈ Z, r ≥ r0, s ∈ N.

Let R be a σ-ε∗-ring and let M and N be filtered σ-ε∗-R-modules withfiltrations (Mr)r∈Z and (Nr)r∈Z, respectively. Then a σ-ε∗-homomorphismf : M → N is said to be a σ-ε∗-homomorphism of filtered σ-ε∗-R-modules iff(Mr) ⊆ Nr for any r ∈ Z.

It is easy to see that if a σ-ε∗-R-module M has an excellent filtration, then Mis a finitely generated F-module. In what follows, a finitely generated F-modulewill be also called a finitely generated σ-ε∗-R-module.

If R is a σ-ε∗-field and z1, . . . , zq is a finite system of generators of a σ-

ε∗-R-module M (that is M =n∑

i=1

Fzi), then

(n∑

i=1

Frzi

)r∈Z

is an excellent

filtration of M called the standard filtration of M associated with the systemof generators z1, . . . , zq. Thus, every vector σ-ε∗-space over a σ-ε∗-field hasexcellent filtrations.


The following result generalizes theorems 3.2.3 and 3.5.2 that introduce char-acteristic polynomials of difference and inversive difference modules.

Theorem 3.6.4 Let R be an Artinian σ-ε∗-ring with a basic set σ ∪ ε whereσ = α1, . . . , αm and ε = β1, . . . , βn are the sets of injective endomorphismsand automorphisms of the ring R, respectively. Let M be a σ-ε∗-R-module andlet (Mr)r∈Z be an excellent filtration of M . Then there exists a numerical poly-nomial Φ(t) in one variable t such that

(i) Φ(r) = lR(Mr) for all sufficiently large r ∈ Z(ii) deg Φ(t) ≤ m + n and the polynomial Φ(t) can be represented as

Φ(t) =m−1∑i=0

ai

(t + i

i

)+

n∑j=0

2jbj

(t + m + j

m + j

). (3.6.2)

where a0, . . . , am−1, b0, . . . , bn ∈ Z.

PROOF. Let us consider the graded ring grF associated with the filtration(Fr)r∈Z of the ring of σ-ε∗-operators F . Let x1, . . . , xm, y1, . . . , yn, yn+1, . . . , y2n

be the canonical images of the elements α1, . . . , αm, β1, . . . , βn, β−11 , . . . , β−1

n ,respectively, in the ring grF , so that xi = αi + F0 ∈ gr1F = F1/F0 yj =βj +F0 ∈ gr1F , and yn+j = β−1

j +F0 ∈ gr1F (1 ≤ i ≤ m, 1 ≤ j ≤ n). It is easyto see that x1, . . . , xm, y1, . . . , y2n generate the ring grF over R, these elementscommute with each other and xia = axi, yja = βj(a)yj , yn+ja = β−1

j (a)yn+j

for i = 1, . . . , m; j = 1, . . . , n. In the rest of the proof the ring grF will be alsodenoted by Rx1, . . . , xm, y1, . . . , y2n. Note that a homogeneous componentgrsF (s ∈ N) is the R-module generated by the set of all power productsxk1

1 . . . xkmm yl1

i1. . . yln

insuch that

∑mµ=1 kµ +

∑nν=1 lν = s (l1, . . . , ln are distinct

integers from the set 1, . . . , 2n such that lp − lq = n for any p = 1, . . . , n; q =1, . . . , n).

Let gr M =⊕

r∈Z grrM be the graded grF-module associated with the ex-cellent filtration (Mr)r∈Z (so that grrM = Mr/Mr−1 for any r ∈ Z). Repeatingthe arguments of the proof of Theorem 3.2.3, one can obtain that gr M is afinitely generated grF-module. Since lR(Mr) =

∑s≤r lR(grsM), the theorem

will be proved if one can prove the existence of a numerical polynomial g(t) inone variable t such that

(a) g(s) = lR(grsM) for all sufficiently large s ∈ Z;(b) deg g(t) ≤ m + n − 1 and the polynomial g(t) can be represented as

g(t) =m−1∑i=0

ai

(t + i

i

)+

n−1∑j=0

2j+1bj

(t + m + j

m + j

)where a0, . . . , am−1, b0, . . . , bn−1 ∈ Z.

(Indeed, if such a polynomial g(t) exists, then Corollary 1.4.7 implies theexistence of the polynomial Φ(t) with the desired properties.) We shall prove


the existence of the polynomial g(t) with the properties (a) and (b) by inductionon n = Card ε.

If n = 0, then gr M is a finitely generated graded module over the Artinianσ-ring R. In this case, the existence of the polynomial g(t) with the desiredproperties is stated by Theorem 3.2.1.

Now, suppose that n > 0. Let us set M = grM and consider the exactsequence of finitely generated Rx1, . . . , xm, y1, . . . , y2n-modules

0 → Ker θn → M θn−→ ynM → 0 (3.6.3)

where θn is the mapping of multiplication by yn, that is θn(z) = ynzfor any z ∈ M. It is easy to see that θn is an additive mapping andθn(az) = αn(a)θn(z) for any z ∈ M, a ∈ R. Repeating the arguments ofthe proof of Theorem 3.5.2 we obtain that there exist numerical polynomi-

als g1(t) =m−1∑i=0

a′i

(t + i

i

)+

n−1∑j=0

2jb′j

(t + m + j

m + j

)and g2(t) =

m−1∑i=0

a′′i

(t + i

i

)+

n−1∑j=0

2jb′′j

(t + m + j

m + j

)(a′

i, a′′i , b′j , b

′′j ∈ Z for i = 0, . . . , m−1; j = 0, . . . , n−1) such

that g1(s) = lR(Ker θn

⋂grsM) and g2(s) = lR(θn(grsM)) for all sufficiently

large s ∈ Z. Furthermore, as in the proof of Theorem 3.5.2, we obtain that there

exist a numerical polynomial g3(t) =m−1∑i=0

ci

(t + i

i

)+

n−2∑j=0

2j+1dj

(t + m + j

m + j

)(ci, dj ∈ Z for i = 0, . . . , m − 1; j = 0, . . . , n − 2) such that g2(t) =g1(t)−g3(t). (The polynomial g3(t) is characterized by the property that g3(s) =lR ((Kerθn|grsM )/y2ngrs−1M)) for all sufficiently large s ∈ Z.) Since the func-tion lR(·) is additive in the class of all finitely generated R-modules, the exact se-quence (3.6.3) implies that the polynomial g(t) = g1(t)+g2(t) = 2g1(t)−g3(t) =m−1∑i=0

(2a′i−ci)

(t + i

i

)+

n−2∑j=0

2j+1(b′j −dj)(

t + m + j

m + j

)+2nb′n−1

(t + m + n − 1

m + n − 1

)satisfies conditions (a) and (b). This completes the proof of the theorem.

Definition 3.6.5 Let R be an Artinian σ-ε∗-ring with a basic set σ ∪ ε whereσ = α1, . . . , αm and ε = β1, . . . , βn are the sets of injective endomor-phisms and automorphisms of R, respectively. Let M be a σ∗-R-module andlet (Mr)r∈Z be an excellent filtration of a M . Then the polynomial Φ(t) whoseexistence is established by Theorem 3.6.4 is called the σ-ε∗-dimension or char-acteristic polynomial of the module M associated with the excellent filtration(Mr)r∈Z.

Example 3.6.6 Let K be a σ-ε∗-field with a basic set σ∪ε where σ = α1, . . . ,αn and ε = β1, . . . , βn are the sets of injective endomorphisms and auto-morphisms of the field K, respectively. Then the ring of σ-ε∗-operators F overK can be considered as a σ-ε∗-K-module equipped with the excellent filtration


(Fr)r∈Z considered above. Let ΦF (t) be the σ-ε∗-dimension polynomial of Fassociated with this filtration. Then

ΦF (r) = dimR(Fr) = Card αk11 . . . αkm

m βl1 . . . βlnn ∈ Θ|

m∑i=1

ki +n∑

j=1

|lj | ≤ r

for all sufficiently large r ∈ Z. In other words, for all sufficiently larger ∈ Z, ΦF (r) is equal to the number of (m + n)-tuples in the set A =

(k1, . . . , km, l1, . . . , ln) ∈ Nm × Zn|m∑

i=1

ki +n∑

j=1

|lj | ≤ r. We shall find this

number by decomposing A into the disjoint union of r + 1 sets Aq (0 ≤ q ≤ r)

such that Aq = (k1, . . . , km, l1, . . . , ln) ∈ Nm×Zn |m∑

i=1

ki = q,n∑

j=1

|lj | ≤ r−q.

By Proposition 1.4.9, CardAq =(

q + m − 1m − 1

) n∑i=0

(−1)n−i2i

(n

i

)(r − q + i

i

)hence CardA=

r∑q=0

(q + m − 1

m − 1

) n∑i=0

(−1)n−i2i

(n

i

)(r − q + i

i

)=

n∑i=0

(−1)n−i2i

(ni

) r∑q=0

(q + m − 1

m − 1

)(r − q + i

i

).

The last expression can be essentially simplified if one notices thatr∑

q=0

(q + m − 1

m − 1

)(r − q + i

i

)=

(m + r + i

m + i

).

Indeed, applying formulas (1.4.13) and (1.4.16), we obtain that(

m + r + i

m + i

)=

Card (x1, . . . , xm+i ∈ Nm+i|m+i∑k=1

xk ≤ r =r∑

q=0

Card (x1, . . . , xm+i ∈

Nm+i |m∑

k=1

xk = q andm+i∑

k=m+1

xk ≤ r − q =r∑

q=0

(q + m − 1

m − 1

)(r − q + i

i

).

It follows that the σ-ε∗-dimension polynomial ΦF (t) can be expressed in theform

ΦF (t) =n∑

i=0

(−1)n−i2i

(n

i

)(t + m + i

m + i

). (3.6.4)

Let R be a σ-ε∗-ring with a basic set σ ∪ ε where σ = α1, . . . , αn andε = β1, . . . , βn are the sets of injective endomorphisms and automorphisms,respectively. Let F be the ring of σ-ε∗-operators over R, F [x] the ring of poly-nomials in one variable x over F , and F ′ the subring

⊕r∈N Fr

⊗R Rxr of the

ring F ⊗R R[x]. Furthermore, for any finitely generated σ-ε∗-R-module M with

a filtration (Mr)r∈Z, let M ′ denote the left F ′-module⊕

r∈N Mr

⊗R Rxr.


The proofs of the following two lemmas and Theorem 3.6.9 are similar tothe proofs of the corresponding statements for difference and inversive differencemodules (see Lemmas 3.2.6, 3.2.7, 3.5.5, 3.5.6 and Theorems 3.2.8, 3.5.7).

Lemma 3.6.7 Let R be a σ-ε∗-ring and let (Mr)r∈Z be a filtration of a σ-ε∗-R-module M such that all its components Mr (r ∈ Z) are finitely generatedR-modules. Then the following conditions are equivalent:

(i) The filtration (Mr)r∈Z is excellent;(ii) M ′ is a finitely generated F ′-module.

Lemma 3.6.8 Let R be a Noetherian σ-ε∗-ring such that elements of the set σare automorphisms of R. Then the ring F ′ is left Noetherian.

Theorem 3.6.9 Let R be a Noetherian σ-ε∗-ring such that elements of the setσ are automorphisms of R. Let ρ : N −→ M be an injective homomorphism offiltered σ-ε∗-R-modules and let the filtration of the module M be excellent. Thenthe filtration of N is also excellent.

Let R be a σ-ε∗-field, M a finitely generated σ-ε∗-R-module, and (Mr)r∈Z

and (M ′r)r∈Z two excellent filtrations of M . As in the case of difference and

inversive difference modules, one can easily see that there exists q ∈ N suchthat M ′

r ⊆ Mr+q and Mr ⊆ M ′r+q for all suffisiently large r ∈ Z. Therefore, if

Φ(t) and Φ1(t) are σ-ε∗-dimension polynomials associated with the filtrations(Mr)r∈Z and (M ′

r)r∈Z, respectively, then Φ(r) ≤ Φ1(r+q) and Φ1(r) ≤ Φ(r+q)for all sufficiently large r ∈ Z. We arrive at the following statement.

Theorem 3.6.10 Let R be a σ-ε∗-field, M a finitely generated σ-ε∗-R-module,

and Φ(t) =m−1∑i=0

ai

(t + i

i

)+

n∑j=0

2jbj

(t + m + j

m + j

)the σ-ε∗-dimension polynomial

of M associated with some excellent filtration of this module (ai, bj ∈ Z for

0 ≤ i ≤ m − 1, 0 ≤ j ≤ n). Then the integers bn =∆m+nΦ(t)

2n, d = deg Φ(t),

and

cd =

⎧⎪⎨⎪⎩∆dΦ(t)2d−m

if d ≥ m

∆dΦ(t) if d < m

do not depend on the choice of the excellent filtration the polynomial Φ(t) isassociated with.

Definition 3.6.11 Let R be a σ-ε∗-field, M a finitely generated σ-ε∗-R-module,and Φ(t) the σ-ε∗-dimension polynomial of M associated with an excellent fil-

tration of M . Then the integers∆m+nΦ(t)

2n, d = deg Φ(t), and cd considered in

Theorem 3.6.10 are called σ-ε∗-dimension, σ-ε∗-type, and typical σ-ε∗-dimensionof the module M . These numbers are denoted by σε∗dimRM , σε∗typeRM , andtσε∗dimRM , respectively.


Exercise 3.6.12 Let R be a σ-ε∗-field with a basic set σ∪ε where σ = α1, α2and ε = β. Let M be a σ-ε∗-R-module with one generator z and the definingrelation α1βz = α2z. (One can treat M as a σ-ε∗-R-module E/F(α1β − α2)ewhere F is the ring of σ-ε∗-operators over R and E is a free left F-modulewith a single free generator e.) Find the σ-ε∗-dimension polynomial of the mod-ule M associated with the excellent filtration (Frz)r∈Z. Determine σε∗dimRM ,σε∗typeRM , and tσε∗dimRM .

Let R be a σ-ε∗-field with a basic set σ ∪ ε, where σ = α1, . . . , αm andε = β1, . . . , βn, and let Θ be the commutative semigroup of all power productsof the form (3.6.1). Elements z1, . . . , zk of a σ-ε∗-R-module M are said to be σ-ε∗-linearly independent over the field R if and only if the family θzi|θ ∈ Θ, 1 ≤i ≤ k is linearly independent over R. If this family is linearly dependent overR, we say that z1, . . . , zk are σ-ε∗-linearly dependent over the field R. (It is easyto see that if F is the ring of σ-ε∗-operators over the σ-ε∗-field R, then elementsz1, . . . , zk are σ-ε∗-linearly independent over R if and only if they are linearlyindependent over F .)

The following two statements can be proven in the same way as the corre-sponding results on difference modules (see Theorems 3.2.11 and 3.2.12).

Theorem 3.6.13 Let R be a σ-ε∗-field such that elements of the set σ areautomorphisms of R, and let 0 −→ N

i−→ Mj−→ P −→ 0 be an exact sequence

of finitely generated σ-ε∗-R-modules ( i and j are σ-ε∗-homomorphisms). Thenσε∗dimRM = σε∗dimRN + σε∗dimRP .

Theorem 3.6.14 Let R be a σ-ε∗-field such that elements of the set σ areautomorphisms of R, and let M be a finitely generated σ-ε∗-R-module. Thenσε∗dimRM is equal to the maximal number of elements of M that are σ-ε∗-linearly independent over the field R.

Exercises 3.6.15 Let K be a σ-ε∗-field with a basic set δ = σ ∪ ε whereσ = α1, . . . , αm and ε = β1, . . . , βn are the sets of injective endomor-phisms and automorphisms of the field K, respectively. Let T be the freecommutative semigroup generated by σ, Γ the free commutative group gen-erated by ε, and Θ the free commutative semigroup of elements of the formθ = αk1

1 . . . αknn βl1

1 . . . βlnn where ki ∈ N, lj ∈ Z (1 ≤ i ≤ m, 1 ≤ j ≤ n). As in

the beginning of this section, for any such an element θ, we set ordσθ =∑m

i=1 ki,ordεθ =

∑nj=1 |lj | and ord θ = ordσθ + ordεθ. Furthermore, for any r, s ∈ N,

let T (r) = θ ∈ Θ | ordσθ ≤ r, Γ(s) = θ ∈ Θ | ordεθ ≤ s and Θ(r, s) = θ ∈Θ | ordσθ ≤ r, ordεθ ≤ s.

Let D, E , and F denote the rings of σ-, ε-, and σ-ε-operators over K, re-spectively, and for any A =

∑θ∈Θ aθθ ∈ F , let ordσA = maxordσθ | aθ = 0

and ordεA = maxordεθ | aθ = 0. Furthermore, for any r ∈ Z, let F (σ)r = A ∈

F | ordσA ≤ r and F (ε)r = A ∈ F | ordεA ≤ r if r ≥ 0, and F (σ)

r = F (ε)r = 0

if r < 0.Let M be a σ-ε-K-module, that is, a left F-module (clearly, M is also a

module over the rings D and E). We say that M is σ- (ε-) filtered if M is a filtered


module over the ring F treated as a filtered ring with the filtration (F (σ)r )r∈Z

(respectively, with the filtration (F (ε)r )r∈Z). The corresponding filtration of M

(called, respectively, a σ- or an ε-filtration) is supposed to be exhaustive anseparated. If (M (σ)

r )r∈Z is a σ-filtration of M such that every component M(σ)r

is a finitely generated E-module and there exists r0 ∈ Z such that F (σ)s M

(σ)r =

M(σ)r+s for all r ≥ r0, s ≥ 0, then this σ-filtration is called excellent. Similarly, an

ε-filtration (M (ε)r )r∈Z is said to be excellent if every component M

(ε)r is a finitely

generated D-module and there exists r0 ∈ Z such that F (ε)s M

(ε)r = M

(ε)r+s for all

r ≥ r0, s ≥ 0.

1. Prove that if a σ-ε-K-module M is finitely generated as a left F-module

by elements x1, . . . , xk, then (k∑

j=1

F (σ)r xj)j∈Z and (

k∑j=1

F (ε)r xj)j∈Z are excellent

σ- and ε-filtration of M , respectively.

2. Prove that if (M (σ)r )r∈Z is an excellent σ-filtration of a σ-ε-K-module M ,

then there exists a numerical polynomial φσ(t) with the following properties:

(i) φσ(r) = ε∗-dimKM(σ)r for all sufficiently large r ∈ N.

(ii) deg φσ ≤ m, so the polynomial φσ(t) can be written in the form φσ(t) =m∑

i=0

aσi

(t + i

i

)with integer coefficients aσi.

(iii) aσm = σε∗dimKM .

3. Prove that if (M (ε)r )r∈Z is an excellent ε-filtration of a σ-ε-K-module M ,

then there exists a numerical polynomial φε(t) with the following properties:

(i) φε(r) = σ-dimKM(ε)r for all sufficiently large r ∈ N.

(ii) deg φε ≤ n, and the polynomial φε(t) can be written in the form φε(t) =n∑

j=0

2jaεj

(t + j

j

)with integer coefficients aεi.

(iii) aεn = σε∗dimKM .

4. Generalize the Grobner basis technique developed in section 3.3 and thetechnique of characteristic sets considered in section 3.5 to the case of freemodules over the ring of σ-ε-operators F and any fixed partition of the set ofoperators σ ∪ ε. Formulate and prove analogs of Theorems 3.3.16 and 3.5.38 forthis case.

Let A be a commutative ring, M an A-module and U a family of A-submodules of M . Furthermore, let BU denote the set of all pairs (N,N ′) ∈ U×Usuch that N ⊇ N ′, and let Z be the augmented set of integers, that is the setobtaining by adjoining a new symbol ∞ to the set Z (Z is considered as alinearly ordered set whose order < is the extension of the natural ordering of Zsuch that a < ∞ for any a ∈ Z).


Proposition 3.6.16 With the above notation, there exists a unique mappingµU : BU → Z with the following properties:

(i) µU (N,N ′) ≥ −1 for every pair (N,N ′) ∈ BU ;(ii) If d ∈ N, then µU (N,N ′) ≥ d if and only if N = N ′ and there exists an

infinite chain N = N0 ⊇ N1 ⊇ · · · ⊇ N ′ such that µU (Ni−1, Ni) ≥ d − 1 for alli = 1, 2, . . . .

PROOF. Let us define the desired mapping µU as follows. If N ∈ U , we setµU (N,N) = −1. If (N,N ′) ∈ BU , N = N ′, and condition (ii) holds for all d =∈ N, we set µU (N,N ′) = ∞. Finally, if a pair (N,N ′) ∈ BU satisfies condition(ii) for some d ∈ N, we define µU (N,N ′) as the greatest of such integers d. Itis easy to see that the mapping µ is well-defined, it satisfies conditions (i) and(ii), and it is iniquely determined by these conditions.

Exercise 3.6.17 Let (N,N ′) ∈ BU and let the mapping µU : BU → Z satisfyconditions (i) and (ii) of Proposition 3.6.16. Prove the following properties ofthis mapping.

(a) µU (N,N ′) = −1 if and only if N = N ′.(b) µU (N,N ′) = 0 if and only if N N ′ and any chain N = N0 ⊇ N1 ⊇

· · · ⊇ N ′ (Ni ∈ U for i = 0, 1, 2, . . . ) stabilizes at some stage (that is, thereexists m ∈ N such that Nm = Nm+1 = · · · = N ′).

(c) µU (N,N ′) ≥ 1 if and only if there exists an infinite strictly descendingchain N = N0 N1 · · · N ′ with Ni ∈ U (i = 0, 1, 2, . . . ).

Definition 3.6.18 With the above notation, the least upper bound of the setµU (N,N ′)|(N,N ′) ∈ BU is called the type of the A-module M over the familyof its submodules U .

Definition 3.6.19 Let A be a ring, M an A-module, and U a family ofA-submodules of M . Then the least upper bound of the lengths of all chainsN0 N1 · · · Np, where Ni ∈ U and µU (Ni−1, Ni) = typeUM (i = 0, . . . , pand the number p is considered as the length of the chain), is called the dimen-sion of the A-module M over the family U .

The type and dimension of an A-module M over a family of its A-submodulesU are denoted by typeUM and dimUM , respectively.

Exercise 3.6.20 Let V be a finite-dimensional vector space over a field K andlet U be the family of all vector K-subspaces of V . Show that typeUV = 0 anddimUV = dimV (where dimV denotes the dimension of the vector K-space Vin the usual sense).

Let M be a module over a ring A and let U be a family of A-submodulesof M . It is easy to see that if typeUM < ∞, then dimUM ≥ 1. At the sametime, if typeUM = ∞, then dimUM can be equal to zero. One can obtainthe corresponding example by taking an infinite direct sum of vector spaces ofdimensions 1, 2, . . . ; we leave the details to the reader as an exercise.


Exercise 3.6.21 Let K be a field and V an infinite-dimensional vector K-spacewith a countable basis. Show that if U is the family of all vector K-subspacesof V , then typeUV = ∞ and dimUV = ∞.

[Hint: Represent a countable basis B of the vector space V as a disjointcountable union

⋃∞i=1 Bi of countable sets. Then consider vector subspaces W

(1)k

generated by the sets B \⋃ki=1 Bi and show that µU (V, 0) = ∞.]

Exercise 3.6.22 Let F be the ring of functions of one real variable t that aredefined and continuous on the whole real line R. It is easy to see that for anytwo real numbers a and b, a < b, the set O(a, b) = f(t) ∈ F |f(t) = 0 for anyt ∈ R \ (a, b) is an ideal of the ring F . Let U be the family of all such ideals.Find typeUF and dimUF .

Exercise 3.6.23 Let A be a commutative ring of finite Krull dimension d and letU be the family of all prime ideals of A. Show that typeUA = 0 and dimUA = d.

Theorem 3.6.24 Let R be a σ-ε∗-field whose basic set is a union of a setσ = α1, . . . , αm of injective endomorphisms of the field R and a set ε =β1, . . . , βn of automorphisms of R. Let M be a finitely generated σ-ε∗-R-module and let U be the family of all σ-ε∗-R-submodules of M . Then:

(i) If σε∗dimRM > 0, then typeUM = m + n and dimUM = σε∗dimRM ;(ii) If σε∗dimRM = 0, then typeUM < m + n.

PROOF. As before, let Θ and F denote the commmutative semigroup ofall power products of the form (3.6.1) and the ring of σ-ε∗-operators over R,respectively. Let (Mr)r∈Z be an excellent filtration of the σ-ε∗-R-module Mand let N,L ∈ U . By Theorem 3.6.9, (N

⋂Mr)r∈Z and (L

⋂Mr)r∈Z are excel-

lent filtrations of the σ-ε∗-R-modules N and L, respectively, so one can considerthe corresponding σ-ε∗-dimension polynomials ΦN (t) and ΦL(t) associated withthese filtrations. It is easy to see that the inclusion N ⊆ L implies the inequal-ity ΦN (t) ≤ ΦL(t) (where ≤ denotes the natural order on the set of numericalpolynomials in one variable: f(t) ≤ g(t) if and only if f(r) ≤ g(r) for all suf-ficiently large r ∈ Z). Furthermore, L = N if and only if ΦN (t) = ΦL(t).(Indeed, if ΦN (t) = ΦL(t), but L N , then there exists r0 ∈ Z such thatL⋂

Mr N⋂

Mr for all r > r0. In this case one would have ΦL(t) < ΦN (t)that contradicts our assumption.)

Let us prove that if L,N ∈ BU = (P,Q) ∈ U×U |P ⊇ Q and µU (N,L) ≥ d(d ∈ Z, d ≥ −1), then deg (ΦN (t) − ΦL(t)) ≥ d.

We proceed by induction on d. Since deg (ΦN (t) − ΦL(t)) ≥ −1 for any pair(P,Q) ∈ BU and deg (ΦN (t) − ΦL(t)) ≥ 0 if N L (as we have noticed, in thiscase ΦN (t) > ΦL(t)), our statement is true for d = −1 and d = 0.

Now, let d > 0 and let for any i ∈ N, i < d, the inequality µU (N,L) ≥ i( (N,L) ∈ BU ) imply that deg (ΦN (t) − ΦL(t)) ≥ i. Suppose that (N,L) ∈ BU

and µU (N,L) ≥ d. Then there exists an infinite strictly descending chain

N = N0 N1 · · · L


of σ-ε∗-R-submodules of M such that µU (Ni−1, Ni) ≥ d − 1 for i = 1, 2, . . . . Ifdeg

(ΦNi−1(t) − ΦNi

(t)) ≥ d for some i ≥ 1, then deg (ΦN (t) − ΦL(t)) ≥ d and

our statement is proved.Suppose that deg

(ΦNi−1(t) − ΦNi

(t))

= d − 1 for all i = 1, 2, . . . . Thenevery polynomial ΦNi−1(t) − ΦNi

(t) (i ∈ N, i ≥ 1) can be written in the

form ΦNi−1(t) − ΦNi(t) =

d−1∑k=0

cik

(t + k

k

)where ci0, . . . , ci,d−1 ∈ Z, ci,d−1 > 0

(see Theorem 3.6.4). In this case, ΦN (t) − ΦNi(t) =

d−1∑k=0

c′ik

(t + k

k

)where

c′i0, . . . , c′i,d−1 ∈ Z and c′i,d−1 =

i∑ν=1

cν,d−1. Therefore, c′1,d−1 < c′2,d−1 < . . .

whence deg (ΦN (t) − ΦL(t)) ≥ d for any pair (N,L) ∈ B with µU (N,L) ≥ d.The last statement, together with the fact that deg (ΦN (t) − ΦL(t)) ≤ m+n

for any (N,L) ∈ B, implies that µU (N,L) ≤ m + n for all pairs (N,L) ∈ B. Itfollows that

typeUM ≤ m + n . (3.6.5)

If σε∗dimRM = q > 0, then Theorem 3.6.14 shows that M contains qelements z1, . . . , zq that are σ-ε∗-linearly independent over the ring of σ-ε∗-operators F . Clearly, if U ′ is the family of all F -submodules of the left F -moduleFz1, then typeU ′Fz1 ≤ typeUM . Thus, in order to prove the first statementof the theorem, it suffices to show that typeU ′Fz1 ≥ m + n. Let U ′′ de-note the family of all F-submodules of Fz1 that can be represented as finitesums of F-modules of the form Fαk1

1 . . . αkmm (β1 − 1)l1 . . . (βn − 1)lnz1 where

k1, . . . , km, l1, . . . , ln ∈ N. We are going to use induction on m + n to show thatµU ′′(Fz1, 0) ≥ m + n. If m + n = 0, inequality is trivial. Let m + n > 0 that isat least one of the numbers m, n is positive. Suppose, first, that m > 0.

Since the element z1 is linearly independent over F , we have the strictlydescending chain

Fz1 Fαmz1 Fα2mz1 · · · 0

of elements of U ′′. Let us show that µU ′′(Fαi−1m z1,Fαi

mz1) ≥ m + n − 1 for alli = 1, 2, . . . . Let L = Fαi−1

m z1/Fαimz1 and let y be the image of the element

αi−1m z1 under the natural epimorphism Fαi−1

m z1 → L. Then αmy = y andαmθ′y = θ′αmy for any element θ′ ∈ Θ′, where

Θ′ = m−1∏ν=1

n∏µ=1

αuνν (βµ − 1)vµ |u1, . . . , um−1, v1, . . . , vn ∈ N.

Setting σ′ = α1, . . . , αm−1, treating R as a σ′-ε∗-field, and denoting the cor-responding ring of σ′-ε∗-operators by F ′, we obtain that L = F ′y and the ele-ment y is linearly independent over F ′. Furthermore, µU ′′(Fαi−1

m z1,Fαimz1) =

µU ′′1(L, 0) where U ′′

1 is the family of all F ′-submodules of L that can be repre-sented as finite sums of the F ′-modules of the form F ′θ′y (θ′ ∈ Θ′). By the induc-tion hypothesis, µU ′′

1(L, 0) ≥ m+n−1 whence µU ′′(Fαi−1

m z1,Fαimz1) ≥ m+n−1

(i = 1, 2, . . . ). Thus, µU ′′(Fz1, 0) ≥ m + n.


Now, let m + n > 0 and n > 0. Let us consider the strictly descending chainof F-modules

Fz1 F(βn − 1)z1 F(βn − 1)2z1 · · · 0.

As in the case m > 0, for every i = 1, 2, . . . , one can consider the σ-ε∗1-R-moduleF(βn − 1)i−1z1/F(βn − 1)iz1 (where ε1 = β1, . . . , βn−1) and obtain that

µU ′′(F(βn − 1)i−1z1,F(βn − 1)iz1) ≥ m + n − 1

whence µU ′′(Fz1, 0) ≥ m + n. Thus,

typeUM ≥ typeU ′Fz1 ≥ typeU ′′Fz1 ≥ m + n.

Combining these inequalities with (3.6.5), we obtain that

typeUM = m + n.

Our proof of the last equality shows that for every ν = 1, . . . , q, typeU ′νFzν =

m + n where U ′ν denotes the family of all F-submodules of the F-module Fzν .

Therefore, µU

(∑kν=1 Fzν ,

∑k−1ν=1 Fzν

)= m + n = typeUM for k = 1, . . . , q

(if k = 0, the sum is 0), so the chain

M =q∑

ν=1

Fzν q−1∑ν=1

Fzν · · · Fz1 0

shows thatdimUM ≥ σε∗dimRM . (3.6.6)

Let N0 N1 · · · Np be a chain of σ-ε∗-R-submodules of M such thatµU (Nk−1, Nk) = typeUM = m + n for k = 1, . . . , p. The arguments used at thebeginning of the proof show that the degree of each polynomial ΦNk−1(t)−ΦNk

(t)(1 ≤ k ≤ p) is equal to m + n, so that such a polynomial can be written as

ΦNk−1(t) − ΦNk(t) =

m−1∑i=0

aki

(t + i

i

)+

n∑j=0

2jbkj

(t + m + j

m + j

)where aki, bkj ∈ Z(0 ≤ i ≤ m − 1, 0 ≤ j ≤ n) and bkn = σε∗dimRNi−1 −σε∗dimRNi ≥ 1. It follows that

ΦN0(t) − ΦNp(t) =

p∑i=1

(ΦNi−1(t) − ΦNi(t)

)=

m−1∑i=0

a′i

(t + i

i

)+

n∑j=0

2jb′j

(t + m + j

m + j

)

where a′i, b

′j ∈ Z(0 ≤ i ≤ m − 1, 0 ≤ j ≤ n) and b′n =

∑pi=1 bkn ≥ p.

On the other hand,

ΦN0(t) − ΦNp(t) ≤ ΦN0(t) ≤ ΦM (t) =

m−1∑i=0

ai

(t + i

i

)+

n∑j=0

2jbj

(t + m + j

m + j

)


where bn = σε∗dimRM. (We write f(t) ≤ g(t) for two numerical polynomialsf(t) and g(t) if f(r) ≤ g(r) for all sufficiently large r ∈ Z.)

It follows that bn ≥ b′n ≥ p whence σε∗dimRM ≥ dimUM . Combiningthe last inequality with (3.6.6) we obtain the desired equality dimUM =σε∗dimRM .

Now, let us prove the last statement of the theorem. If σε∗dimRM = 0, thenfor any pair (N,L) ∈ BU , we have the equality σε∗dimRN = σε∗dimRL = 0whence deg ΦN (t) < m+n, deg ΦL(t) < m+n, and deg (ΦN (t) − ΦL(t)) < m+n.As it has been shown, the last inequality implies the inequality µU (N,L) <m + n. Thus, if σε∗dimRM = 0, then typeUM < m + n.

Exercise 3.6.25 Let K be a G-field, F the ring of G-operators over K, M afinitely generated G-K-module, and U the set of all F-submodules of M . Provethe following statements:

(a) If δG(M) > 0, then typeUM = rank G and dimUM = δG(M).(b) If δG(M) = 0, then typeUM ≤ tG(M).

Chapter 4

Difference Field Extensions

4.1 Transformal Dependence. DifferenceTranscendental Bases and DifferenceTranscendental Degree

Let K be a difference field with a basic set σ = α1, . . . , αn and let T be thefree commutative semigroup generated by σ (this semigroup was introduced inSection 2.1). If L is a difference (σ-) field extension of K and A ⊆ L, then anelement v ∈ L is said to be transformally dependent or σ-algebraically dependenton a set A over K if v is σ-algebraic over the field K〈A〉. Otherwise, v is saidto be transformally (or σ-algebraically) independent on A over K. The fact thatan element v ∈ L is σ-algebraically dependent on a set A over K will be writtenas v ≺K A (if this is not the case, we write v ⊀K A).

Exercises 4.1.1 Let K be a difference field with a basic set σ and L a σ-fieldextension of K.

1. Prove that an element v ∈ L is σ-algebraically dependent on a set A ⊆ L ifand only if there exists a finite family η1, . . . , ηs ⊆ A such that v is σ-algebraicover K〈η1, . . . , ηs〉.

2. Prove that a set S ⊆ L is σ-algebraically dependent over K if and only ifthere exists an element s ∈ S such that s ≺K S \ s.

Theorem 4.1.2 Let K be a difference field with a basic set σ = α1, . . . , αn,let L be a σ-field extension of K and let η and ζ be two elements in L. Then

(i) If ζ is σ-algebraic over K〈η〉 and η is σ-algebraic over K, then ζ isσ-algebraic over K.

(ii) If ζ is σ-algebraic over K〈η〉 and η is σ-transcendental over K〈ζ〉, then ζis σ-algebraic over K.(iii) The set of all elements of L that are σ-algebraic over K is an intermediate

σ-field of L/K.

245

246 CHAPTER 4. DIFFERENCE FIELD EXTENSIONS

(iv) If X ⊆ L and every element of X is σ-algebraic over K, then K〈X〉 is aσ-algebraic extension of K.

(v) Let M be a σ-field extension of L. Then M is σ-algebraic over K if andonly if M is σ-algebraic over L and L is σ-algebraic over K.(vi) The relation ≺K is the dependence relation from L to the power set of L.

In other words, ≺K satisfies conditions (i) - (iv) of Definition 1.1.4.

PROOF. (i) Let us consider a ring of σ-polynomials in one σ-indeterminatey over L and let us fix an orderly ranking < of the set of terms Ty = τy | τ ∈ T.(The concepts of ranking and orderly ranking were introduced at the beginningof Section 2.4). Since η is σ-algebraic over K, there exists a σ-polynomial f ∈Ky such that f(ζ) = 0. Let u = τ1y be the leader of f and let r1 = ord τ1.Then τ1η ∈ K(τη | τy < τ1y) hence τ ′τ1η ∈ K(τη | τy < τ ′τ1y) for everyτ ′ ∈ T . It follows that for any r ≥ r1, every element of the set T (r)η belongs tothe field K((T (r) \ T (r − r1))η), so that

K((T (r)η) = K( (T (r) \ T (r − r1))τ1η). (4.1.1)

(As in Section 3.1, for any r ∈ N, T (r) denotes the set of all power productsτ = αk1

1 . . . αknn ∈ T such that ord τ =

∑ni=1 ki ≤ r; also, for any set T ′ ⊆ T

and for any ξ ∈ L, T ′ξ denotes the set τ(ξ)|τ ∈ T ′.) Similarly, the factthat ζ is σ-algebraic over K〈η〉 implies that there exists τ2 ∈ T such thatτ2ζ ∈ K〈η〉(θζ | θy < τ2y).

It follows that there exists q ∈ N such that τ2ζ ∈ K(T (q)η ∪θζ | θy < τ2y),and therefore τ ′τ2ζ ∈ K(T (q + ord τ ′)η ∪ θζ | θy < τ ′τ2y) for every τ ′ ∈ T .

Let r2 = ord τ2. Then the last inclusion implies that for any s ≥ r2

K(T (s)ζ) ⊆ K((T (s + q)η ∪ (T (s) \ T (s − r2)τ2)ζ).

Taking into account (4.1.1) we obtain that

K(T (s)ζ) ⊆ K((T (s + q) \ T (s + q − r1)τ1)η ∪ (T (s) \ T (s − r2)τ2)ζ) (4.1.2)

for any s > maxr1−q, r2. By formula (1.4.16), the number of generators τζ in

the left-hand side of (4.1.2) is equal to(

s + n

n

)=

sn

n!+o(sn), while the number

of generators in the right-hand side of (4.1.2) is given by the alternating sum(s + q + n

n

)−(

s + q + n − r1

n

)+(

s + n

n

)−(

s − r2 + n

n

), which is a numerical

polynomial in s of degree less than n. Applying Proposition 1.6.28(iii), we obtainthat for sufficiently large s, the set T (s)ζ is algebraically dependent over K henceζ is σ-algebraic over K.

(ii) If ζ is σ-algebraic over K〈η〉, then there exists τ0 ∈ T such that τ0η isalgebraic over the field N = K〈η〉(τζ | τy < τ0y). It follows that there is apolynomial f(X) in one variable X with coefficients in N such that f(τ0ζ) = 0.

4.1. DIFFERENCE TRANSCENDENTAL BASES 247

If some coefficient of f contains a transform τη, then the last equality wouldimply that η is σ-algebraic over K〈ζ〉. Since this is not true, all coefficients ofthe polynomial f(X) belong to K(τζ | τy < τ0y). Therefore, τ0ζ is algebraicover the last field, hence ζ is σ-algebraic over K.(iii) Let L0 = u ∈ L |u is σ-algebraic over K. Let η, ζ ∈ L0 and let ξ denote

one of the elements η + ζ, ηζ, ηζ−1 (if ζ = 0), or αiη (1 ≤ i ≤ n). Sinceξ is σ-algebraic over K〈η, ζ〉 = K〈η〉〈ζ〉, ζ is σ-algebraic over K〈η〉 and η isσ-algebraic over K, part (i) implies that ξ is σ-algebraic over K, that is, ξ ∈ L0.(iv) If X ⊆ L and every element of X is σ-algebraic over K, then part (iii)

shows that K〈X〉 ⊆ L0, so that K〈X〉 is a σ-algebraic extension of K.(v) Clearly, if K ⊆ L ⊆ M is a sequence of σ-field extensions and M/K is

σ-algebraic, then M/L and L/K are also σ-algebraic. Conversely, suppose thatL is a σ-algebraic extension of K and M is a σ-algebraic extension of L. Ifu ∈ M , then there exist finitely many elements v1, . . . , vk ∈ L such that u isσ-algebraic over K〈v1, . . . , vk〉. Since for every i = 2, . . . , k, vi is σ-algebraic overK〈v1, . . . , vi−1〉, one can apply part (i) and obtain that u is σ-algebraic over K.(vi) The fact that the relation ≺K satisfies the first two conditions of Defini-

tion 1.1.4 is obvious. Let S, U and V be three subsets of L such that S ≺K Uand U ≺K V , that is, every element of S is σ-algebraic over K〈U〉 and everyelement of U is σ-algebraic over K〈V 〉. Applying parts (iv) and (v) to the se-quence of σ-field extensions K〈V 〉 ⊆ K〈V ∪ U〉 ⊆ K〈V ⋃

U⋃

S〉 we obtainthat the extension K〈V ⋃

U⋃

S〉/K〈V 〉 is σ-algebraic. Therefore every elementof S is σ-algebraic over K〈V 〉, so the relation ≺K satisfies condition (iii) ofDefinition 1.1.4. In order to prove that ≺K satisfies the last condition of thedefinition as well, suppose that a set S ⊆ L and elements s ∈ S, u ∈ L havethe property that u ≺K S, but u ⊀K S \ s. Then u is σ-algebraic over K〈S〉and σ-transcendent over K〈S \ s〉. It follows that there exists a σ-polynomialf in one σ-indeterminate y with coefficients in K〈S〉 such that f(u) = 0. Sincesome of the coefficients of f contain s (otherwise, u would be σ-algebraic overK〈S\s〉), the last equality implies that s is σ-algebraic over K〈(S\s)⋃u〉.This completes the proof of the theorem.

The last part of Theorem 4.1.2 and Proposition 1.1.6 imply the followingproperties of σ-algebraic dependence. We leave the proof of the correspondingstatements to the reader as an exercise.

Corollary 4.1.3 Let K be a difference field with a basic set σ, L a σ-fieldextension of K, S ⊆ L and u ∈ L. Then

(i) If the set S is σ-algebraically independent over K and u is σ-transcendentalover K〈S〉, then the set S

⋃u is σ-algebraically independent over K.(ii) Suppose that u ≺K S and S ≺K V for some set V ⊆ L (that is, every

element of S is σ-algebraic over K〈V 〉). Then u ≺K V

(iii) Let v1, . . . , vm ∈ L (m ≥ 2) and let u ≺K v1, . . . , vm, but u ⊀K

v1, . . . , vm−1. Then vm ≺K u, v1, . . . , vm−1


(iv) Let S′ ⊆ S and let a finite set s1, . . . , sk of elements of S be σ-algebraicallyindependent over K. Furthermore, suppose that si ≺K S′ for i = 1, . . . , k.Then there exist elements u1, . . . , uk ∈ S′ such that ui ≺K (S′ \ u1, . . . , uk) ∪s1, . . . , sk (1 ≤ i ≤ k).

Definition 4.1.4 Let K be a difference field with a basic set σ, L a σ-fieldextension of K, and A ⊆ L. A set B ⊆ A is called a basis for transformaltranscendence or a difference (or σ-) transcendence basis of A over K if B is amaximal σ-algebraically independent over K subset of A. In other words, a setB ⊆ A is a σ-transcendence basis of A over K if B is σ-algebraically independentover K and any subset of A containing B is σ-algebraically dependent over K. IfA = L, the set B is called a basis for transformal transcendence or a difference(or σ-) transcendence basis of L over K. (In this case we also say that B is aσ-transcendence basis of the extension L/K.)

By Zorn’s Lemma (see Proposition 1.1.3(v)), every subset of the σ-field L(we use the notation of the last definition) has a σ-transcendence basis overK. Furthermore, the last statement of Theorem 4.1.2 implies the following twoconsequences of Proposition 1.1.8.

Proposition 4.1.5 Let K be a difference field with a basic set σ, L a σ-fieldextension of K, and A ⊆ L. Then the following conditions on a set B ⊆ A areequivalent.

(i) B is a σ-transcendence basis of A over K.(ii) The set B is σ-algebraically independent over K and every element of A

is σ-algebraic over K〈B〉.(iii) B is a minimal subset of A with respect to the property A ≺K B. (In

other words, every element of A is σ-algebraic over K〈B〉, but not over K〈B0〉if B0 B.)

Proposition 4.1.6 Let K be a difference field with a basic set σ and L a σ-fieldextension of K. Then any two σ-transcendence bases of L over K have the samecardinality.

Definition 4.1.7 Let K be a difference field with a basic set σ, L a σ-fieldextension of K, and A ⊆ L. Then the difference (or σ-) transcendence degreeof A over K is the number of elements of any σ-transcendence basis of A overK, if this number is finite, or infinity in the contrary case.

With the notation of Definition 4.1.7, the σ-transcendence degree of A over Kis denoted by σ-trdegKA (in particular, if A does not have finite σ-transcendencebases over K, we write σ-trdegKA = ∞). If A = L, then the σ-transcendencedegree of L over K is also called a σ-transcendence degree of the extension L/K.

A difference (σ-) field extension L/K is said to be purely σ-transcendental ifthere is a σ-algebraically independent over K set B ⊆ L such that L = K〈B〉.


Theorem 4.1.8 Let K be a difference field with a basic set σ and L a σ-fieldextension of K.

(i) Any family of σ-generators of L over K contains a σ-transcendence basisof this difference field extension. If the σ-field K is inversive and L a σ∗-overfieldof K, then any system of σ∗-generators of L over K contains a σ-transcendencebasis of L over K.

(ii) If a set A ⊆ L is σ-algebraically independent over K, then it can beextended to a σ-transcendence basis B ⊇ A of L/K.(iii) Let η1, . . . , ηm ∈ L. Then σ-trdegKK〈η1, . . . , ηm〉 ≤ m. If K is inversive

and L a σ∗-overfield of K, then σ-trdegKK〈η1, . . . , ηm〉∗ ≤ m.(iii) Let η1, . . . , ηm and ζ1, . . . , ζs be two finite subsets of L such that

K〈η1, . . . , ηm〉 = K〈ζ1, . . . , ζs〉 (or K〈η1, . . . , ηm〉∗ = K〈ζ1, . . . , ζs〉∗ if K is in-versive and L a σ∗-overfield of K). If the set ζ1, . . . , ζs is σ-algebraicallyindependent over K, then s ≤ m.(iv) Let S = K〈y1, . . . , ys be the ring of difference (σ-) polynomials in σ-

indeterminates y1, . . . , ys over K. If k = s, then S cannot be a ring of σ-polynomials in k σ-indeterminates over K.

PROOF. Statement (ii) and the fact that every family of σ-generators ofL over K contains a σ-transcendence basis of L/K are direct consequences ofProposition 1.1.8(iii). If the σ-field K is inversive and L = K〈S〉∗, then the firstpart of statement (i) shows that S contains some σ-transcendence basis B ofthe σ-field extension K〈S〉/K. It is easy to see that B is also a σ-transcendencebasis of L/K. Indeed, if a ∈ L, then there exists τ ∈ T such that τ(a) ∈ K〈S〉.It follows that τ(a) is σ-algebraic over K〈B〉 hence a is also σ-algebraic over thelast σ-field.

The first part of statement (iii) follows from (i), since the system of σ-generators η1, . . . , ηm of K〈η1, . . . , ηm〉/K contains a σ-transcendence basisof this extension which consists of σ-trdegKK〈η1, . . . , ηm〉 elements. Similarly,the second part of (iii) immediately follows from (ii). The last statement of thetheorem is a direct consequence of Proposition 4.1.6.

The following theorem shows that the difference transcendence degree is additiveover a tower of fields.

Theorem 4.1.9 Let K be a difference field with a basic set σ, L a σ-field ex-tension of K, and M a σ-field extension of L. Let A be a σ-transcendence basisof the extension L/K and B a σ-transcendence basis of M/L. Then A∪B is aσ-transcendence basis of M/K and therefore,

σ-trdegKM = σ-trdegLM + σ-trdegKL.

PROOF. Since the set B is σ-algebraically independent over L, it is σ-algebraically independent over K〈A〉. Therefore, the set A

⋃B is σ-algebraically

independent over K. To complete the proof one has to show that if u ∈ M , thenu is σ-algebraic over K〈A⋃

B〉. Since B is a σ-transcendence basis of M/L,


there exists a non-zero difference (σ-) polynomial f(y) in one σ-indeterminate ywith coefficients in L〈B〉 such that f(u) = 0. If S denotes the set of coefficientsof f , then u is σ-algebraic over K〈S ⋃

B〉, so that u is σ-algebraically dependenton the set S

⋃B over K (as before, we write this as u ≺K S

⋃B). Since S ≺K A,

we have S⋃

B ≺K A⋃

B whence u ≺K A⋃

B (see Corollary 4.1.3(ii)). Thiscompletes the proof.

The following statement is an analogue of Proposition 1.6.31 for differencefield extensions. We leave its proof to the reader as an exercise.

Proposition 4.1.10 Let K be a difference field with a basic set σ and N aσ-field extension of K.

(i) If F and G are two intermediate σ-fields of N/K, then their composi-tum FG is a σ-subfield of N and σ-trdegF FG ≤ σ-trdegKG. Furthermore,σ-trdegKFG ≤ σ-trdegKF + σ-trdegKG.

(ii) Let K ⊆ L ⊆ M ⊆ N be a sequence of difference (σ-) field extensions andA ⊆ M \L. Then σ-trdegK〈A〉L〈A〉 ≤ σ-trdegKL. If either A is σ-algebraicallyindependent over L or every element of A is σ-algebraic over K, the last in-equality becomes an equality.

Exercises 4.1.11 1. Find an example of a purely σ-transcendental difference(σ-) field extension L/K with two σ-transcendental bases B and C such thatL = L〈B〉 but L〈C〉 is a proper σ-subfield of L.

2. Let K be a difference field with a basic set σ, M a σ-field extension of Kand L an intermediate σ-field of M/K. Furthermore, let S be a σ-algebraicallyindependent over L subset of M . Prove that the σ-field extension L〈S〉/K〈S〉is σ-algebraic if and only if L/K is σ-algebraic.

We conclude this section with an alternative description of the differencetranscendence degree of a finitely generated difference field extension. This de-scription is due to P. Evanovich [61].

Let K be a difference field with a basis set σ = α1, . . . , αn and let σk =α1, . . . , αk for k = 1, . . . , n − 1. If S is any set in a σ-overfield of K andσ′ ⊆ σ, then K〈S〉σ′ will denote the σ′-overfield of K generated by S. (Ifσ′ = αi1 , . . . , αik

, then K〈S〉σ′ coincides with the field K(αd1i1

. . . αdkik

(a) | a ∈S, d1, . . . , dk ∈ N).) In the case σ′ = σ we write K〈S〉 instead of K〈S〉σ. Asusual, if Σ is a subset of a ring L and φ an endomorphism of L, then φ(Σ)denotes the set φ(a) | a ∈ Σ. Furthermore, if α is an endomorphism and Φ isa set of endomorphisms of L , then αΦ = αφ |φ ∈ Φ and Φ(Σ) = φ(a) |φ ∈Φ, a ∈ Σ.

Let L = K〈S〉 be a σ-field extension of K generated by a finite set S

and for every k = 1, 2, . . . , let Lk = K〈⋃kj=0 αj

n(S)〉σn−1 and δk = σn−1-trdegLk−1Lk. (A). Applying Proposition 4.1.10, one can easily see that δk =σn−1-trdegαn(Lk−1)αn(Lk) ≥ σn−1-trdegLk

Lk+1 for k = 1, 2, . . . . (The fields Lk

and Lk+1 are obtained from the fields αn(Lk−1) and αn(Lk), respectively, byadjoining (in the sense of σn−1-field extensions) the same set [K \ αn(K)]∪ S.)


Theorem 4.1.12 With the above notation, σ-trdegKL = minδk | k = 1.2, . . . (= lim

k→∞δk).

PROOF. It follows from the definition of the non-increasing sequence δkthat there exists k0 ∈ N such that δk = minδk | k = 1.2, . . . for all k ≥ k0.Since αk0

n (S) is a set of σn−1-generators of the σn−1-field extension Lk0/Lk0−1,we can choose a set B ⊆ S such that αk0

n (B) is a σn−1-transcendence ba-sis of Lk0/Lk0−1. Then for every p ∈ N, αk0+p

n (B) is a σn−1-transcendencebasis of αp

n(Lk0)/αpn(Lk0−1) and so must contain a σn−1-transcendence ba-

sis of Lk0+p/Lk0+p−1. Since δk0 = δk0+p, we obtain that αk0+pn (B) itself is

a σn−1-transcendence basis of Lk0+p/Lk0+p−1. Denoting the free commuta-tive semigroup of all power products αi1

1 . . . αin−1n−1 (i1, . . . , in−1 ∈ N) by Tσn−1 ,

one can easily observe that the set∞⋃

p=0

αk0+pn Tσn−1(B) is algebraically inde-

pendent over K, so that αk0n (B) is σ-algebraically independent over K. Since

αk0n (B) can be extended to a σ-transcendence basis of L/K, we obtain that

minδk | k = 1.2, . . . = Cardαk0n (B) ≤ σ-trdegKL. (As it will follow from the

rest of the proof, αk0n (B) is a σ-transcendence basis of L/K.)

To prove the opposite inequality, let ∆ = σ-trdegKL and let c = σn−1-trdegKLk0−1. Then for any p ∈ N, σn−1-trdegKLk0+p = (p + 1)δk0 + c.

If Z is a σ-transcendence basis of L/K contained in S, then, for every p ∈

N, the setk0+p⋃i=0

αin(Z) is σn−1-algebraically independent over K. Since this set

contains (k0 + p + 1)∆ elements, (p + 1)δk0 + c ≥ (k0 + p + 1)∆. Letting p → ∞we see that δk0 ≥ ∆ whence δk0 = σ-trdegKL.

Corollary 4.1.13 Let K be a difference field with a basic set σ = α1, . . . , αnand let L = K〈S〉 be a σ-field extension of K generated by a finite set S. As

before, let Lk = K〈k⋃

j=0

αjn(S)〉σn−1 and δk = σn−1-trdegLk−1Lk for k = 1, 2, . . .

(as we have seen, δ1 ≥ δ2 ≥ . . . ). Then there exists a finite set Z ⊆ S suchthat Z is a σ-transcendence basis of L/K and if k0 = minδk | k = 1, 2, . . . (so that σn−1-trdegLk−1Lk = σ-trdegKL for all k ≥ k0), then αk

n(Z) is a σn−1-transcendence basis of Lk over Lk−1 for k = k0, k0 + 1, . . . .

PROOF. It follows from the prof of Theorem 4.1.12 that if we choose Z ⊆ Ssuch that αk

n(Z) is a σn−1-transcendence basis of Lk/Lk−1 for all k ≥ k0, thenαk0

n (Z) is a σn−1-transcendence basis of L/K. Clearly, CardZ = Cardαk0n (Z) =

σ-trdegKL and any relation of σ-algebraic dependence of elements of Z over Kyields a relation of σ-algebraic dependence of the corresponding elements ofαk0

n (Z) over K. Therefore, Z is a σ-transcendence basis of L over K.

Let K be a difference field with a basic set σ = α1, . . . , αn. In whatfollows, for any set σ′ = αi1 , . . . , αiq

⊆ σ, Tσ′ will denote the free commutative


semigroup generated by σ′. Furthermore, let σj = α1, . . . , αj (0 ≤ j ≤ n withσ0 = ∅, σn = σ) and for any set S ⊆ K and any integers in, . . . , it ∈ N (1 ≤ t ≤n), let

S∗(in, . . . , it) =in−1⋃j=0

αjnTσn−1(S)

⋃ in−1−1⋃j=0

αinn αj

n−1Tσn−2(S)⋃

. . .

it−1⋃j=0

αinn . . . α

it+1t+1 αj

tTσt−1(S) and S(in, . . . , it) = S∗(in, . . . , it + 1).

Let L be a σ-field extension of K generated by a finite set S. It followsfrom the proof of Theorem 4.1.12. that one can choose kn ∈ N such that σ-trdegKL = σn−1-trdegK〈S(in)〉σn−1

K〈S∗(in)〉σn−1 for any in ≥ kn.

Using the same argument we obtain that there exists kn−1 ∈ N such thatσ-trdegKL = σn−2-trdegK〈S(kn,in−1)〉σn−2

K〈S∗(kn, in−1)〉σn−2 for any in−1 ≥kn−1. Since the endomorphism αn is injective and both fields K〈S(in, in−1)〉σn−2

and K〈S∗(in, in−1)〉σn−2 can be obtained by adjoining the same set to theσn−2-fields αin−kn

n (K〈S(kn, in−1)〉σn−2) and αin−knn (K〈S∗(kn, in−1)〉σn−2), re-

spectively, we have

σ-trdegKL = σn−1-trdegK〈S(in)〉σn−1K〈S∗(in)〉σn−1

≤ σn−2-trdegK〈S∗(in,in−1)〉σn−2K〈S(in, in−1)〉σn−2

≤ σn−2-trdegαin−knn (K〈S∗(in,in−1)〉σn−2 )α

in−knn

×(K〈S(in, in−1)〉σn−2)≤ σ-trdegKL.

Thus, if in ≥ kn and in−1 ≥ kn−1, then

σ-trdegKL = σn−2-trdegK〈S∗(in,in−1)〉σn−2K〈S(in, in−1)〉σn−2 .

Furthermore, if Z ⊆ S is chosen so that αknn α

kn−1n−1 (Z) is a σn−2-transcendence

basis of K〈S(kn, kn−1)〉σn−2 over K〈S∗(kn, kn−1)〉σn−2 , then αinn α

in−1n−1 (Z) is a

σn−2-transcendence basis of K〈S(in, in−1)〉σn−2 over K〈S∗(in, in−1)〉σn−2 forany in ≥ kn and in−1 ≥ kn−1. Corollary 4.1.13 shows that αin

n (Z) is a σn−1-transcendence basis of K〈S(in)〉σn−1 over K〈S∗(in)〉σn−1 and thus, Z is a σ-transcendence basis of L over K.

Extending these arguments by induction we obtain that there exist positiveintegers k1, . . . , kn and a set Z ⊆ S such that

(i) Z is a σ-transcendence basis of L/K;(ii) if t ∈ 1, . . . , n and iν ≥ kν for ν = t, . . . , n, then αin

n . . . αitt (Z) is a

σt−1-transcendence basis of K〈S(in, . . . , it)〉σt−1 over K〈S∗(in, . . . , it)〉σt−1 .(As above, σt−1 = α1, . . . , αt−1.) In particular,

σt−1-trdegK〈S∗(in,...,it)〉σt−1K〈S(in, . . . , it)〉σt−1 = σ-trdegKL.


Definition 4.1.14 With the above notation, a σ-transcendence basis Z of L/Ksatisfying conditions (i) and (ii) is called a limit σ-transcendence basis of L overK (or a limit σ-transcendence basis of the σ-field extension L/K).

The following example shows that not every σ-transcendence basis is a limitone.

Example 4.1.15 Let us consider the field of complex numbers C as an ordinarydifference field with a basic set σ = α where α is the identity automorphismof C. Let X = x0, x1, . . . and Y = y0, y1, . . . be two denumerable setsof elements in some overfield of C such that the set X

⋃Y is algebraically

independent over C. Furthermore, let D = d1, d2, . . . be the set of elements inthe algebraic closure of the field C(X

⋃Y ) such that d2

j = xj +yj (j = 1, 2, . . . )and let L = C(X

⋃Y ∪D). Then the field L can be treated as a σ-field extension

of C if we define the action of α on the set of generators X⋃

Y⋃

D by α(xj) =xj+1, α(yj) = yj+1, and α(dj+1) = dj+2 (j = 0, 1, . . . ). Clearly, L = C〈S〉,where S = x0, y0, d1, and Z = x0, y0 ⊆ S is a σ-transcendence basis of L/C.However, Z is not a limit σ-transcendence basis of L over C. Indeed, for any

k ≥ 1, αk(Z) is not a transcendence basis of C(k⋃

i=0

αi(S))/C(k−1⋃i=0

αi(S)), since

the set α(x0), α(y0) is algebraically dependent over C(x0, y0, d1) (α(x0) +α(y0) = d2

1).

Exercise 4.1.16 Let K be a difference field with a basic set σ = α1, . . . , αnand let L = K〈S〉 be a σ-field extension of K generated by a finite set S. Let Zbe a limit σ-transcendence basis of L/K that satisfy condition (ii) (stated beforeDefinition 4.1.14) for some positive integers k1, . . . , kn. Prove that if i1, . . . , in ∈N and iν ≥ kν (ν = 1, . . . , n), then the set Tσ(Z) \Z(i1, . . . , in) is algebraicallyindependent over K(S(i1, . . . , in)).

In the formulation of the next theorem we use the following convention. LetK be a difference field with a basic set σ = α1, . . . , αn and let Ky1, . . . , ysbe a ring of σ-polynomials in σ-indeterminates y1, . . . , ys over K. By the orderof a term u = αk1

1 . . . αknn yj (1 ≤ j ≤ s) with respect to αi (1 ≤ i ≤ n) we

mean the integer ki; it will be denoted by ordαiu. If f is a σ-polynomial in

Ky1, . . . , ys and u1, . . . , um are all terms that appear in f , then by the orderof f with respect to αi (denoted by ordαi

f) we mean maxordαiuν |1 ≤ ν ≤ m.

Theorem 4.1.17 Let K be a difference field with a basic set σ, α ∈ σ, andσα = σ \ α. Let Ky be the ring of σ-polynomials in one σ-indeterminatey over K and let L be a σ-overfield of K. Furthermore, let an element η ∈ Lbe σ-algebraic over K and let r ∈ N be the smallest integer such that there isa σ-polynomial A ∈ Ky that vanishes at η and whose order with respect to αis r. Then σα-trdegKK〈η〉 = r.

PROOF. By the condition of the theorem, elements η, . . . , αr−1(η) areσα-algebraically independent over K and αr(η) is σα-algebraic over the field


K〈η, . . . , αr−1(η)〉σα. (In this case we treat K as a difference field with the

basic set σα and consider its σα-field extension.) Let us show that every ele-ment αk(η), k ≥ r, is σα-algebraic over K〈η, . . . , αk−1(η)〉σα

. We proceed byinduction on k. As we have seen, our statement is true for k = r. Suppose itis true for some k > r, so that αk(η) is σα-algebraic over K〈η, . . . , αk−1(η)〉σα

.Then αk+1(η) is σα-algebraic over the σα-field α(K)〈α(η), . . . , αk(η)〉σα

andhence over its overfield K〈η, . . . , αk(η)〉σα

.It follows (see Theorem 4.1.2(v)) that every element αk(η) k ≥ r, is σα-

algebraic over K〈η, . . . , αr−1(η)〉σα, so that the field extension

K〈η〉/K〈η, . . . , αr−1(η)〉σαis σα-algebraic, whence σα-trdegKK〈η〉 = r.

We conclude this section with some some results on ordinary difference fieldextensions. The following statement is a direct consequence of Theorem 4.1.17.

Corollary 4.1.18 Let K be an ordinary difference field with a basic set σ =α, Ky the ring of σ-polynomials in one σ-indeterminate y over K, and Lan overfield of K. Furthermore, let an element η ∈ L be σ-algebraic over K andlet r ∈ N be the smallest integer such that there is a σ-polynomial A ∈ Ky oforder r that vanishes at η. Then trdegKK〈η〉 = r.

Let K be an ordinary difference field with a basic set σ = α and letL = K〈S〉 be a σ-field extension of K generated by a set S. Then one has thedescending chain of σ-fields L = K〈S〉 ⊇ K〈α(S)〉 ⊇ K〈α2(S)〉 ⊇ . . . (asusual, for any automorphism τ of L and any set Σ ⊆ L, τ(L) denotes the setτ(a) | a ∈ S). Suppose that trdegKL < ∞ (hence σ-trdegKL = 0) and letri = trdegKK〈αi(S)〉 (i = 0, 1, 2, . . . ). Then r0 = trdegKL ≥ r1 ≥ r2 ≥ . . . ,hence there exists a finite limit s = limi→∞ ri = minri | i ∈ N and there existsk ∈ N such that s = rk.

Proposition 4.1.19 With the above notation and conventions, s = trdegK∗L∗

where ∗ stands for inversive closure.

PROOF. Let elements η1, . . . , ηp ∈ L∗ be algebraically independent overK∗. Then there exists m ∈ N such that αm(η1), . . . , αm(ηp) lie in L. Thenαk+m(η1), . . . , αk+m(ηp) form a subset of K〈αk(S)〉 which is algebraically inde-pendent over K∗ and hence over K. It follows that s = trdegKK〈αk(S)〉 ≥ p.Therefore, s ≥ trdegK∗L∗.

To prove the opposite inequality, let us choose a transcendence basis ζ1, . . . ,ζs of K〈αk(S)〉 over K and show that the elements of this basis are algebraicallyindependent over K∗.

Suppose that ζ1, . . . , ζs are algebraically dependent over K∗. Then there isa finite set Σ ⊂ K∗ such that ζ1, . . . , ζs are algebraically dependent over K〈Σ〉.Furthermore, one can choose d ∈ N such that αd(Σ) ⊆ K. Now it is easyto see that αd(ζ1), . . . , αd(ζs) are algebraically dependent over αd(K)〈αd(Σ)〉and hence over K. On the other hand, since every element of K〈αk(S)〉 is al-gebraic over K(ζ1, . . . , ζs), every element of αd(K)〈αk+d(S)〉 is algebraic overαd(K)(αd(ζ1), . . . , αd(ζs) ) and hence over K. Since trdegKK〈αk+q(S)〉 = s,the elements αd(ζ1), . . . , αd(ζs) are algebraically independent over K (actually,

4.2. DIMENSION POLYNOMIALS OF σ-FIELD EXTENSIONS 255

they form a transcendence basis of K〈αk+d(S)〉 over K). We have arrived at acontradiction that implies the algebraic dependence of ζ1, . . . , ζs over K∗ andhence the equality s ≥ trdegK∗L∗.

Corollary 4.2.20 Let K be an ordinary difference field with a basic set σ =α, Ky the ring of σ-polynomials in one σ-indeterminate y over K, and Lan overfield of K. Let an element η ∈ L be σ-algebraic over K and let r ∈ Nbe the smallest integer such that there is a σ-polynomial A ∈ Ky of effectiveorder r that vanishes at η. Then trdegK∗(K〈η〉)∗ = r.

PROOF. Let s = trdegK∗(K〈η〉)∗. It follows from our considerations beforeProposition 4.1.19 that trdegKK〈αi(η)〉 ≥ s for every i ∈ N, and the equalityoccurs for all sufficiently large i. Therefore, (see Corollary 4.1.18), s is the min-imum of the orders of nonzero σ-polynomials C ∈ Ky such that some αk(η)(k ∈ N) is a solution of C.

Now, in order to complete the proof, i.e., to show that r = s, it remains tonotice that Ky contains a nonzero σ-polynomial C of effective order m withsolution η if and only if Ky contains a nonzero σ-polynomial C of order msuch that C(αk(η)) = 0 for some k ∈ N. Indeed, given C with the describedproperties, let ordC = m + q (q ≥ 0) and let C be the σ-polynomial obtainedfrom C by replacing each αi(y) (i = q, q +1, . . . ) with αi−q(y). Clearly, ord C =m and C(αq(η)) = 0. Conversely, if 0 = C ∈ Ky, ord C = m and C(αk(η)) =0 for some k ∈ N, then one can consider a σ-polynomial C obtained from Cby replacing each αi(y) (i = 0, 1, . . . ) with αi+k(y). Clearly, EordC = m andC(η) = 0.

4.2 Dimension Polynomials of Differenceand Inversive Difference Field Extensions

In this section we introduce and study certain numerical polynomials associ-ated with finitely generated difference and inversive difference field extensions.Properties of these polynomials give us an important technique for the studyof difference fields and system of algebraic difference equations. Furthermore,as we shall see, such polynomials carry invariants of difference and inversivedifference field extensions, that is, numbers that do not depend on the systemsof generators of extensions. Unless otherwise is indicated, we assume that allfields considered below have zero characteristic.

The following theorem is a difference version of the classical Kolchin’s resulton differential dimension polynomial [103].

Theorem 4.2.1 Let K be a difference field with a basic set σ = α1, . . . , αn,T the free commutative semigroup generated by σ, and for any r ∈ N, T (r) =τ ∈ T |ord τ ≤ r. Furthermore, let L = K〈η1, . . . , ηs〉 be a σ-overfield of Kgenerated by a finite family η = η1, . . . , ηs. Then there exists a polynomialφη|K(t) ∈ Q[t] with the following properties.


(i) φη|K(r) = trdegKK(τηj |τ ∈ T (r), 1 ≤ j ≤ s) for all sufficiently larger ∈ N.

(ii) deg φη|K(t) ≤ n and the polynomial φη|K(t) can be written as

φη|K(t) =n∑

i=0

ai

(t + i

i

)where a0, . . . , an ∈ Z.(iii) The integers an, d = deg φη|K(t) and ad are invariants of the polynomial

φη|K(t), that is, they do not depend on the choice of a system of σ-generators η.Furthermore, an = σ-trdegKL.(iv) Let P be the defining σ-ideal of (η1, . . . , ηs) in the ring of σ-polynomials

Ky1, . . . , ys and let A be a characteristic set of P with respect to some or-derly ranking of y1, . . . , ys. Furthermore, for every j = 1, . . . , s, let Ej =(k1, . . . , kn) ∈ Nn|αk1

1 . . . αknn yj is the leader of a σ-polynomial in A. Then

φη|K(t) =s∑

i=1

ωEj(t)

where ωEj(t) is the Kolchin polynomial of the set Ej.

PROOF. As in Section 2.4, let TY denote the set of all terms τyi (τ ∈T, 1 ≤ i ≤ s), and let V = u ∈ TY |u is not a transform of any leaderuA of a σ-polynomial A ∈ A. Furthermore, for every r ∈ N, let V (r) =u ∈ V | ord u ≤ r.

If A∈A, then A(η) = 0, hence uA(η) is algebraic over the field K(τηj | τyj <uA (τ ∈ T, 1 ≤ j ≤ s)). (The symbol < denotes our orderly ranking ofy1, . . . , ys.) It follows that for every r ∈ N, the field Lr = K(τηj | τ ∈T (r), 1 ≤ j ≤ s) is an algebraic extension of the field K(v(η) | v ∈ V (r)). ByProposition 2.4.4, the ideal P does not contain nonzero difference polynomialsreduced with respect to A. It follows that for every r ∈ N, the set Vη(r) =v(η) | v ∈ V (r) is algebraically independent over K, hence Vη(r) is a transcen-dence basis of Lr over K and trdegKLr = CardVη(r). For every j = 1, . . . , s,the number of terms αk1

1 . . . αknn yj in V (r) is equal to the number of n-tuples

(k1, . . . , kn) ∈ Nn such that∑n

i=1 ki ≤ r and (k1, . . . , kn) does not exceed anyn-tuple in Ej with respect to the product order on Nn. By Theorem 1.5.2, forall sufficiently large r ∈ N this number is equal to ωEj

(r) where ωEj(t) is the

Kolchin polynomial of the set Ej . Thus, trdegKLr = CardVη(r) =s∑

j=1

ωEj(r)

for all sufficiently large r ∈ N, so the numerical polynomial φη|K(t) =s∑

i=1

ωEj(t)

satisfies conditions (i), (ii) and (iv) of the theorem. (Since degωEj≤ n for

j = 1, . . . , s, deg φη|K ≤ n, so that statement (ii) is a direct consequence ofCorollary 1.4.5.)


Let us show that if we represent the polynomial φη|K(t) in the form φη|K(t) =n∑

i=0

ai

(t + i

i

)with a0, . . . , an ∈ Z, then an, d = deg φη|K and ad do not de-

pend on the choice of a system of σ-generators η of L over K. Indeed, letζ = ζ1, . . . , ζq be another system of σ-generators of L/K and let φζ|K(t) =

n∑i=0

bi

(t + i

i

)(b0, . . . , bn ∈ Z) be the numerical polynomial such that φζ|K(r) =

trdegKK(τζj |τ ∈ T (r), 1 ≤ k ≤ q) for all sufficiently large r ∈ N. Then thereexists h ∈ N such that ηj ∈ K(τζk | τ ∈ T (h), 1 ≤ k ≤ q) for j = 1, . . . , sand ζk ∈ K(τηj | τ ∈ T (h), 1 ≤ j ≤ s) for k = 1, . . . , q. It follows thatφη|K(r) ≤ φζ|K(r +h) and φζ|K(r) ≤ φη|K(r +h) for all sufficiently large r ∈ Nwhence the polynomials φη|K(t) and φζ|K(t) have the same degrees and thesame leading coefficients.

Now, in order to complete the proof we need to show that an is equal tothe difference transcendence degree of L/K. Let e = σ-trdegKL. Without lossof generality we can assume that η1, . . . , ηe form a σ-transcendence basis of Lover K. Then for any r ∈ N, the set τηj | τ ∈ T (r), 1 ≤ j ≤ e is algebraically

independent over K hence φη|K(r) ≥ eCard T (r) = e

(r + n

n

)for all sufficiently

large r ∈ N. Therefore, an ≥ e.Let us prove the opposite inequality (and therefore, the desired equality

an = e). For every j = e + 1, . . . , s, the element ηj is σ-algebraic over the σ-field K〈η1, . . . , ηe〉, so there exists a term vj = τjyj (τj ∈ T ) such that τjηj isalgebraic over the field K〈η1, . . . , ηe〉(τηi | τ ∈ T, e + 1 ≤ i ≤ s and τyi < vj).Therefore, there exists h ∈ N such that the elements τe+1ηe+1, . . . , τsηs arealgebraic over the field K(τηi | τ ∈ T (h), 1 ≤ i ≤ e⋃τηi | τ ∈ T, e+1 ≤ i ≤ sand τyi < vj). Let rj = ord τj (e + 1 ≤ j ≤ s), let τ ′yj be a transform of vj

and r′ = ord τ ′. Then the element τ ′ηj is algebraic over the field K(τηi | τ ∈T (h + r′ − rj), 1 ≤ i ≤ e⋃τηi | τ ∈ T, e + 1 ≤ i ≤ s and τyi < τ ′yj). Itfollows that for all r ≥ maxre+1, . . . , rs, every element of the field Lr =K(τηj | τ ∈ T (r), 1 ≤ j ≤ s) is algebraic over the field L′

r = K(τηi | τ ∈T (r+h), 1 ≤ i ≤ e∪τηj | τ ∈ T (r)\⋃s

j=e+1 T (r−rj)τj). Therefore, φη|K(r) =trdegKLr ≤ trdegKK(τηi | τ ∈ T (r + h), 1 ≤ i ≤ e⋃τηj | τ ∈ T (r), e + 1 ≤j ≤ s) = trdegKL′

r ≤ e · CardT (r +h)+s∑

j=d+1

[CardT (r)−CardT (r− rj)] =

e

(r + h + n

n

)+

s∑j=d+1

[(r + n

n

)−

(r + n − rj

n

)]for all sufficiently large r ∈ N.

Since the last sum is a polynomial of r of degree less than n, we have an ≤ e.Thus, an = σ-trdegKL.

Definition 4.2.2 The polynomial φη|K(t) whose existence is established byTheorem 4.2.1 is called the difference (or σ-) dimension polynomial of the dif-ference field extension L of K associated with the system of σ-generators η. Theintegers d = deg φη|K(t) and ad are called, respectively, the difference (or σ-)


type and typical difference (or σ-) transcendence degree of L over K. Theseinvariants of φη|K(t) are denoted by σ-typeKL and σ-t.trdegKL, respectively.

Example 4.2.3 With the notation of Theorem 4.2.1, suppose that the el-ements η1, . . . , ηs are σ-algebraically independent over K. If φη|K(t) is thecorresponding difference dimension polynomial of L/K, then φη|K(r) =

trdegKK(τηj | τ ∈ T, 1 ≤ j ≤ s) = s · CardT (r) = s

(r + n

n

)for

all sufficiently large r ∈ N (see formula (1.4.16)). Therefore, in this case

φη|K(t) = s

(t + n

n

).

Theorem 4.2.4 Let K be a difference field with a basic set σ = α1, . . . , αnand let L be a finitely generated σ-field extension of K with a set of σ-generatorsη = η1, . . . , ηs.

(i) If d = σ-trdegKL and η1, . . . , ηd is a σ-transcendence basis of L over

K, then φ(ηd+1,...,ηs)|K〈η1,...,ηd〉(t) φη|K(t)−d

(t + n

n

). ( is the natural order

on the set of numerical polynomials introduced in Definition 1.5.10.)

(ii) φη|K(t) = m

(t + n

n

)for some m ∈ N if and only if

σ-trdegKL = trdegKK(η1, . . . , ηs) = m.

PROOF. For every r ∈ N, let A1(r) = τηj | τ ∈ T (r), 1 ≤ j ≤ d,A2(r) = τηj | τ ∈ T (r), d + 1 ≤ j ≤ s, and A(r) = A1(r) ∪ A2(r) = τηj | τ ∈T (r), 1 ≤ j ≤ s. Then the definition of the polynomial φ(ηd+1,...,ηs) |K〈η1,...,ηd〉and Proposition 1.6.31 show that

φ(ηd+1,...,ηs)|K〈η1,...,ηd〉(r) = trdegK〈η1,...,ηd〉K〈η1, . . . , ηd〉(A2(r))≤ trdegK(A1(r))K(A(r))= trdegKK(A(r)) − trdegKK(A1(r))

= φη |K(r) − d

(r + n

n

)for all sufficiently large r ∈ N. Therefore,

φ(ηd+1,...,ηs)|K〈η1,...,ηd〉(t) φη|K(t) − d

(t + n

n

).

Suppose that φη|K(t) = m

(t + n

n

)for some m ∈ N. Then m = σ-trdegKL

and without loss of generality we can assume that elements η1, . . . , ηm form aσ-transcendence basis of L over K. In this case, φ(ηm+1,...,ηs) |K〈η1,...,ηm〉(t) = 0,since φ(ηm+1,...,ηs) |K〈η1,...,ηm〉(r) = trdegKK(τηj | τ ∈ T (r), 1 ≤ j ≤ s) −trdegKK(τηj | τ ∈ T (r), 1 ≤ j ≤ m) = m

(r + n

n

)− m

(r + n

n

)= 0 for all

sufficiently large r ∈ N.


Let P be the defining ideal of the (s−m)-tuple (ηm+1, . . . , ηs) in the ring ofσ-polynomials K〈η1, . . . , ηm〉y1, . . . , ys−m over the σ-field K〈η1, . . . , ηm〉 andlet A be a characteristic set of P with respect to some orderly ranking < ofy1, . . . , ys−m such that y1 < y2 < · · · < ys−m. For every j = 1, . . . , s−m, let Ej

denote the set of all n-tuples (k1, . . . , kn) ∈ Nn such that the term αk11 . . . αkn

n yj

is a leader of some (obviously, unique) σ-polynomial in A. By Theorem 4.2.1(iv),

φ(ηm+1,...,ηs) |K〈η1,...,ηm〉(t) =s−m∑j=1

ωEj(t) = 0, hence ωEj

(t) = 0 for j = 1, . . . ,

s−m. It follows that every set Ej (1 ≤ j ≤ s−m) consists of a single element(0, . . . , 0) (see Theorem 1.5.2). Since y1 < yj for j = 2, . . . , s−m, a σ-polynomialin A with the leader y1 is a usual polynomial in one indeterminate y1 withcoefficients in K〈η1, . . . , ηm〉. It follows that ηm+1, as well as any element τηm+1

(τ ∈ T ), is algebraic over K〈η1, . . . , ηm〉. Since φ(ηm+2,...,ηs) |K〈η1,...,ηm〉(t) φ(ηm+1,...,ηs) |K〈η1,...,ηm〉(t), φ(ηm+2,...,ηs) |K〈η1,...,ηm〉(t) = 0, so one can repeat theabove reasoning and obtain that all elements τηj with τ ∈ T, m+1 ≤ j ≤ s arealgebraic over K〈η1, . . . , ηm〉.

Since ηm+1, . . . , ηs are algebraic over K〈η1, . . . , ηm〉, there exists r0 ∈ N suchthat ηj (m + 1 ≤ j ≤ s) are algebraic over the field K(τηi | τ ∈ T (r0), 1 ≤ i ≤m). Therefore, for every r ≥ r0, the field K(τηj | τ ∈ T (r), 1 ≤ j ≤ s) isalgebraic over K(τηj | τ ∈ T (r), 1 ≤ j ≤ m).

Suppose that ηm+1 is not algebraic over K(η1, . . . , ηm). Let p be the mini-mal number in the set of all q ∈ N such that ηm+1 is algebraic over the fieldK(τηj | τ ∈ T (q), 1 ≤ j ≤ m) (according to our assumption, p ≥ 1). Sinceηm+1 is transcendental over K(τηj | τ ∈ T (p − 1), 1 ≤ j ≤ m), there existsan element v = τ0ηh (1 ≤ h ≤ m) such that ord τ0 = p and v is algebraic overthe field K(τηj | τ ∈ T (p), 1 ≤ j ≤ m ∪ ηm+1 \ v) (see Theorem 1.6.26and Proposition 1.1.6). It is easy to see that if τ ′ ∈ T and ord τ ′ = r ≥ r0, thenthe element τ ′v = τ ′τ0ηh is algebraic over the field K(τηj | τ ∈ T (r + p), 1 ≤j ≤ m∪τ ′ηm+1 \ τ ′v). Furthermore, τ ′ηm+1 is algebraic over K(τηj | τ ∈T (r), 1 ≤ j ≤ m) and therefore, over K(τηj | τ ∈ T (r + p), 1 ≤ j ≤ m),whence τ ′v is algebraic over K(τηj | τ ∈ T (r + p), 1 ≤ j ≤ m \ τ ′v). Thus,the set τηj | τ ∈ T (r + p), 1 ≤ j ≤ m is algebraically dependent over Kthat contradicts the fact that η1, . . . , ηm are σ-algebraically independent overK. It follows that ηm+1 is algebraic over the field K(η1, . . . , ηm) and similarlyevery element ηm+2, . . . , ηs is algebraic over K(η1, . . . , ηm), so that m = σ-trdegKL = trdegKK(η1, . . . , ηs).

Conversely, suppose that the last equality holds and η1, . . . , ηm is a σ-transcendence basis of K〈η1, . . . , ηs〉 over K. Then the elements η1, . . . , ηm arealgebraically independent over K and K(η1, . . . , ηs) is an algebraic extension ofK(η1, . . . , ηm). It follows that if τ ∈ T (r) (r ∈ N) and 1 ≤ i ≤ s, then theelement τηi is algebraic over the field K(τηj | τ ∈ T (r), 1 ≤ j ≤ m). Therefore,trdegKK(τηj | τ ∈ T (r), 1 ≤ j ≤ s) = trdegKK(τηj | τ ∈ T (r), 1 ≤ j ≤m) = m

(t+nn

)for all sufficiently large r ∈ N, so that φη|K(t) = m

(t+nn

).

The following theorem gives versions of Theorem 4.2.1 for finitely generatedinversive difference field extensions.


Theorem 4.2.5 Let K be an inversive difference field with a basic set σ =α1, . . . , αn, Γ the free commutative group generated by σ and for any r ∈N, Γ(r) = γ = αk1

1 . . . αknn ∈ Γ|ord γ =

∑ni=1 |ki| ≤ r. Furthermore, let

L = K〈η1, . . . , ηs〉∗ be a σ∗-overfield of K generated by a finite family η =η1, . . . , ηs. Then there exists a polynomial ψη|K(t) ∈ Q[t] with the followingproperties.

(i) ψη|K(r) = trdegKK(γηj |γ ∈ Γ(r), 1 ≤ j ≤ s) for all sufficiently larger ∈ N.

(ii) deg ψη|K(t) ≤ n and the polynomial ψη|K(t) can be written as

ψη|K(t) =2na

n!tn + o(tn) (4.2.3)

where a ∈ Z and o(tn) is a numerical polynomial of degree less than n.(iii) The integers a, d = deg ψη|K(t) and the coefficient of td in the polynomial

ψη|K(t) do not depend on the choice of a system of σ-generators η. Furthermore,a = σ-trdegKL.(iv) Let P be the defining σ-ideal of (η1, . . . , ηs) in the ring of σ∗-polynomials

Ky1, . . . , ys∗ and let A be a characteristic set of P with respect to some or-derly ranking of y1, . . . , ys. Furthermore, for every j = 1, . . . , s, let Fj =(k1, . . . , kn) ∈ Zn|αk1

1 . . . αknn yj is a leader of a σ∗-polynomial in A. Then

ψη|K(t) =s∑

i=1

φFj(t) where φFj

(t) is standard Z-dimension polynomial of the

set Fj (see Definition 1.5.18).

PROOF. Let Ky1, . . . , ys∗ be the ring of inversive difference (σ∗-) polyno-mials over K and let ≤ be an orderly ranking of the family of σ∗-indeterminatesy1, . . . , ys (see Definition 2.4.6 and the terminology introduced after the de-finition). Furthermore, let W denote the set of all terms γyj (γ ∈ Γ, 1 ≤j ≤ s) which are not transforms of any leader uA of a σ∗-polynomial A ∈A. (The concept of transform is understood in the sense of Definition 2.4.5.)As in the proof of Theorem 4.2.1 we obtain that for any r ∈ N, the setW (r) = γyj ∈ W | ord γ ≤ r, 1 ≤ j ≤ s is a transcendence basis of thefield K(γηj γ ∈ Γ(r), 1 ≤ j ≤ s) over K. Furthermore, for every j = 1, . . . , sand for every r ∈ N, the number of terms αk1

1 . . . αknn yj in W (r) is equal to

the number of n-tuples (k1, . . . , kn) ∈ Zn \ Fj such that∑n

i=1 |ki| ≤ r and(k1, . . . , kn), is not greater than any element of Fj with respect to the order introduced in Section 1.5. It follows from Theorem 1.5.17 that the polynomial

ψη|K(t) =s∑

j=1

φFj(t) satisfies condition (i) of the theorem, deg ψη|K ≤ n, and

statement (iv) holds. Furthermore, the polynomial ψη|K can be written in theform (4.2.3) with some a ∈ Q.

The first part of (iii) can be easily obtained by repeating the proof of thecorresponding part of Theorem 4.2.1(iii). Let us show that if ψη|K(t) is writtenin the form (4.2.3), then a is the difference (σ-) transcendence degree of L over


K (in particular, a is an integer). Let d = σ-trdegKL. Without loss of generalitywe can assume that η1, . . . , ηd form a σ-transcendence basis of L/K, so that

ψη|K(r) ≥ trdegKK(γyj | γ ∈ Γ(r), 1 ≤ j ≤ d) = dCard Γ(r)

= d

n∑i=1

(−1)n−i2i

(n

i

)(r + i

i

)

for all sufficiently large r ∈ N (see formula (1.4.17)). It follows that a =∆nψη|K(t)

2n≥ d (as before, ∆nf(t) denotes the nth finite difference of a poly-

nomial f(t), see Section 1.4).

To prove the opposite inequality let us denote the kth ortant of Zn byZ(n)

k (we used this notation in Section 1.5) and set Γk = γ = αl11 . . . αln

n ∈

Γ | (l1, . . . , ln) ∈ Z(n)k (clearly, Γ =

2n⋃i=1

Γk). Since the elements ηd+1, . . . , ηs

are σ-algebraic over K〈η1, . . . , ηd〉, for every j = d + 1, . . . , s and for everyk = 1, . . . , 2n, there exists an element γ

(k)j ∈ Γk such that γ

(k)j ηj is algebraic

over K〈η1, . . . , ηd〉(γηj | γ ∈ Γk, γyj < γ(k)j yj). Therefore, there is an element

r0 ∈ N such that γ(k)j ηj is algebraic over the field K(γηi | γ ∈ Γ(r0), 1 ≤ i ≤

d⋃γηj | γ ∈ Γk, γyj < γ(k)j yj) for j = d + 1, . . . , s.

Let rkj = ord γ

(k)j (d + 1 ≤ j ≤ s, 1 ≤ k ≤ 2n) and let γ′yj (γ ∈ Γk,

d + 1 ≤ j ≤ s) be a transform of γ(k)j yj (in the sense of Definition 2.4.5).

Clearly, if r′ = ord γ′, then r′ ≥ rkj and the element γ′ηj is algebraic over the

field K(γηi | γ ∈ Γ(r0 + r′ − rkj ), 1 ≤ i ≤ d ∪ γηj | γ ∈ Γ, γyj < γ′yj). Thus,

for any r ≥ maxj,krkj , every element of the field K(γηi | γ ∈ Γ(r), 1 ≤ j ≤ s)

is algebraic over the field

K(γηi | γ ∈ Γ(r0 + r), 1 ≤ i ≤ d⋃

γηj | γ ∈ Γ(r) \2n⋃

k=1

Γk(r − rkj )γ(k)

j ,

d + 1 ≤ j ≤ s) where Γk(q) (q ∈ N) denotes the set γ ∈ Γk | ord γ ≤ q.

Now, repeating the arguments of the proof of part (iii) of Theorem 4.2.1, weobtain that

ψη|K(r) = trdegKK(γηj |γ ∈ Γ(r), 1 ≤ j ≤ s) ≤ d · Card Γ(r + r0)

+s∑

j=d+1

2n∑k=1

Card(Γk(r) − Γk(r − r

(k)j )

)


for all sufficiently large r ∈ N. Since Card(Γk(r − r(k)j ) =

(r − r

(k)j + n

n

)(see

formula (1.4.16)),

ψη|K(t) ≤ dCard Γ(r) d

n∑i=1

(−1)n−i2i

(n

i

)(t + i

i

)

+s∑

j=d+1

2n∑k=1

[(t + n

n

)−

(t − r

(k)j + n

n

)]+ o(tn).

The last inequality and the already proven inequality a = ∆nψη|K(t)

2n ≥ d, implythat a = d = σ∗-trdegKL and the polynomial ψη|K(t) can be written in theform (4.2.3). This completes the proof of the theorem.

Definition 4.2.6 The polynomial ψη|K(t) whose existence is established byTheorem 4.2.5 is called the σ∗-dimension polynomial of the σ∗-field extensionL of K associated with the system of σ∗-generators η.

Example 4.2.7 With the notation of Theorem 4.2.5, suppose that the elementsη1, . . . , ηs are σ-algebraically independent over K, and let φη|K(t) denote thecorresponding σ∗-dimension polynomial of L/K. As in Example 4.2.3, we obtainthat

ψη|K(r) = s · Card Γ(r) = s

n∑k=0

(−1)n−k2k

(n

k

)(r + k

k

)for all sufficiently large r ∈ N (see formula (1.4.17)), whence

ψη|K(t) = s

n∑k=0

(−1)n−k2k

(n

k

)(t + k

k

).

Now, with the notation of Theorem 4.2.5, we are going to show that the mod-ule of differentials ΩL|K is a vector σ∗-L-space and the σ∗-dimension polynomialψη|K(t) of the extension L/K coincides with the σ∗-dimension polynomial asso-ciated with the natural excellent filtration of ΩL|K . First, we need the followinglemma.

Lemma 4.2.8 Let A be an inversive difference ring with a basis set σ and let Bbe a σ∗-A-algebra (see Definition 2.2.8). Then the module of differential ΩB|Ahas a canonical structure of a σ∗-B-module such that for every α ∈ σ∗ andb ∈ B, α(db) = dα(b). This structure is unique.

PROOF. The uniqueness is clear since the B-module ΩB|A is generated bythe elements db for b ∈ B. To show the existence of the desired structure of aσ∗-B-module on the B-module ΩB|A, observe that B

⊗A B is a σ∗-A-algebra

(where α(x⊗

y) = α(x)⊗

α(y) for every α ∈ σ∗ and every generator x⊗

y ofthe A-algebra B

⊗A B) and µ : B

⊗A B → B (x

⊗y → xy for x, y ∈ B) is


a σ-epimorphism of σ∗-A-algebras. Then I = Ker µ is a σ∗-ideal of B⊗

A B,I/I2 is a σ∗-B-module and if b ∈ B, then α((b

⊗1−1

⊗b)+I2) = (α(b)

⊗1−

1⊗

α(b)) + I2 for every α ∈ σ∗. The last formula is equivalent to the equalityα(db) = dα(b) and this completes the proof.

Let K be an inversive difference field with a basic set σ = α1, . . . , αn andlet L be a finitely generated σ∗-field extension of K with a set of generatorsη = η1, . . . , ηs. Let ΩL|K be a vector σ∗-space of differentials (with the struc-ture defined in the last lemma) and for every r ∈ N, let (ΩL|K)r denote thevector L-subspace of ΩL|K generated by the set dγ(ηi)|γ ∈ Γ(r), 1 ≤ i ≤ s.Furthermore, let (ΩL|K)r = 0 for any r ∈ Z, r < 0.

Theorem 4.2.9 With the above notation,(i) ((ΩL|K)r)r∈Z is an excellent filtration of the σ∗-L-module ΩL|K .(ii) dimK(ΩL|K)r = trdegKK(γηj |γ ∈ Γ(r), 1 ≤ j ≤ s) for all r ∈ Z.(iii) The σ∗-dimension polynomial ψη|K(t) is equal to the σ∗-dimension poly-

nomial of ΩL|K associated with the filtration ((ΩL|K)r)r∈Z.

PROOF. Let E be the ring of σ∗-operators over L and (Er)r∈Z the standardfiltration of E . Furthermore, let Lr = K(γηj |γ ∈ Γ(r), 1 ≤ j ≤ s) (r ∈ N)and Lr = K for r ∈ Z, r < 0. Since α(dξ) = dα(ξ) for any ξ ∈ L, α ∈ σ∗

and Lr+q = Lr(γηj | γ ∈ Γ(q), 1 ≤ j ≤ s) for all r, q ∈ N, ((ΩL|K)r)r∈Z isa filtration of the σ∗-L-module ΩL|K (see Definition 3.5.1) and Eq(ΩL|K)r =(ΩL|K)r+q for all r, q ∈ N.

Since the finite set γdηj | γ ∈ Γ(r), 1 ≤ j ≤ s generates (ΩL|K)r as a vectorL-space, the filtration ((ΩL|K)r)r∈Z is excellent.

By Proposition 1.7.13, if ξ1, . . . , ξkr (r ∈ N, kr ≥ 0) is a transcendence

basis of the field Lr over K, then dξ1, . . . , dξkr is a basis of the vector L-space

(ΩL|K)r. This implies statements (ii) and (iii). Let K be an inversive difference field with a basic set σ = α1, . . . , αn,

L a finitely generated σ∗-field extension of K with a set of σ∗-generators η =η1, . . . , ηs, and E the ring of σ∗-operators over L. Since the ring E is leftNoetherian (see Theorem 3.4.2) and the left E-module ΩL|K is finitely generated,there exists a finite resolution

0 → Mqdq−1−−−→ Mq−1 → . . .

d0−→ M0ρ−→ ΩL|K (4.2.4)

where each Mi (0 ≤ i ≤ q) is a free filtered E-module (with respect to thestandard filtration of E) and ρ, d0, . . . , dq−1 are σ-homomorphisms of filteredE-modules (see [176, Theorem 10.4.9]). As we have seen in Example 3.5.4, theσ∗-dimension polynomial χMi

(t) of the module Mi (0 ≤ i ≤ q) can be expressedas one of the sums in formula (3.5.5). The last statement of Theorem 4.2.9 andthe exact sequence (4.2.4) show that the σ∗-dimension polynomial ψη|K(t) ofthe extension L/K can be expressed as

φη|K(t) =q∑

i=1

(−1)iχMi(t). (4.2.5)


Applying the last formula in (3.5.5) and writing each(t+i−l

i

)in the canonical

form (1.4.9) we can represent the polynomial ψη|K(t) in the form

ψη|K(t) =n∑

i=0

ai2i

(t + i

i

)(4.2.6)

where a0, . . . , an ∈ Z. The following definition is based on Theorem 4.2.5(iii)and the last observation.

Definition 4.2.10 Let K be an inversive difference field with a basic set σ, L aσ∗-field extension of K generated by a finite family η = η1, . . . , ηs, and ψη|K(t)the corresponding σ∗-dimension polynomial of the extension L/K written in the

form (4.2.6). Then d = deg ψη|K and the coefficient ad =∆dψη|K(t)

2dare called

the inversive difference (or σ∗-) type and typical inversive difference(or typicalσ∗-) transcendence degree of L over K. They are denoted by σ∗-typeKL and σ∗-t.trdegKL, respectively. (By Theorem 4.2.5, these characteristics do not dependon the choice of the set of σ∗-generators of L over K; if d = n, then ad = σ-trdegKL.)

The proof of Theorem 4.2.9 shows that one can naturally generalizeTheorem 4.2.5 to inversive difference fields of arbitrary characteristic.

Definition 4.2.11 Let K be an inversive difference field with a basic set ofautomorphisms σ = α1, . . . , αn and let L be a σ∗-field extension of K. Anascending sequence Lrr∈Z of intermediate fields of L/K is called a filtrationof this σ∗-field extension if it satisfies the following conditions:

(i) If η ∈ Lr and α ∈ σ∗, then α(η) ∈ Lr+1.

(ii)⋃r∈Z

Lr = L.

(iii) Lr = K for all sufficiently small r ∈ Z.

A filtration Lrr∈Z of L/K is called excellent if every Lr is a finitely gen-erated field extension of K (a filtration with this property is said to be finite)and there exists r0 ∈ Z such that Lr = Lr0(γ(η) | γ ∈ Γ(r − r0), η ∈ Lr0) forevery r ∈ Z, r > r0 (a filtration with the last property is called good). Finally,a filtration Lrr∈Z of L/K is called separable if every Lr is a separable fieldextension of K.

Theorem 4.2.12 Let K be an inversive difference field of arbitrary character-istic with a basic set of automorphisms σ = α1, . . . , αn. Let L be a σ∗-fieldextension of K and let Lrr∈Z be an excellent and separable filtration of L/K.Then there exists a numerical polynomial ψ(t) in one variable t such that

(i) ψ(r) = trdegKLr for all sufficiently large r ∈ Z.(ii) deg ψ ≤ n and the polynomial ψ(t) can be represented as

ψ(t) =n∑

i=0

ai2i

(t + i

i

)where a0, . . . , an ∈ Z.


(iii) an = σ∗-trdegKL.Furthermore, if the degree d of the polynomial ψ(t) isless than n, then d and the coefficient ad do not depend on the excellent andseparable filtration of L/K the polynomial ψ(t) is associated with.

The proof of this theorem is similar to the proof of Theorem 4.2.9 (with theuse of Proposition 1.7.13(iv) and Proposition 1.6.36). We leave the details tothe reader as an exercise.

The polynomial whose existence is established by Theorem 4.2.12 is calleda σ∗-dimension polynomial of the σ∗-field extension L/K associated with thegiven excellent and separable filtration of this extension.

The following statement is an analog of Theorem 4.2.4 for inversive differencefield extensions. The proof of this result is similar to the proof of Theorem 4.2.4,we leave the details to the reader an exercise.

Theorem 4.2.13 Let K be an inversive difference field with a basic set σ =α1, . . . , αn and let L be a finitely generated σ∗-field extension of K with a setof σ∗-generators η = η1, . . . , ηs. Then

(i) If d = σ-trdegKL and η1, . . . , ηd is a σ-transcendence basis of L overK, then ψ(ηd+1,...,ηs)|K〈η1,...,ηd〉(t) φη|K(t) − d

(t+nn

).

(ii) ψη|K(t) = m

n∑i=0

(−1)n−i2i

(n

i

)(t + i

i

)for some m ∈ N if and only if

σ-trdegKL = trdegKK(η1, . . . , ηs) = m.

Based on Theorem 3.5.15 and the corresponding result for differential fieldextensions of zero differential transcendence degree (see [97, Section 3, Corollary2] or [110, Theorem 5.6.3]) one can expect that if L is a finitely generated σ∗-fieldextension of an inversive difference field K with a basic set σ = α1, . . . , αn,then there exists a set σ1 = β1, . . . , βn of pairwise commuting automorphismsof L such that σ1 is equivalent to σ (in the sense of Definition 3.5.14) and L isa finitely generated σ∗

2-field extension of K if L and K are treated as inversivedifference fields with the basic set σ2 = β1, . . . , βn−1∗. The following exampleshows that this is not so.

Example 4.2.14 Let us consider the field of real numbers R as an inversivedifference field whose basic set σ consists of the identical automorphism α. LetA be the ring of all functions f(x) (x ∈ R) with real values which are definedfor all x ∈ R except for possibly finitely many points. Then A can be treated asa σ∗-overring of R such that αf(x) = f(x+1) for every f(x) ∈ A. Let η denotethe function 22x ∈ A and let L = R〈η〉∗ be the σ∗-field generated by η over R(clearly, L is a σ∗-subring of A). Since α(η) = 22x+1

= η2, L = R(η,√

η, 4√

η, . . . )where

√η = 22x−1

= α−1(η), 4√

η = 22x−2= α−2(η), . . . .

With the notation of Theorem 4.2.5 we have

ψη |R(r) = trdegRR(η,√

η, . . . , 2r√η) = 1


for every r ∈ N, so ψη |R(t) = 1 and σ∗-trdegRL = 0. At the same timeL is not a finitely generated field extension of R. (Otherwise, we would haveL = R(η,

√η, . . . , 2s√η) for some s ∈ N that contradicts the obvious fact that

2s+1√η /∈ R(η,√

η, . . . , 2s√η).) We see that despite the equality σ∗-trdegRL = 0,L cannot be considered as a finitely generated inversive difference field extensionof R with respect to an empty basic set. (Notice that the only sets of automor-phisms that are equivalent to a basic set σ = α of an ordinary inversivedifference field are σ and α−1.)

The last example shows that there is no direct analog of Theorem 3.5.16 forinversive difference field extensions. However, one can obtain the following weakversion of this theorem.

Theorem 4.2.15 Let K be an inversive difference field with a basic set σ =α1, . . . , αn, let L be a finitely generated σ∗-field extension of K, and let d =σ∗-typeKL. Then there exists a set σ1 = β1, . . . , βn of mutually commutingautomorphisms of L and a finite family ζ = ζ1, . . . , ζq of elements of L suchthat σ1 is equivalent to σ and if σ2 = β1, . . . , βd, then L is an algebraicextension of the field H = K〈ζ1, . . . , ζq〉∗σ2

. (The last field is a finitely generatedσ∗

2-field extension of K when K is treated as an inversive difference field withthe basic set σ2.)

PROOF. By Theorem 4.2.9, the module of differentials ΩL|K is a finitely gen-erated vector σ∗-L-space and it(ΩL|K) = d (in accordance with Definition 3.5.9,it(M) denote the σ∗-type of a vector σ∗-L-space M). By Theorem 3.5.16, thereexists a set σ1 = β1, . . . , βn of mutually commuting automorphisms of Land a finite family ζ = ζ1, . . . , ζq of elements of L such that σ1 is equiva-lent to σ and the elements dζ1, . . . , dζq generate ΩL|K as a vector σ∗

2-L-space,where σ2 = β1, . . . , βd. If H = K〈ζ1, . . . , ζq〉∗σ2

and η ∈ L, then the elementdη ∈ ΩL|K is a linear combination of elements dζ1, . . . , dζq with coefficients in thering of σ∗

2-operators over L, that is, dη is a linear combination of finitely many el-ements d(γijζi) (1 ≤ i ≤ q, 1 ≤ j ≤ ki for some k1, . . . , kq ∈ N) where all γij be-long to the free commutative group Γσ2 generated by σ2. By Proposition 1.7.13,the element η is algebraic over the field K(γijζi) | 1 ≤ i ≤ q, 1 ≤ j ≤ ki),hence it is algebraic over H. This completes the proof.

Theorem 4.2.1(iv) and Theorem 4.2.5(iv), together with Theorem 1.5.11(ii)and Proposition 1.5.19, show that the set of all dimension polynomials of finitelygenerated difference field extensions coincides with the set of all dimension poly-nomials of finitely generated difference field extensions; moreover, each of thesesets coincides with the set W of all Kolchin polynomials. By Theorem 1.5.11(vi),the set W is well-ordered with respect to the order ≺ on the set of all numericalpolynomials (see Definition 1.5.10).

Let K be a difference field with a basic set σ and L a finitely generated σ-field extension of K. Let Wσ(L,K) denote the set of all σ-dimension polynomialsof the extension L/K associated with various finite systems of σ-generators of


L over K. Since Wσ(L,K) ⊆ W , there exists a system of σ-generators η =η1, . . . , ηs of L over K such that φη|K(t) is the minimal element of the setWσ(L,K) (well-ordered by ≺). This σ-dimension polynomial φη|K(t) is calledthe minimal σ-dimension polynomial of the extension L/K; it is denoted byφL|K(t). Similarly, if K is an inversive difference field with a basic set σ andL a finitely generated σ∗-field extension of K, then the set of all σ∗-dimensionpolynomials of L over K, denoted by W ∗

σ (L,K), is a well-ordered subset of W(with respect to ≺). The minimal element of this set (which is a σ∗-dimensionpolynomial ψη|K(t) asociated with some finite system of σ∗-generators η of Lover K) is called the minimal σ∗-dimension polynomial of the extension L/K;it is denoted by ψL|K(t).

We conclude this section with two theorems on multivariable dimension poly-nomials of difference and inversive difference field extensions.

Let K be a difference field of with a basic set σ = α1, . . . , αn and letT be the free commutative semigroup generated by σ. As in Section 3.3, let apartition of the set σ into a disjoint union of its subsets be fixed:

σ = σ1 ∪ · · · ∪ σp (4.2.7)

where p ∈ N, and σ1 = α1, . . . , αn1, σ2 = αn1+1, . . . , αn1+n2, . . . , σp =αn1+···+np−1+1, . . . , αn (ni ≥ 1 for i = 1, . . . , p ;n1 + · · · + np = n). As insection 3.3, for any element τ = αk1

1 . . . αknn ∈ T we consider the orders of τ

with respect to each set σi (ordiτ =∑n1+···+ni

ν=n1+···+ni−1+1 kν with n0 = 0) andset T (r1, . . . , rp) = τ ∈ T |ord1τ ≤ r1, . . . , ordpτ ≤ rp for any r1, . . . , rp ∈ N.Also, if Σ is any subset of Np, we set Σ′ = e ∈ Σ|e is a maximal element of Σwith respect to one of the p! lexicographic orders <j1,...,jp

where j1, . . . , jp is apermutation of 1, . . . , p.

Theorem 4.2.16 With the above notation, let L = K〈η1, . . . , ηs〉 be a difference(σ-) field extension of K generated by a finite set η = η1, . . . , ηs. Then thereexists a polynomial φη|K(t1, . . . , tp) in p variables with rational coefficients suchthat

(i) φη(r1, . . . , rp) = trdegKK(τηi | τ ∈ T (r1, . . . , rp), 1 ≤ j ≤ s) for allsufficiently large (r1, . . . , rp) ∈ Np

(ii) degtiφη ≤ ni (1 ≤ i ≤ p), so that deg φ ≤ n and the polynomial

φη(t1, . . . , tp) can be represented as

φη(t1, . . . , tp) =n1∑

i1=0

. . .

np∑ip=0

ai1...ip

(t1 + i1

i1

). . .

(tp + ip

ip

)


(iii) d = deg φη, an1...np, p-tuples (j1, . . . , jp) ∈ Σ′, the corresponding coef-

ficients aj1...jp, and the coefficients of the terms of total degree d do not de-

pend on the choice of the system of σ-generators η of L over K. Furthermore,an1...np

= σ-trdegKL.


Theorem 4.2.17 Let K be an inversive difference field and let a partition(4.2.7) of its basic set σ = α1, . . . , αn be fixed. Let Γ be the free commu-tative group generated by σ and for any r1, . . . , rn ∈ N let Γ(r1, . . . , rp) =γ ∈ Γ|ord1τ ≤ r1, . . . , ordpτ ≤ rp ( ordiγ is the order of an element γwith respect to the set σi defined in the last part of Section 3.5). Furthermore,let L = K〈η1, . . . , ηs〉 be a σ∗-field extension of K generated by a finite setη = η1, . . . , ηs. Then there exists a polynomial ψη|K(t1, . . . , tp) in p variableswith rational coefficients such that

(i) ψη(r1, . . . , rp) = trdegKK(γηi | γ ∈ Γ(r1, . . . , rp), 1 ≤ j ≤ s) for allsufficiently large (r1, . . . , rp) ∈ Np.

(ii) degtiψη ≤ ni (1 ≤ i ≤ p), so that deg ψ ≤ n and the polynomial

ψη(t1, . . . , tp) can be represented as

ψη(t1, . . . , tp) =n1∑

i1=0

. . .

np∑ip=0

ai1...ip

(t1 + i1

i1

). . .

(tp + ip

ip

)where ai1...ip

∈ Z for all i1, . . . , ip.(iii) d = deg ψη, an1...np

, p-tuples (j1, . . . , jp) ∈ Σ′, the corresponding coef-ficients aj1...jp

, and the coefficients of the terms of total degree d do not de-pend on the choice of the system of σ∗-generators η of L over K. Furthermore,2n | an1...np

andan1...np

2n= σ-trdegKL.

Definition 4.2.18 Numerical polynomials φη(t1, . . . , tp) and ψη(t1, . . . , tp)whose existence is established by Theorems 4.2.16, 4.2.17, are called dimensionpolynomials of the σ- (respectively, σ∗-) field extension L/K associated with thegiven system of σ- (respectively, σ∗-) generators η and with the given partitionof the basic set σ into p disjoint subsets σ1, . . . , σp.

PROOF OF THEOREMS 4.2.16 and 4.2.17. Let ΩL|K be the module ofdifferentials associated with the extension L/K described in Theorem 4.2.17and let for any r1, . . . , rp ∈ N, (ΩL|K)r1,...,rp

denote the vector L-subspace ofΩL|K generated by the set dγ(ηi) | γ ∈ Γ(r), 1 ≤ i ≤ s. Furthermore, let(ΩL|K)r1,...,rp

= 0 whenever ri < 0 for at least one i. By Lemma 4.2.8, ΩL|Kis a vector σ∗-L-space, and it is easy to see (cf. the proof of Theorem 4.2.9)that (ΩL|K)r1,...,rp

| (r1, . . . , rp) ∈ Zp is an excellent p-dimension filtrationof ΩL|K and dimL(ΩL|K)r1,...,rp

= trdegKK(γηi | τ ∈ Γ(r1, . . . , rp), 1 ≤ j ≤s) for all r1, . . . , rp ∈ Z. Now all statements of Theorem 4.2.17 follow fromTheorem 3.5.38.

In order to prove Theorem 4.2.16, we consider the following generalizationof the method of characteristic sets used in the proof of Theorem 4.2.1.

Let Ky1, . . . , ys be the ring of difference (σ-) polynomials in σ-indeter-minates y1, . . . , ys over K and let TY denote the set of all elements τyi (τ ∈T, 1 ≤ i ≤ s) called terms. Let us consider p orders <1, . . . , <p on the set TYthat correspond to the orders on the semigroup T introduced in Section 3.3(we use the same symbols <i for the orders on T and TY). These orders aredefined as follows: τyj <i τ ′yk (τ, τ ′ ∈ T, 1 ≤ j, k ≤ s, 1 ≤ i ≤ p) if and only


if τ <i τ ′ in T or τ = τ ′ and j < k. By the ith order of a term u = τyj

(1 ≤ i ≤ p, τ ∈ T, 1 ≤ j ≤ s) we mean the number ordiu = ordiτ . Thenumber ord u = ord τ is called the order of the term u. We say that a termu = τyi is divisible by a term v = τ ′yj and write u | v, if i = j and τ ′ | τ . Forany terms u1 = τ1yj , . . . , uq = τqyj containing the same σ-indeterminate yj

(1 ≤ j ≤ s), the term lcm(τ1, . . . , τq)yj will be called the least common multipleof u1, . . . , uq; it is denoted by lcm(u1, . . . , uq). If u = τyi, v = τ ′yj and i = j,we set lcm(u, v) = 0.

If A ∈ Ky1, . . . , ys, A /∈ K, and 1 ≤ k ≤ p, then the highest with respect tothe ordering <k term that appears in A is called the k-leader of the σ-polynomialA. It is denoted by u

(i)A . If A is written as a polynomial in one variable u

(1)A ,

A = Id(u(1)A )

d+ Id−1(u

(1)A )

d−1+ · · · + I0 (σ-polynomials Id, Id−1, . . . , I0 do not

contain u(1)A ), then Id is called the leading coefficient of A; it is denoted by IA.

Let A and B be two σ-polynomials in Ky1, . . . , yn. We say that A haslower rank then B and write rk A < rk B if either A ∈ K, B /∈ K, or the vec-tor (u(1)

A , degu

(1)A

A, ord2u(2)A , . . . , ordpu

(p)A ) is less than the vector (u(1)

B , degu

(1)B

B,

ord2u(2)B , . . . , ordpu

(p)B ) with respect to the lexicographic order (where u

(1)A and

u(1)B are compared with respect to <1 and all other coordinates of the vectors

are compared with respect to the natural order on N). If the two vectors areequal (or A ∈ K and B ∈ K) we say that the σ-polynomials A and B are of thesame rank and write rk A = rk B.

A σ-polynomial B is said to be reduced with respect to a σ-polynomial A ifthe following two conditions hold.

(i) B does not contain any term τu(1)A (τ ∈ T, τ = 1) such that ordi(τu

(i)A ) ≤

ordiu(i)B for i = 2, . . . , p.

(ii) If B contains u(1)A , then either there exists j, 2 ≤ j ≤ p, such that

ordju(j)B < ordju

(j)A or ordju

(j)A ≤ ordju

(j)B for all j = 2, . . . , p and deg

u(1)A

B <

degu

(1)A

A.

A σ-polynomial B is said to be reduced with respect to a set Σ ∈ Ky1, . . . , ysif B is reduced with respect to every element of Σ.

A set of ∆-polynomials Σ ⊆ Ky1, . . . , yn is called autoreduced if Σ⋂

K = ∅and every element of Σ is reduced with respect to any other element of this set.

It follows from Lemma 1.5.1 that if S is any infinite set of terms τyj (τ ∈T, 1 ≤ j ≤ s), then there exists an index j (1 ≤ j ≤ s) and an infinite sequenceof terms τ1yj , τ2yj , . . . , τkyj , . . . such that τk|τk+1 for all k = 1, 2, . . . . This fact,in turn, implies that every autoreduced set is finite.

Indeed, suppose that Σ is an infinite autoreduced subset of Ky1, . . . , ys.Then Σ must contain an infinite set Σ′ ⊆ Σ such that all σ-polynomials in Σ′

have different 1-leaders. (If it is not so, then there exists an infinite set Σ1 ⊆ Σsuch that all σ-polynomials from Σ1 have the same 1-leader u. By Lemma 1.5.1,the infinite set (ord2u

(2)A , . . . , ordpu

(p)A |A ∈ Σ1 contains a nondecreasing

infinite sequence (ord2u(2)A1

, . . . , ordpu(p)A1

) ≤P (ord2u(2)A2

, . . . , ordpu(p)A2

) ≤P . . .


(A1, A2, · · · ∈ Σ1, ≤P is the product order on Np). Since the sequence of degreesdeguAi|i = 1, 2, . . . cannot be strictly decreasing, there exists two indices iand j such that i < j and deguAi ≤ deguAj . We obtain that Aj is reduced withrespect to Ai that contradicts the fact that Σ is an autoreduced set.)

Thus, one can assume that all 1-leaders of our infinite autoreduced set Σ aredifferent. Then, as we have noticed, there exists an infinite sequence B1, B2, . . .

of elements of Σ such that u(1)Bi

|u(1)Bi+1

for all i = 1, 2, . . . . Let kij = ordju(1)Bi

and lij = ordju(i)Bi

(2 ≤ j ≤ p). Obviously, lij ≥ kij (i = 1, 2, . . . ; j = 2, . . . , p),so that (li2 − ki2, . . . , lip − kip)|i = 1, 2, . . . ⊆ Np−1. By Lemma 1.5.1, thereexists an infinite sequence of indices i1 < i2 < . . . such that (li12−ki12, . . . , li1p−ki1p) ≤P (li22 −ki22, . . . , li2p −ki2p) ≤P . . . . Then for any j = 2, . . . , p, we have

ordj

⎛⎝ u(1)Bi2

u(1)Bi1

u(j)Bi1

⎞⎠ = ki2j − ki1j + li1j ≤ ki2j − li2j − ki2j = li2j = ordju(j)Bi2

,

so that Bi2 contains a term τu(1)Bi1

= u(1)Bi2

such that τ = 1 and ordj(τu(j)Bi1

) ≤ordju

(j)Bi2

for j = 2, . . . , p. Since the sequence of degrees of Bikwith respect to

u(1)ik

cannot be infinite, Σ contains two σ-polynomials which are reduced withrespect to each other, contrary to the fact that the set Σ is autoreduced. Thus,every autoreduced set is finite.

Let A1, . . . , Ar be an autoreduced set in the ring Ky1, . . . , ys, let Ik

denote the initial of Ak with respect to the ranking <1 of TY (1 ≤ k ≤ r),and let I(Σ) = B ∈ Ky1, . . . , ys | either B = 1 or B is a product of finitelymany elements of the form τ(Ik), τ ∈ T, 1 ≤ k ≤ r. Repeating the proofof Theorem 2.4.1, one can obtain that for any σ-polynomial B, there existB0 ∈ Ky1, . . . , ys and J ∈ I(Σ) such that B0 is reduced with respect to Σ,B0 has lower rank than B (in the sense of Section 2.4 with the order <1 on theset TY ), and JB ≡ B0 (mod[Σ]) (that is, JB − B0 ∈ [Σ]).

As in Section 2.4, while considering autoreduced sets in the ring Ky1,. . . ,ynwe shall always assume that their elements are arranged in order of increasingrank. Given two autoreduced sets Σ = A1, . . . , Ar and Σ′ = B1, . . . , Bs, wesay that Σ has lower rank than Σ′ if one of the following two cases holds.

(1) There exists k ∈ N such that k ≤ minr, s, rk Ai = rk Bi for i =1, . . . , k − 1 and rk Ak < rk Bk.

(2) r > s and rk Ai = rk Bi for i = 1, . . . , s.If r = s and rk Ai = rk Bi for i = 1, . . . , r, then Σ is said to have the same

rank as Σ′.

Repeating the proof of Theorem 2.4.3 we obtain that in every nonemptyfamily of autoreduced sets of differential polynomials there exists an autoreducedset of lowest rank.

If Q is an ideal of the ring Ky1, . . . , ys, then an autoreduced subset of Qof lowest rank is called a characteristic set of this ideal. (Clearly, the set of allautoreduced subsets of Q is not empty.) As in the proof of Proposition 2.4.4we obtain that if Σ is a characteristic set of difference ideal Q of Ky1, . . . , ys,


then a σ-polynomials B ∈ Q is reduced with respect to the set Σ if and only ifB = 0.

Now we can complete the proof of Theorem 4.2.16 using the scheme of theproof of Theorem 3.3.16. Let P be the defining difference ideal of the σ-fieldextension L = K〈η1, . . . , ηs〉 and let Σ = A1, . . . , Ad be a characteristic setof P . If p > 1, then for any r1, . . . , rp ∈ N, let Ur1...rp

= u ∈ TY | ordiu ≤ ri

for i = 1, . . . , p and either u is not a transform of any u(1)Ai

(i. e., u = τu(1)Ai

for

any τ ∈ T ; i = 1, . . . , d), or for every τ ∈ T, A ∈ Σ such that u = τu(1)A , there

exists i ∈ 2, . . . , p such that ordi(τu(i)A ) > ri. If p = 1, we set Ur1 = u ∈

TY | ord1u ≤ r1 and u is not a transform of any u(1)Ai

.We are going to show that the set Ur1...rp

= u(η)|u ∈ Ur1...rp is a tran-

scendence basis of the field K(τηi | τ ∈ T (r1, . . . , rp), 1 ≤ j ≤ s over K.First of all, let us prove that the set Ur1...rp

is algebraically independentover K. Let g be a polynomial in k variables (k ∈ N, k ≥ 1) such thatg(u1(η), . . . , uk(η)) = 0 for some elements u1, . . . , uk ∈ Ur1...rp

. Then the σ-polynomial g = g(u1, . . . , uk) is reduced with respect to Σ (if p > 1 and g

contains uj = τu(1)Ai

for some i, 1 ≤ i ≤ d, there exists i ∈ 2, . . . , p such that

ordi(τu(j)Ai

) > ri ≥ ordiu(i)g ). Since g ∈ P , we obtain that g = 0, so the set

Ur1...rpis algebraically independent over K.

Now, we are going to show that every element τηj (1 ≤ j ≤ s, τ ∈T (r1, . . . , rp) is algebraic over the field K(Ur1,...,rp

). Let τηj /∈ Ur1,...,rp(if

τηj ∈ Ur1,...,rp, the statement is obvious). Then τyj /∈ Ur1,...,rp

whence τyj

is equal to some term of the form τ ′u(1)Ai

(τ ′ ∈ T , 1 ≤ i ≤ d) such that

ordk(τ ′u(k)Ai

) ≤ rk for all 1 < k ≤ p. Let us represent Ai as a polynomial in u(1)Ai

:

Ai = I0(u(1)Ai

)e

+ I1(u(1)Ai

)e−1

+ · · · + Ie, where I0, I1, . . . Ie do not contain u(1)Ai

(therefore, all terms in these σ-polynomials are lower than u(1)Ai

with respect tothe order <1). Since Ai ∈ P ,

Ai(η) = I0(η)(u(1)Ai

)(η)e+ I1(η)(u(1)

Ai)(η)

e−1+ · · · + Ie(η) = 0. (4.2.8)

Since I0 is reduced with respect to Σ, I0 /∈ P , so that I0(η) = 0. If we apply τ ′

to the both sides of the equation (4.2.8), the resulting equation will show thatthe element τ ′u(1)

Ai(η) = τηj is algebraic over the field K(τ ηl|ordiτ ≤ ri for

i = 1, . . . , p; 1 ≤ l ≤ s, and τ yl <1 τ ′u(1)Ai

= τyj. Now, the induction on theset of terms TY ordered by the relation <1 completes the proof of the fact thatUr1...rp

(η) is a transcendence basis of the field K(τηi | τ ∈ T (r1, . . . , rp), 1 ≤j ≤ s over K.

Let U(1)r1...rp = u ∈ TY |ordiu ≤ ri for i = 1, . . . , p and u = τu

(1)Aj

for any

τ ∈ T ; j = 1, . . . , d and U(2)r1...rp = u ∈ ΘY |ordiu ≤ ri for i = 1, . . . , p and

there exists at least one pair i, j (1 ≤ i ≤ p, 1 ≤ j ≤ d) such that u = θu(1)Aj

and ordi(θu(i)Aj

) > ri. (If p = 1, then we set U(2)r1...rp = ∅). Clearly, Ur1...rp

=

U(1)r1...rp

⋃U

(2)r1...rp and U

(1)r1...rp

⋂U

(2)r1...rp = ∅.


By Theorem 1.5.2, there exists a numerical polynomial ω1(t1, . . . , tp) in p

variables t1, . . . , tp such that ω1(r1, . . . , rp) = CardU(1)r1...rp for all sufficiently

large (r1 . . . rp) ∈ Np and degtiω1 ≤ ni (i = 1, . . . , p). Furthermore, repeat-

ing the last part of the proof of Theorem 3.3.16, we obtain that there ex-ists a numerical polynomial ω2(t1, . . . , tp) in p variables t1, . . . , tp such thatω2(r1, . . . , rp) = CardU

(2)r1...rp for all sufficiently large (r1 . . . rp) ∈ Np and

degtiω2 ≤ ni (i = 1, . . . , p). Clearly, the polynomial φη = ω1 + ω2 satisfies

conditions (i) and (ii) of Theorem 4.2.16. In order prove the last statement ofthe theorem, one just need to notice that if ζ = ζ1, . . . , ζq is another system ofσ-generators of L over K and φζt1, . . . , tp) is the corresponding dimension poly-nomial, then there exist positive integers s1, . . . , sp such that ηi ∈ K(τζk | τ ∈T (r1, . . . , rp), 1 ≤ k ≤ q) and ζk ∈ K(τηi | τ ∈ T (r1, . . . , rp), 1 ≤ i ≤ s) forany i = 1, . . . , s and k = 1, . . . , q. This observation immediately implies the laststatement of Theorem 4.2.16.

Exercise 4.2.19 Let K be a difference field whose basic set σ is a union oftwo disjoint subsets, σ1 = α1, . . . , αm and σ2 = β1, . . . , βn (m ≥ 1, n ≥ 1).Let T , T1 and T2 be the free commutative semigroups generated by the sets σ,σ1 and σ2, respectively, and for any element τ = αk1

1 . . . αkmm βl1

1 . . . βlnn ∈ T , let

ord1τ =∑m

i=1 ki, ord2τ =∑n

j=1 lj and ord τ = ord1τ + ord2τ . Furthermore, ifr, s ∈ N, then Ti(r) will denote the set τ ∈ Ti | ordiτ ≤ r (i = 1, 2) and T (r, s)will denote the set τ ∈ T | ord1τ ≤ r, ord2τ ≤ s.

Let L = K〈η〉 be a σ-field extension of K generated by a finite set η =η1, . . . , ηq and for any r ∈ N let L′

r = K〈T1(r)η〉σ2 and L′′r = K〈T2(r)η〉σ1

(as before, if Φ ⊆ T and A ⊆ L, then φ(A) = φ(a) | a ∈ A,φ ∈ Φ). Provethat there exist two numerical polynomials, χ1(t) and χ2(t) with the followingproperties:

(i) χ1(r) = σ2-trdegKL′r and χ2(r) = σ1-trdegKL′′

r for all sufficiently larger ∈ N.

(ii) deg χ1 ≤ m, deg χ2 ≤ n, so the polynomials χ1(t) and χ2(t) can be written

as χ1(t) =m∑

i=0

ai

(t + i

i

)and χ2(t) =

n∑j=0

bj

(t + j

j

)with integer coefficients ai

and bj .(iii) am = bn = σ-trdegKL.

Hint : Apply the results of Exercises 3.6.15

Exercise 4.2.20 With the notation of the preceding exercise, assume that theelements of σ are automorphisms of K whose extensions to L are automorphismsof L. Let Γ1 and Γ2 denote the free commutative groups generated by thesets σ1 and σ2, respectively, and for any r ∈ N let L∗

r = K〈Γ1(r)η〉∗σ2and

L∗∗r = K〈Γ2(r)η〉∗σ1

. Prove that there exist two numerical polynomials, χ∗1(t)

and χ∗2(t) with the following properties:


(i) χ∗1(r) = σ2-trdegKL∗

r and χ∗2(r) = σ1-trdegKL∗∗

r for all sufficiently larger ∈ N.

(ii) deg χ∗1 ≤ m, deg χ∗

2 ≤ n, so the polynomials χ1(t) and χ2(t) can be written

as χ∗1(t) =

2ma

m!+ o(tm) and χ∗

2(t) =2nb

n!+ o(tn) where a = b = σ-trdegKL.

Formulate and prove an analog of this result and the result of Exercise 4.2.19for the case when only elements of σ2 are automorphisms of K whose extensionsare automorphisms of L.

Theorems 4.2.16 and 4.2.17 allow us to assign a numerical polynomial to aprime inversive difference ideal in an algebra of difference or inversive differencepolynomials.

Definition 4.2.21 Let K be a difference field with a basic set σ = α1, . . . , αn,Ky1, . . . , ys an algebra of σ-polynomials in σ-indeterminates y1, . . . , ys

over K, and P a prime inversive difference ideal in Ky1, . . . , ys. Letη = (η1, . . . , ηs) be a generic zero of the ideal P . Then the σ-dimensionpolynomial φη|K(t) of the σ-field extension K〈η1, . . . , ηs〉/K is called the dif-ference (or σ-) dimension polynomial of the ideal P . It is denoted by φP (t). If

this polynomial is written in the canonical form (1.4.9), φP (t) =n∑

i=0

ai

(t + i

i

),

then the coefficient an (which is equal to σ-trdegKK〈η1, . . . , ηs〉/K) is called thedifference (or σ-) dimension of the ideal P ; it is denoted by σ-dimP . Further-more, d = deg φP (t) and the coefficient ad are called the difference (or σ-) typeand the typical difference (or the typical σ-) dimension of P .

Similarly, if K is an inversive difference field with a basic set σ =α1,. . . ,αn, Q is a prime σ∗-ideal in an algebra of σ∗-polynomials Ky1,. . ., ys∗and ζ = (ζ1, . . . , ζs) is a generic zero of Q, then the σ∗-dimension polynomialψζ|K(t) of the σ∗-field extension K〈ζ1, . . . , ζs〉∗/K is called the σ∗-dimensionpolynomial of the ideal Q and denoted by ψQ(t). If this polynomial is written

in the form (4.2.6), ψQ(t) =n∑

i=0

bi2i

(t + i

i

), then the coefficient bn (which is

equal to σ-trdegKK〈η1, . . . , ηs〉∗/K) is called the inversive difference (or σ∗-)dimension of Q; it is denoted by σ∗-dimQ. Furthermore, d = deg ψQ(t) andthe coefficient bd are called the inversive difference (or σ∗-) type and the typicalinversive difference (or the typical σ∗-) dimension of Q.

Note that since every two generic zeros of a prime inversive difference idealP in an algebra of σ- or σ∗- polynomials difference are equivalent (see Proposi-tion 2.2.4), so the σ- and σ∗- dimension polynomials of P are well-defined.

Proposition 4.2.22 Let K be a difference field with a basic set σ=α1,. . . ,αnand Ky1, . . . , ys an algebra of σ-polynomials in σ-indeterminates y1, . . . , ys

over K. Let P and Q be prime inversive difference ideals in Ky1, . . . , ys suchthat P Q. Then φQ(t) ≺ φP (t).


Similarly, if the σ-field K is inversive and P and Q are two prime σ∗-idealsin an algebra of σ∗-polynomials Ky1, . . . , ys∗ over K such that P Q, thenψQ(t) ≺ ψP (t).

PROOF. Let P and Q be prime inversive difference ideals of Ky1, . . . , yssuch that P Q. Furthermore, for every r ∈ N, let Ar denote the polynomialring K[τyi | τ ∈ T (r), 1 ≤ i ≤ s] and let Pr and Qr,denote the prime idealsP ∩ Ar and Q ∩ Ar of this ring, respectively. Finally, let Fr and Gr denote thedifference fields of quotients of the σ-rings Ar/Pr and Ar/Qr, respectively. SinceP Q, Pr Qr for all sufficiently large r ∈ N. It follows that trdegKFr <trdegKGr for all sufficiently large r ∈ N (see Theorem 1.2.25(ii) ) hence φQ(t) ≺φP (t). The statement about the inequality of σ∗-dimension polynomials can beobtained in the same way.

Exercise 4.2.23 Let K be a G-field of zero characteristic (where G is a finitelygenerated commutative group acting on the field K) and let L = K〈η1, . . . , ηs〉Gbe a G-field extension of K generated by a family η = (η1, . . . , ηs) (we use theterminology and assumptions considered in the last part of section 3.5). Provethat there exists a numerical polynomial θ(t) with the following properties:

(i) θ(t) = trdegKK(G(r)η1

⋃ · · ·⋃G(r)ηs) for all sufficiently large r ∈ Z.(ii) deg θ(t) ≤ n where n = rank G, and the polynomial θ(t) can be written

as θ(t) =n∑

i=0

ai

(t + i

i

)where a0, . . . , an ∈ Z and 2n|an.

(iii) Let G0 be any free commutative subgroup of G such that rank G0 =rank G. Then

an

2nis equal to the maximal number of of elements ζ1, . . . , ζp ∈ L

such that the set g(ζi) | g ∈ G0, 1 ≤ i ≤ p is algebraically independent over K.(This number is called the G-transcendence degree of L over K and is denotedby G-trdegKL.)

4.3 Limit Degree

In this section we define and study an invariant of a difference field extensionthat play an important role in the theory of systems of algebraic difference equa-tions. This invariant, called the limit degree of the extension, was introduced byR. Cohn [39] for ordinary case; the correspondent concept for partial differencefield extensions was defined by P. Evanovich [61]. In what follows we present amodified version of the Evanovich’s approach.

Let K be a difference field with a basic set σ = α1, . . . , αn and let Tdenote the free commutative multiplicative semigroup generated by the elementsα1, . . . , αn. In what follows we shall consider an order on T such that

τ = αk11 . . . αkn

n τ ′ = αl11 . . . αln

n if and only if (kn, . . . , k1) ≤lex (ln, . . . , l1).

Obviously, T is well-ordered with respect to .

4.3. LIMIT DEGREE 275

For any r1, . . . , rn ∈ N, we set T(r1, . . . , rn) = τ ∈ T | τ αr11 . . . αrn

n and extend this notation to the case when the symbol ∞ replaces some ri (withthe condition k < ∞ for any k ∈ N).

Let L = K〈S〉 be a difference (σ-) field extension of a difference field Kgenerated by a finite set S. As usual, if T ′ ⊆ T and A ⊆ L, then T ′(A) (or T ′A )denotes the set τ(a) | a ∈ A, τ ∈ T ′. Furthermore, if σ′ ⊂ σ, then K can benaturally treated as a difference field with a basic set σ′. A σ′-field extension ofthis σ′-field generated by a set S will be denoted by K〈S〉σ′ .

If (r1, . . . , rn) ∈ Nn and r1 ≥ 1, then the degree of the field extensionK(T(r1,. . ., rn)(S))/K(T(r1−1,. . ., rn)(S)) will be denoted by d(S; r1,. . ., rn).(Obviously, d(S; r1, . . . , rn) is either a non-negative integer or ∞.)

Lemma 4.3.1 With the above notation, d(S; r1, . . . , rn) ≥ d(S; r1+p1, . . . , rn+pn) for every (p1, . . . , pn) ∈ Nn.

PROOF. Clearly, it is sufficient to prove that d(S; r1, . . . , rn) ≥ d(S; r1 +1, . . . , rn) and d(S; r1, . . . , rn) ≥ d(S; r1, . . . , ri−1, ri + 1, ri+1, . . . , rn) forevery i = 2, . . . , n. To prove the first inequality, notice that d(S; r1, . . . , rn)= α1(K)(α1T(r1, . . . , rn)S) : α1(K)(α1T(r1 − 1, . . . , rn)S) ≥ K(α1

T(r1, . . . , rn)S⋃

T (0, r2, . . . , rn)S) : K(α1T(r1 − 1, . . . , rn)S⋃

T(0, r2, . . . ,rn)S) = K(T(r1 + 1, r2, . . . , rn)S) : K(T(r1, . . . , rn)S) (we use the inequali-ties from Theorem 1.6.2(iv)), so that d(S; r1, . . . , rn)≥ d(S; r1 + 1, . . . , rn). Similarly, if 2 ≤ i ≤ n, then d(S; r1, . . . , rn)= αi(K)(αiT(r1, . . . , rn)S) : αi(K)(αiT(r1 − 1, r2 . . . , rn)S) ≥ K(αi

T(r1, . . . , rn)S)⋃

T(∞, . . . ,∞, 0, ri+1, . . . , rn)S) :K(αiT(r1−1, r2, . . . , rn)S)⋃T(∞, . . . ,∞, 0, ri+1, . . . rn)S) = d(S; r1, . . . , ri−1, ri + 1, ri+1, . . . , rn). This

completes the proof.

The last lemma implies that if d(S; r1, . . . , rn) is finite for some (r1, . . . , rn) ∈Nn, then mind(S; r1, . . . , rn) | (r1, . . . , rn) ∈ Nn is finite. In this case we de-note this minimum value of d(S; r1, . . . , rn) by d(S).

If d(S; r1, . . . , rn) = ∞ for all (r1, . . . , rn) ∈ Nn, we set d(S) = ∞.The following statement shows that d(S) does not depend on the system

of generators S of L/K. Therefore, d(S) can be considered as a characteristicof this finitely generated difference extension; it is called the limit degree ofL/K and denoted by ld(L/K). If a difference field extension L/K is not finitelygenerated, then its limit degree ld(L/K) is defined as the maximum of limitdegrees of finitely generated difference subextensions N/K (K ⊆ N ⊆ L) ifthis maximum exists, or ∞ if it does not. As it follows from the multiplicativeproperty of limit degree proved below, if a difference field extension L/K isfinitely generated, then ld(L/K) is also the maximum of the limit degrees of thefinitely generated subextensions of L/K (including L/K itself.)

Lemma 4.3.2 Let K be a difference field with a basic set σ = α1, . . . , αn,and let S and S′ be two finite systems of generators of a difference (σ-) fieldextension L/K, that is, L = K〈S〉 = K〈S′〉. Then d(S) = d(S′).


PROOF. Let d=d(S), e=d(S′), and let d(S; r1, . . . , rn) and d(S′; r1, . . . , rn)denote the degrees K(T(r1, . . . , rn)(S)) : K(T(r1 − 1, . . . , rn)S) andK(T(r1, . . . , rn)S′) : K(T(r1 − 1, r2 . . . , rn)S′), respectively.

It is easy to see that there exists a positive integer h such that K(S) ⊆K(T(h, . . . , h)S′) and K(S′) ⊆ K(T(h, . . . , h)S). Since T(k1, . . . , kn)T(l1,. . . , ln) = T(k1 + l1, . . . , kn + ln) for any (k1, . . . , kn), (l1, . . . , ln) ∈ Nn, the lasttwo inclusions imply that

K(T(r1, . . . , rn)S) ⊆ K(T(r1 + h, . . . , rn + h)S′) (4.3.1)

andK(T(r1, . . . , rn)S′) ⊆ K(T(r1 + h, . . . , rn + h)S) (4.3.2)

for every (r1, r2 . . . , rn) ∈ Nn.Assume, first, that d < ∞ and e < ∞. Then there exists m ∈ N such that d =

K(T(m, . . . , m)S) : K(T(m − 1, m, . . . , m)S) and e = K(T(m, . . . , m)S′) :K(T(m − 1, m, . . . , m)S′) whence

d = K(T(m + p1, . . . , m + pn)S) : K(T(m + p1 − 1,m + p2, . . . , m + pn)S)(4.3.3)

and

e = K(T(m + p1, . . . , m + pn)S) : K(T(m + p1 − 1,m + p2, . . . , m + pn)S)(4.3.4)

for every (p1, . . . , pn) ∈ Nn.Therefore, for every integer k > h we have

K(T(m + k + h, . . . , m + k + h)S) : K(T(m, . . . , m)S) = dn(k+h) (4.3.5)

and

K(T(m + k, . . . , m + k)S′) : K(T(m + h, . . . ,m + h)S′) = en(k−h) . (4.3.6)

Furthermore, the inclusions (4.3.1) and (4.3.2) imply that

K(T(m + k, . . . , m + k)S′) ⊆ K(T(m + k + h, . . . ,m + k + h)S) (4.3.7)

andK(T(m, . . . , m)S) ⊆ K(T(m + h, . . . ,m + h)S′). (4.3.8)

Combining (4.3.7) and (4.3.8) with the equalities (4.3.5) and (4.3.6) we ob-tain that

dn(k+h) ≥ en(k−h) (4.3.9)

for all k > h. Allowing k → ∞ in the last inequality we arrive at the inequalityd ≥ e. Similarly we can obtain that e ≥ d whence e = d. Furthermore, inclusions(4.3.7) and (4.3.8) (and inclusions obtained from (4.3.7) and (4.3.8) by inter-changing S and S′) imply that if one of the values d(S), d(S′) is finite, thenthe other value is also finite and d(S) = d(S′). This completes the proof of thelemma.

The following proposition gives some relationships between the limit degreeand difference transcendence degree.


Proposition 4.3.3 Let K be a difference field with a basic set σ = α1, . . . , αnand L a σ-field extension of K. Then

(i) If σ-trdegKL > 0, then ld(L/K) = ∞.(ii) If the σ-field extension L/K is finitely generated and σ-trdegKL = 0, then

ld(L/K) < ∞.

PROOF. (i) Suppose that σ-trdegKL > 0, but ld(L/K) = d < ∞.Then L contains an element η which is σ-transcendental over K. Obviously,ld(K〈η〉/K) ≤ ld(L/K) < ∞, so there exists (r1, . . . , rn) ∈ Nn such thatK(T(r1, . . . , rn)η) : K(T(r1−1, r2, . . . , rn)η) < ∞. In this case αr1 . . . αrn

n (η)is algebraic over the field K(T(r1 − 1, . . . , rn)η) that contradicts the fact thatη is σ-transcendental over K.

(ii) Let L = K〈S〉 for some finite set S ⊆ L and let σ-trdegKL = 0. Assume,first that CardS = 1, S = η. The order on T naturally determines a well-ordering of the set of all terms τy (τ ∈ T ) in the ring of difference polynomialsKy in one difference indeterminate y. (Using the same symbol for thiswell-ordering, we have τy τ ′y if and only if τ τ ′.) Since η is σ-algebraicover K, there exists a difference polynomial P ∈ Ky such that P (η) = 0(as before, P (η) denotes the element of L obtained by replacing every τy inP by τη). Let u = τy be the greatest term of P with respect to , and letτ = αr1

1 . . . αrnn where r1, . . . , rn ∈ N, r1 ≥ 1 (if r1 = 1, we can replace P by

α1P ). Then the element τ(η) is algebraic over K(T(r1 − 1, . . . , rn)η) henceK(T(r1, . . . , rn)η) : K(T(r1 − 1, . . . , rn)η) < ∞, hence ld(L/K) < ∞.

Now let CardS > 1, S = η1, . . . , ηm. As we have seen, for everyi = 1, . . . , m, there exist ri1, . . . , rin ∈ N such that K(T (ri1, . . . , rin)ηi) :K(T (ri1 − 1, . . . , rin)ηi) < ∞. By Lemma 4.3.1, there exists a sufficientlylarge r ∈ N such that K(T(r, . . . , r)ηi) : K(T(r − 1, r, . . . , r)ηi) < ∞ for

i = 1, . . . , m. Then K(m⋃

i=1

T(r, . . . , r)ηi) : K(m⋃

i=1

T(r − 1, r, . . . , r)ηi) < ∞(see Theorem 1.6.2(iii)), hence ld(L/K) < ∞.

Theorem 4.3.4 Let K be a difference field with a basic set σ, M a σ-field ex-tension of K and L an intermediate σ-field of this extension. Then ld(M/K) =[ld(M/L)][ld(L/K)].

PROOF. If σ-trdegKM > 0, then ld(M/K) = ∞ (see Proposition 4.3.3) andat leastoneof thenumbersσ-trdegLM ,σ-trdegKL is positive, so thecorrespondinglimit degree is ∞ and the statement of the theorem is true. Thus, we can assumethat σ-trdegKM = 0, so that every element of M is σ-algebraic over K.

Suppose, first, that L and M are finitely generated σ-field extensions of K.Let L = K〈X〉, M = K〈Y 〉 (X and Y are some finite sets), and let d = ld(L/K),e = ld(M/L), and f = ld(M/K). (Because of our assumption, d, e, and f arefinite). Obviously, M = K〈X ⋃

Y 〉 and there exist p1, . . . , pn ∈ N such that


d = K(T(r1, . . . , rn)X) : K(T(r1 − 1, r2, . . . , rn)X),

e = L(T(r1, . . . , rn)Y ) : L(T(r1 − 1, r2, . . . , rn)Y ),

and

f = K(T(r1, . . . , rn)(X ∪ Y )) : K(T(r1 − 1, r2, . . . , rn)(X ∪ Y ))

for all r1, . . . , rn with (p1, . . . , pn) ≤P (r1, . . . , rn). (≤P denotes the productorder on Nn). It follows that for any (h1, . . . , hn) ∈ Nn,

K(T(r1 + h1, . . . , rn + hn)X) : K(T(r1, . . . , rn)X) = dh1+···+hn ,

L(T(r1 + h1, . . . , rn + hn)Y ) : L(T(r1, . . . , rn)Y ) = eh1+···+hn ,

and

K(T(r1 + h1, . . . , rn + hn)(X ∪ Y )) : K(T(r1, . . . , rn)(X ∪ Y )) = fh1+···+hn .

Let us show that there exists a finitely generated field (not necessarily a σ-field)subextension N/K of L/K such that

N(T (p1 + 1, p2, . . . , pn)Y ) : N(T (p1, . . . , pn)Y ) = e. (4.3.10)

Indeed, applying Theorem 1.6.2(v) to the sequence of field extensions K((T(p1, . . . , pn)Y ) ⊆ L((T(p1, . . . , pn)Y ) ⊆ L(T(p1+1, . . . , pn)Y ), we obtain thatthere exists a finite set W ⊆ L(T(p1, . . . , pn)Y ) such that

K(W ∪ T(p1 + 1, . . . , pn)Y ) : K(W ∪ T(p1, . . . , pn)Y )

= L(T(p1 + 1, . . . , pn)Y ) : L(T(p1, . . . , pn)Y ) = e.

Since W ⊆ L((T(p1, . . . , pn)Y ) = K(⋃τ∈T

τ(X) ∪ T(p1, . . . , pn)Y ), there

exists a finite set Λ ⊆ L such that W ⊆ K(Λ⋃

(T(p1, . . . , pn)Y ). ByTheorem 1.6.2(iv),

e = L(T(p1 + 1, p2 . . . , pn)Y ) : L(T(p1, . . . , pn)Y )

≤ K(Λ⋃

T(p1 + 1, p2, . . . , pn)Y ) : K(Λ ∪ T(p1, . . . , pn)Y )

≤ K(W⋃

T(p1 + 1, p2, . . . , pn)Y ) : K(W⋃

T(p1, . . . , pn)Y ) = e.

Setting N = K(Λ) we satisfy condition (4.3.10). Furthermore, since Λ ⊆ K〈X〉,there exist positive integers q1, . . . , qn such that (p1, . . . , pn) ≤P (q1, . . . , qn) andN ⊆ K(T(q1, . . . , qn)X). Applying Theorem 1.6.2(iv) we obtain that

K(T(q1+1, q2, . . . , qn)X∪T(p1+1, p2, . . . , pn)Y ) : K(T(q1+1, q2, . . . , qn)X∪T(p1, . . . , pn)Y ) ≤ N(T(p1 + 1, p2, . . . , pn)Y ) : N(T(p1, . . . , pn)Y ) = e.

(4.3.11)


and

K(T(q1 + 1, q2, . . . , qn)X ∪ T(p1, . . . , pn)Y ) : K(T(q1, . . . , qn)X)∪T(p1, . . . , pn)Y ) ≤ K(T(q1+1, q2, . . . , qn)X) : K(T(q1, . . . , qn)X) ≤ d.

(4.3.12)Combining (4.3.11) and (4.3.12) we arrive at the inequality

K(T(q1 + 1, q2 . . . , qn)X) ∪ T(p1 + 1, p2 . . . , pn)Y ) : K(T(q1, . . . , qn)X)∪T(p1, . . . , pn)Y ) ≤ de. (4.3.13)

Let U = T(q1, . . . , qn)X⋃

T(p1, . . . , pn)Y . Since X⋃

Y ⊆ U , M = K〈U〉hence

f ≤ K(U ∪α1(U)) : K(U) = K(T(q1+1, . . . , qn)X∪T(p1+1, p2, . . . , pn)Y ) :

K(T(q1, . . . , qn)X ∪ T(p1, . . . , pn)Y ) ≤ de (4.3.14)

In order to prove the opposite inequality f ≥ de, let us consider a tran-scendence basis B of the set T(p1, . . . , pn)Y over K(T(p1, . . . , pn)X) such

that B =n⋃

i=1

Bi where B1 is a transcendence basis of T(p1, . . . , pn)Y over

K(T(p1, . . . , pn)X⋃

T(0, p2, . . . , pn)Y ) and for every i = 2, . . . , n, Bi is atranscendence basis of the field K(T(p1, . . . , pn)X

⋃T(0, . . . , 0, pi, . . . , pn)Y )

over K(T(p1, . . . , pn)X⋃

T(0, . . . , 0, pi−1, . . . , pn)Y ) contained in T(0, . . . , 0,pi, . . . , pn)Y .

Clearly, m = K(T(p1, . . . , pn)X ∪ T(p1, . . . , pn)Y ) : K(T (p1, . . . , pn)X ∪B) is finite, and if h ∈ N then every element of the set T(p1 + h, p2, . . . , pn)Xis algebraic over K(T(p1, . . . , pn)X) (it follows from the fact that d < ∞).Therefore, for any h ∈ N, the set B is algebraically independent over K(T(p1+h, p2, . . . , pn)X) and

K(T(p + h, p2, . . . , pn)X ∪ B) : K(T(p1, . . . , pn)X ∪ B)

= K(T(p1 + h, p2, . . . , pn)X) : K(T(p1, . . . , pn)X) = dh (4.3.15)

Now, using the inclusions

K(T(p1, . . . , pn)X ∪ B) ⊆ K(T(p1, . . . , pn)X ∪ T(p1, . . . , pn)Y )

⊆ K(T(p1 + h, p2, . . . , pn)X ∪ T(p1, . . . , pn)Y )

andK(T(p1, . . . , pn)X) ∪ B) ⊆ K(T(p1 + h, p2, . . . , pn)X ∪ B)

⊆ K(T(p1 + h, p2, . . . , pn)X ∪ T(p1, . . . , pn)Y )

we obtain that

[K(T(p1 + h, p2, . . . , pn)X ∪ T(p1, . . . , pn)Y ) : K(T(p1, . . . , pn)X)∪


T(p1, . . . , pn)Y )] · [K(T(p1 + h, p2, . . . , pn)X ∪B) : K(T(p1, . . . , pn)X ∪B)]

≥ K(T(p1 + h, p2, . . . , pn)X ∪ B) : K(T(p1, . . . , pn)X ∪ B).

The last inequality, together with (4.3.15), implies that

K(T(p1 + h, p2, . . . , pn)X ∪ T(p1, . . . , pn)Y ) : K(T(p1, . . . , pn)X∪

T(p1, . . . , pn)Y ) ≥ dh

m. (4.3.16)

Furthermore, Theorem1.6.2(iv) yields the inequality

K(T(p1+h, p2, . . . , pn)X∪T(p1+h, p2, . . . , pn)Y ) : K(T(p1+h, p2, . . . , pn)Y

∪T(p1, . . . , pn)X) ≥ L(T(p1+h, p2, . . . , pn)Y ) : L(T(p1, . . . , pn)Y ) = eh .(4.3.17)

Combining inequalities (4.3.16) and (4.3.17) we obtain that

fh = K(T(p1 + h, p2, . . . , pn)X ∪ T(p1 + h, p2, . . . , pn)Y ) : K(T(p1, . . . , pn)

× Y ∪ T(p1, . . . , pn)Y ) ≥ dheh

m(4.3.18)

for all h ∈ N. Letting h → ∞ we see that the last inequality can hold only ifde ≤ f . Thus, f = de for i = 1, . . . , n that completes the proof of the theorem forthe case when L/K and M/K are finitely generated difference field extensions.

Now suppose that K ⊆ L ⊆ M is any chain of difference fields andσ-trdegKM = 0. (As we have seen, the general case can be reduced to the casewith this condition.) Let d, e and f be as before. Then the first part of the proofshows that if X and Y are finite subsets of L and M , respectively, then

f ≥ ld(K〈X ∪ Y 〉/K) = [ld(K〈X ∪ Y 〉/K〈X〉)][ld(K〈X〉K)] (4.3.19)

It is clear that K〈X〉(T(r1 + 1, r2, . . . , rn)Y ) : K〈X〉(T(r1, . . . , rn)Y ) ≥L(T(r1 + 1, r2, . . . , rn)Y ) : L(T(r1, . . . , rn)Y ) for any r1, . . . , rn whenceld(L〈Y 〉/L) ≤ ld(K〈X ∪ Y 〉/K〈X〉). This inequality, together with (4.3.19),implies that

f ≥ [ld(L〈Y 〉/L)][ld(K〈X〉/K)] . (4.3.20)

If d < ∞ and e < ∞, one can choose X and Y such that the factors in theright-hand side of (4.3.20) are d and e. If either d or e is ∞, one can choose Xor Y to make the corresponding factor arbitrarily large. In either case we obtainthat f ≥ de.

In order to prove the opposite inequality (and complete the proof of thetheorem), it is sufficient to show that de ≥ f in the case d < ∞, e < ∞(otherwise, de = ∞ ≥ f). In other words, we should prove that if U is a finitesubset of M , then

ld(K〈U〉/K) ≤ de . (4.3.21)

If M/L is finitely generated, then the first part of the proof shows thatld(L〈U〉/L) ≤ ld(M/L) = e. If M/L is not finitely generated, then we also


have ld(L〈U〉/L) ≤ e by the definition of ld(M/L). It follows that there existr1, . . . , rn ∈ N such that L(T(r1 + 1, r2, . . . , rn)U) : L(T(r1, . . . , rn)U) ≤ e.Now, as in the first part of the proof, we obtain that there exists a finite setP ⊆ L such that K(P ∪T(r1 +1, r2, . . . , rn)U) : K(P ∪T(r1, . . . , rn)U) ≤ e.Therefore,

ld(K〈P ∪ U〉/K〈P 〉) ≤ e . (4.3.22)

From the first part of the proof, if L/K is finitely generated, or by the definitionof ld(L/K) otherwise, we have

ld(K〈P 〉/K) ≤ ld(L/K) = d . (4.3.23)

Using the already proven statement for finitely generated extensions and(4.3.22), (4.3.23), we can evaluate ld(K〈P ∪ U〉/K) in two ways as follows:

ld(K〈P ∪ U〉/K) = [ld(K〈P ∪ U〉/K〈P 〉)][ld(K〈P 〉/K)] ≤ de (4.3.24)

and

ld(K〈P ∪ U〉/K) = [ld(K〈P ∪ U〉/K〈U〉)][ld(K〈U〉/K)]. (4.3.25)

Combining (4.3.24) and (4.3.25) we obtain that ld(K〈U〉/K) ≤ de. Thiscompletes the proof of the theorem.

Corollary 4.3.5 Let K be a difference field with a basic set σ = α1, . . . , αnand K∗ the inversive closure of K. Then ld(K∗/K) = 1.

PROOF. If S is any finite subset of K∗, then there exist k1, . . . , kn ∈ Nsuch that αk1

1 . . . αknn (S) ⊆ K. It follows that for every (r1, . . . , rn) ∈ Nn with

ri > ki (i = 1, . . . , n), K(T(r1, . . . , rn)S) : K(T(r1 − 1, r2, . . . , rn)S) = 1.Therefore, ld(K∗/K) = 1.

Corollary 4.3.6 Let K be a difference field with a basic set σ = α1, . . . , αnand L a σ-field extension of K. Let L∗ be the inversive closures of L and K∗ ⊆L∗ the inversive closure of K. Then

ld(L∗/K) = ld(L∗/K∗) = ld(L/K).

PROOF. Applying Theorem 4.3.4 to the chains of difference fields K ⊆L ⊆ L∗ and K ⊆ K∗ ⊆ L∗ and using the result of Corollary 4.3.5, weobtain that ld(L∗/K) = [ld(L/K)][ld(L∗/L)] = ld(L/K) and ld(L∗/K) =[ld(K∗/K)][ld(L∗/K∗)] = ld(L∗/K∗).

Using the separable degree of field extensions and its properties (see The-orem 1.6.23), one can define another characteristic of a difference field exten-sion. Let K be a difference field with a basic set σ = α1, . . . , αn and L〈S〉a σ-field extension of K generated by a finite set S. Setting e(S; r1, . . . , rn) =[K(T(r1, . . . , rn)(S)) : K(T(r1 − 1, r2 . . . , rn)(S))]s for any r1, . . . , rn ∈ N(e(S; r1, . . . , rn) is either a non-negative integer or ∞), we can repeat the proof


of Lemmas 4.3.1 and show that e(S; r1, . . . , rn) ≥ e(S; r1 + p1, . . . , rn + pn)for any n-tuples (r1, . . . , rn), (p1, . . . , pn) ∈ Nn. Now we can set e(S) =mind(S; r1, . . . , rn) | (r1, . . . , rn) ∈ Nn and use the arguments of the proof ofLemma 4.3.2 to show that e(S) = e(S′) for any other finite set of σ-generatorsof L/K. It follows that e(S) can be considered as a characteristic of the differ-ence field extension; it is called the reduced limit degree of L/K and denoted byrld(L/K).

If a difference field extension L/K is not finitely generated, then its reducedlimit degree rld(L/K) is defined as the maximum of reduced limit degrees offinitely generated difference subextensions N/K (K ⊆ N ⊆ L) if this maximumexists, or ∞ if it does not. Of course, in the case of difference fields of zerocharacteristic the concepts of reduced limit degree and limit degree are identical.Furthermore, it is easy to see that if Char K = p > 0, then d(S; r1, . . . , rn)(considered in the definition of the limit degree of L = K〈S〉 (CardS < ∞)over K) is a multiple of e(S; r1, . . . , rn) by a (non-negative integer) power of pand the same can be said about d(S) and e(S). Therefore, in this case ld(L/K)is a multiple of rld(L/K) by a power of p.

Using the arguments of the proof of Theorem 4.3.4 one can easily obtainthe following multiplicative property of the reduced limit degree and its conse-quence.

Theorem 4.3.7 Let K be a difference field with a basic set σ, M a σ-field ex-tension of K and L an intermediate σ-field of this extension. Then rld(M/K) =[rld(M/L)][rld(L/K)].

Corollary 4.3.8 Let K be a difference field with a basic set σ and L a σ-fieldextension of K. Furthermore, let L∗ be the inversive closures of L and K∗ ⊆ L∗

the inversive closure of K. Then

rld(L∗/K) = rld(L∗/K∗) = rld(L/K).

Let K be a difference field with a basic set σ = α1, . . . , αn and let ∆ =αi1 , . . . , αik

(1 ≤ k ≤ n) be a subset of σ whose elements are automorphismsof K. If we replace each αj ∈ ∆ by α−1

j and do not change elements of σ \ ∆,then the resulting set will be denoted by σ∆. Obviously, K can be treated as adifference field with the basic set σ∆. If L is a σ-field extension of K such thatthe extensions of αj ∈ ∆ are automorphisms of L, then L can be also treatedas a σ∆-field extension of the σ∆-field K. The limit degree and reduced limitdegree of this extension are called, respectively, the σ∆-limit degree and reducedσ∆-limit degree of L/K. They are denoted by σ∆-ld(L/K) and σ∆-rld(L/K),respectively.

Exercise 4.3.9 With the above notation, prove that if M is a σ-field extensionof L such that the extensions of αj ∈ ∆ are automorphisms of M , then

[σ∆-ld(M/K) = [σ∆-ld(L/K)][σ∆-ld(M/L)] and[σ∆-rld(M/K) = [σ∆-rld(L/K)][σ∆-rld(M/L)].


The following example illustrates that the definition of the limit degree of apartial difference field extension essentially depends on the order of translationsin the basic set.

Example 4.3.10 Let K be a difference field with a basic set σ = α1, α2 andlet Ky be the ring of σ-polynomials in one difference indeterminate y over K.In what follows we consider the set of terms Ty = τy | τ ∈ T together withits two rankings ≺1 and ≺2 such that αk1

1 αk22 y ≺i αl1

1 αl22 y if and only if ki < li

or ki = li and k2−i < l2−i (i = 1, 2).Let f = (α1y)2 + α1y + α2y + y ∈ Ky. Then u = α1y and v = α2y

are leaders of f with respect to the rankings ≺1 and ≺2, respectively. Sincef is linear with respect to v, this σ-polynomial is irreducible and so is anyσ-polynomial τf (τ ∈ T ). Furthermore, it is easy to see that the ideal [f ] isprime. Indeed, suppose that g1g2 ∈ [f ] where g1, g2 ∈ Ky \ [f ]. Consideringthe set of terms Ty together with the ranking ≺2 and applying the reductiontheorem (Theorem 2.4.1), we obtain that there exist nonzero σ-polynomialsg′1, g

′2 ∈ Ky such that g′i is reduced with respect to f and gi − g′i ∈ [f ]

(i = 1, 2). It follows that no term of the form αr1α

s2y with s ≥ 1 appears in g′1

or g′2, hence such a term cannot appear in g′1g′2. On the other hand, g′1g

′2 ∈ [f ]

hence degvg′1g′2 ≥ 1 (see Theorem 2.4.15). This contradiction shows that the

difference ideal [f ] is prime.Let L = K〈η〉 be a σ-field extension of K generated by a single element η

with the defining equation

(α1η)2 + α1η + α2η + η = 0.

(In other words, [f ] is the defining ideal of the generator η over K.) In whatfollows, if S is a set of elements in a σ-field extension of K, then K〈S〉αi willdenote the ordinary difference field extension of K generated by the set S whenK is treated as an ordinary difference field with the basic set αi (i = 1, 2).

Let us compute the limit degrees of L over K with respect to two orders ofthe set T that correspond to the rankings ≺1 and ≺2 of Ty (we shall use thesame symbols for these orders): αk1

1 αk22 ≺i αl1

1 αl22 y if and only if ki < li or

ki = li and k2−i < l2−i (i = 1, 2).For any k, l ∈ N, let Ti(k, l) = τ ∈ T | τ ≺i αk

1αl2 and denote the limit

degree of the extension L/K with respect to ≺i by ldi(L/K) (i = 1, 2). Thenthere exist r, s ∈ N+ such that ld1(L/K) = K(T2(r, s)η) : K(T2(r−1, s)η) andld2(L/K) = K(T1(r, s)η) : K(T1(r, s − 1)η).

Since f(η) = (α1η)2 + α1η + α2η + η = 0, α2η ∈ K〈η〉α1 and αr1α

s2η ∈

K〈η〉α1. Therefore, ld1(L/K) = 1. Let us show that ld2(L/K) = 2. First weapply the endomorphism αr−1

1 αs2 to the left-hand side of the defining equation

for η and obtain that αr1α

s2η satisfy a polynomial equation of second degree

over the field K(T1(r, s− 1)η). Now it remains to show that αr1α

s2η does not lie

in K(T1(r, s − 1)η). Suppose that αr1α

s2η ∈ K(T1(r, s − 1)η) and let w denote

the term αr1α

s2y ∈ Ty. Then there exists a σ-polynomial h = Aw + B ∈ Ky

such that A,B ∈ K[T (r, s − 1)y], A /∈ [f ] (that is, A(η) = 0), and h(η) = 0,so that h ∈ [f ]. Let us apply the reduction process described in the proof of


the reduction theorem (Theorem 2.4.1) to h. We obtain a σ-polynomial h1 ∈ [f ]which is reduced with respect to f and can be written as h1 = A1w+B1 ∈ Kywhere A1, B1 ∈ K[T (r, s−1)y] and A1 /∈ [f ]. On the other hand, Theorem 2.4.15shows that [f ] contains no nonzero σ-polynomial reduced with respect to f .Thus, αr

1αs2η /∈ K(T1(r, s − 1)η) whence ld2(L/K) = 2.

In what follows we consider some peculiar results on limit degree of ordinarydifference field extensions.

Let K be an ordinary difference field with a basic set σ = α and L =K〈S〉 a σ-field extension of K generated by a finite set S. Let S0 = S andfor every k = 1, 2, . . . , let Sk denote the set αi(s) | s ∈ S, 0 ≤ i ≤ k anddk = K(Sk) : K(Sk−1). Then Lemma 4.3.1 shows that dk = α(K)(α(Sk) ) :α(K)(α(Sk−1) ) ≥ K(S

⋃α(Sk) ) : K(S

⋃α(Sk−1) ) = dk+1 for k = 1, 2, . . . .

Let d(S) = mindk | k = 1, 2, . . . if some dk is finite, or d(S) = ∞ if all dk

are infinite. By Lemma 4.3.2, d(S) does not depend on the system of differencegenerators S of L/K; as before, this invariant of the extension L/K is called thelimit degree of the extension and denoted by ld(L/K). As in the case Card σ > 1,if L/K is not finitely generated, its limit degree ld(L/K) is defined to be themaximum of the limit degrees of all finitely generated difference subextensionsof L/K, if this maximum exists, or ∞ if it does not. Of course, limit degreeof ordinary difference field extensions has the properties listed in Proposition4.3.3, the multiplicative property (see Theorem 4.3.4), and the properties statedin Corollaries 4.3.5 and 4.3.6. Also, the use of the separable degree of fieldextensions instead of the degree, leads to the concept of reduced limit degree ofan ordinary field extension L/K and its properties formulated in Theorem 4.3.7and Corollary 4.3.8.

If K is an inversive difference field with a basic set σ = α, then K canbe also treated as a difference field with the basic set σ′ = α−1 called theinverse difference field of K. It is denoted by K ′. Let L be a σ-field extensionof K and L′ the inverse difference field of L (so that L′ is a σ′-field extension ofK ′). The inverse limit degree of L over K is defined to be ld L′/K ′ (it is denotedby ildL/K), and the inverse reduced limit degree of L over K is defined to berldL′/K ′ (it is denoted by irldL/K).

Exercises 4.3.11 Let K be an ordinary difference field with a basic set σ.1. Prove that if M is a σ-field extension of K and L an intermediate

σ-field of M/K, then ild(M/K) = [ild(M/L)][ild(L/K)] and irld(M/K) =[irld(M/L)][irld(L/K)] (cf. Exercise 4.3.9).

2. Let L be a σ-field extension of K, L∗ the inversive closures of L and K∗ ⊆L∗ the inversive closure of K. Prove that rld(L∗/K) = rld(L∗/K∗) = rld(L/K).

Proposition 4.3.12 Let K be an ordinary difference field with a basic setσ = α.

(i) If L is a finitely generated σ-field extension of K, then ld(L/K) = 1 ifand only if L = K(V ) for some finite set V ⊆ L.

(ii) The following two statements are equivalent:


(a) L/K is a finitely generated σ-field extension, L is algebraic over K, andld(L/K) = 1.

(b) L : K is finite.

(iii) If B is a set of elements in some σ-overfield of L, then ld(L〈B〉/K〈B〉) ≤ld(L/K) and ld(L〈B〉/L) ≤ ld(K〈B〉/K).

PROOF. (i) Let L = K〈S〉 and ld(L/K) = 1. Then there exists a non-negative integer p such that K(

⋃p+h+1i=0 αi(S)) : K(

⋃p+hi=0 αi(S)) = 1, that is,

K(⋃p+h+1

i=0 αi(S)) = K(⋃p+h

i=0 αi(S)) for all h ∈ N. It follows that L = K〈S〉 =K(

⋃∞i=0 αi(S)) = K(

⋃pi=0 αi(S)), so that L is generated over K by the finite

set V =⋃p

i=0 αi(S).Conversely, if L = K(V ) for some finite set V , then K(V

⋃α(V )) : K(V ) =

1, hence ld(L/K) = 1.(ii) It is easy to see that statement (b) implies (a). Conversely, if a σ-field

extension L/K is finitely generated, L is algebraic over K and ld(L/K) = 1,then the first part of the theorem shows that L = K(V ) for some finite set V .Since L/K is algebraic, L : K < ∞ (see Theorem 1.6.4(ii)).The last statement of the proposition is a direct consequence of the definitionof limit degree and Theorem 1.6.2(iv). Proposition 4.3.13 Let K be an ordinary difference field with a basic set σ =α and let L be an algebraic σ-field extension of K. Then ld(L/K) = ild(L/K)and rld(L/K) = irld(L/K).

PROOF. It is easy to see that if L∗ and K∗ ⊆ L∗ are the inversive closuresof L and K, respectively, then the field extension L∗/K∗ is algebraic. Sinceld(L∗/K∗) = ld(L/K), rld(L∗/K∗) = rld(L/K) (see Corollaries 4.3.6, 4.3.8)and similar equalities hold for ild(L/K) and irld(L/K) (see Exercises 4.3.11),we can assume from the very beginning that the difference fields K and L areinversive. Furthermore, without loss of generality we can assume that L is afinitely generated σ-field extension of K, L = K〈S〉 for some finite set S.

Let K ′ denote the inverse difference field of K, and let L′ be the inversedifference field of the inversive closure of L. Then L′ is the inversive closure ofK ′〈S〉, so ld(L′/K ′) = ld(K ′〈S〉/K ′). Let d = ld(L/K) and d′ = ild(L/K) =ld(L′/K ′) = d(K ′〈S〉/K ′).

By the definition of the limit degree, there exists p ∈ N such that forevery h = 1, 2, . . . we have d = K(

⋃p+hi=0 αi(S)) : K(

⋃p+h−1i=0 αi(S)). Therefore,

K(⋃p+h

i=0 αi(S)) : K(⋃p

i=0 αi(S)) = dh and K(⋃p+h

i=0 α−i(S)) : K(⋃p

i=0 α−i(S))= (d′)h.

Setting e = K(⋃p

i=0 αi(S)) : K and e′ = K(⋃p

i=0 α−i(S)) : K (as degreesof finitely generated algebraic field extensions, e and e′ are finite), we obtain thatfor every positive integer h, K(

⋃p+hi=0 αi(S)) : K = edh and K(

⋃p+hi=0 α−i(S)) :

K = e′(d′)h. Since αp+h is an automorphism of K and an isomorphism ofK(

⋃p+hi=0 α−i(S)) onto K(

⋃p+hi=0 αi(S)), the last two equalities imply that edh =

e′(d′)h. Letting h → ∞ in the last equation we obtain that d = d′. The proof ofthe equality of the reduced limit degrees is similar.


Proposition 4.3.14 Let K be an ordinary inversive difference field with a basicset σ = α and let L be an algebraic difference field extension of K such thatld(L/K) = 1. Then L is inversive.

PROOF. Clearly, it is sufficient to consider the case that L is a finitelygenerated σ-field extension of K. Then L : K = α(L) : α(K) = α(L) : K <∞. It follows that α(L) = L, so the σ-field L is inversive.

Exercise 4.3.15 Let K be an ordinary difference field with a basic set σ andlet L be a finitely generated σ-field extension of K such that L/K is algebraic.Prove that rld(L/K) = 1 if and only if [L : K]s < ∞.

The following example shows that there are proper algebraic ordinary dif-ference field extensions L/K such that Char K = 0 and ld(L/K) = 1.

Example 4.3.16 Let us consider the field of rational numbers Q as an ordinarydifference field whose basic set σ consists of the identity automorphism α. If oneadjoins to Q an element i such that i2 = −1, then the resulting field L = Q(i)together with the identity automorphism (also denoted by α) can be treated asa σ-field extension of Q. Clearly, L/Q is an algebraic finitely generated σ-fieldextension such that ld(L/Q) = 1 (i is the set of σ-geneartors of L/Q and forevery positive integer h, Q(

⋃h+1k=0 αk(i)) : Q(

⋃hk=0 αk(i)) = 1).

Definition 4.3.17 Let K be an ordinary difference field and L a differencefield extension of K. Then the core LK of L over K is defined to be the set ofelements a ∈ L algebraic and separable over K and such that ld(K〈a〉/K) = 1.

It follows from Theorem 4.3.4 and its corollary that the core LK is an inver-sive intermediate σ-field of the extension L/K such that ld(LK/K) = 1. Fur-thermore, Example 4.3.16 shows that LK need not to be K. Also, if Char K = 0or the field extension L/K is separable, Proposition 4.3.12 implies that L = LK

if and only if L : K is finite.

Exercise 4.3.18 With the above notation, show that if L is a finitely generatedσ-field extension of K, then the extension L/LK is purely inseparable if and onlyif [L : K]s < ∞.

Theorem 4.3.19 Let K be an ordinary inversive difference field with a basicset σ = α. Let L be an algebraic difference field extension of K and LK thecore of L over K. Then a ∈ LK if and only if a ∈ K〈α(a)〉. In particular, theσ-field LK is inversive.

PROOF. Let a ∈ LK . Then K〈a〉 : K < ∞ hence there exists r ∈ N suchthat K〈a〉 = K(a, α(a), . . . , αr(a)). Since α is an automorphism of the field K,K(α(a), α2(a), . . . , αr+1(a)) : K = K(a, α(a), . . . , αr(a)) : K = K〈a〉 : K. Itfollows that K(α(a), α2(a), . . . , αr+1(a)) = K〈a〉 whence a ∈ K〈α(a)〉.


Conversely, suppose that a ∈ K〈α(a)〉. Then there exists n ∈ N such thata ∈ K(α(a), . . . , αn+1(a)). Therefore,

K(a, α(a), . . . , αn+1(a)) = K(α(a), . . . , αn+1(a)) . (4.3.26)

Since α is an automorphism of the field K, K(a, α(a), . . . , αn(a)) : K =K(α(a), . . . , αn+1(a)) : K. Furthermore, equality (4.3.26) shows that K(α(a),. . . , αn+1(a)) ⊇ K(a, α(a), . . . , αn(a)). Therefore,

K(α(a), . . . , αn+1(a)) = K(a, α(a), . . . , αn(a)) (4.3.27)

whence K(a, α(a), . . . , αn(a)) = K(a, α(a), . . . , αn(a), αn+1(a)).It follows that ld (K〈a〉 /K) = 1 and a ∈ LK .

Theorem 4.3.20 Let K and L be as in Theorem 4.3.19, Char K = 0, and letL = K〈S〉 where S is a finite subset of L. Then

LK =∞⋂

n=0

K〈αn(S)〉 . (4.3.28)

PROOF, Let a ∈ LK . Repeatedly applying Theorem 4.3.19 we obtain that

a ∈ K〈αn(a)〉 for n = 1, 2, . . . , so that a ∈∞⋂

n=0

K〈αn(a)〉. Since a ∈ K〈S〉,

K〈αn(a)〉 ⊆ K〈αn(S)〉 for all n ∈ N whence∞⋂

n=0

K〈αn(a)〉 ⊆∞⋂

n=0

K〈αn(S)〉 and

a ∈∞⋂

n=0

K〈αn(S)〉. Thus, LK ⊆∞⋂

n=0

K〈αn(S)〉.Now let us prove the opposite inclusion. Without loss of generality we can

assume that the set S consists of a single element b (that is, L = K〈b〉) andK(b, α(b)) : K = d where d denotes the limit degree of L over K. Indeed,let ld(L/K) = K(S, α(S), . . . , αm+1(S)) : K(S, α(S), . . . , αm(S)) for somem ∈ N,m ≥ 1. By the theorem on a primitive element, there exists an ele-ment b ∈ K(S, α(S), . . . , αm(S)) such that K(b) = K(S, α(S), . . . , αm(S)). Itis easy to see that K〈b〉 = K〈S〉 = L, K(b, α(b)) = K(S, α(S), . . . , αm+1(S))

(hence K(b, α(b)) : K(b) = d), and∞⋂

n=0

K〈αn(b)〉 =∞⋂

n=0

K〈αn(S)〉. (Clearly,

K〈αn(b)〉 = K〈αn(S)〉 for n = 1, 2, . . . .)Let s denote the degree of b over K, that is, the degree of the minimal

polynomial of b in the polynomial ring K[X]. Then every element αi(b) (i ∈ N)is algebraic of degree s over K. Furthermore, for any t ∈ N, t ≥ 1,

K(αi(b), αi+1(b), . . . , αi+t(b)) : K(αi(b), αi+1(b), . . . , αi+t−1(b))

= K(b, α(b), . . . , αt(b)) : K(b, α(b), . . . , αt−1(b)) = d.

Therefore, K(αi(b), αi+1(b), . . . , αi+t(b)) : K = sdt and the set

αi(b)k0αi+1(b)k1 . . . αi+t(b)kt | 0 ≤ k0 ≤ s, 0 ≤ kν ≤ d − 1 for ν = 1, . . . , tis a basis of the vector K-space K(αi(b), αi+1(b), . . . , αi+t(b)).


Now, suppose that a ∈∞⋂

n=0

K〈αn(b)〉. Since a ∈ K〈b〉, there exists r ∈ N

such that a ∈ K(b, α(b), . . . , αr(b)), so that a is a finite linear combination withcoefficients from K of elements of the form bk0α(b)k1 . . . αr(b)kr where 0 ≤ k0 ≤s and 0 ≤ kν ≤ d−1 for ν = 1, . . . , r. At the same time, if p ∈ N, p ≥ r+1, thena ∈ K〈αp(b)〉, hence a ∈ K(αp(b), αp+1(b), . . . , αp+q(b)) for some non-negativeinteger q. We assume that q is the smallest such a number.

The last inclusion shows that a can be represented as a linear combinationwith coefficients from K of elements of the form αp(b)k0αp+1(b)k1 . . . αp+q(b)kq

where 0 ≤ k0 ≤ s and 0 ≤ kν ≤ d − 1 for ν = 1, . . . , q. Comparing thetwo representations of a we obtain that if q ≥ 1, then αp+q(b) is a root ofa polynomial of degree at most d − 1 with coefficients from the field K ′ =K(b, α(b), . . . , αp+q−1(b)). This contradicts the fact that K ′(αp+q(b)) : K ′ = d,that is, the minimal polynomial of αp+q(b) over K ′ has degree d. Therefore,

q = 0 and a ∈ K(αp(b)). We arrive at the inclusion a ∈∞⋂

j=p+1

K(αj(b)) which

implies that α(a) ∈∞⋂

j=p+2

K(αj(b)), α2(a) ∈∞⋂

j=p+3

K(αj(b)), . . . .

Since K(αp+s+1(b)) : K = s and a, α(a), . . . , αs(a) ∈∞⋂

j=p+s+1

K(αj(b)) ⊆

K(αp+s+1(b)), elements a, α(a), . . . , αs(a) are linearly dependent over K. Itfollows that there exists the smallest positive integer k such that the ele-ments a, α(a), . . . , αk(a) are linearly dependent over K. Then αk(a) is a linearcombination of a, α(a), . . . , αk−1(a) over K hence K(a, α(a), . . . , αk(a)) =K(a, α(a), . . . , αk−1(a)). It follows that ld(K〈a〉/K) = 1 and a ∈ LK . Thiscompletes the proof of the theorem.

Example 4.3.21 Let K = Q(x, y0, y1, y−1, y2, y−2, . . . ) be the field of rationalfractions in a denumerable set of indeterminates x, y0, y1, y−1, y2, y−2, . . . overQ. Let us consider K as an inversive ordinary difference field with a basic setσ = α where α acts on Q as the identity mapping, α(x) = x, and α(yr) = yr+1

for every r ∈ Z. Let u and ti (i ∈ N) denote√

x and√

yi, respectively, andlet L = K(u, t0, t1, . . . ). (Thus, the field L is obtained by adjoining to K aroot of the polynomial X2 − x ∈ K[X] and the denumerable set of roots of thepolynomials X2 − yi, i ∈ N.)

The field L can be naturally considered as a difference field extension of Kwhere α(u) = u and α(ti) = ti+1 for i = 0, 1, 2, . . . . It is easy to see that theset S = u, t0 generates this σ-field extension, L = K〈S〉 = K〈u + t0〉 and

K〈αn(S)〉 = K〈u + tn〉 for n = 0, 1, 2, . . . . In this case LK =∞⋂

n=0

K〈αn(S)〉 =

∞⋂n=0

K〈u + tn〉 = K(u). (With the notation of the proof of the last theorem,

b = u + t0, s = 4, and d = 2.)


Theorem 4.3.22 Let K(CharK = 0) and L = K〈S〉 (CardS < ∞) be as inTheorem 4.3.19. Then

(i) If ld(L/K) = K(S) : K, then∞⋂

n=0

K〈αn(S)〉 = K, that is, LK = K.

(ii) L = LK if and only if L : K is finite.

PROOF. (i) As in the proof of Theorem 4.3.20, without loss of generalitywe can assume that the set S consists of a single element b such that K(b) :K = K(b, α(b)) : K(b) = d where d = ld (L/K). Furthermore, the argumentsof the same proof show that the set

B = αi1(b)k1 . . . αim(b)km |m ∈ N, 0 ≤ i1 < i2 < · · · < im and0 ≤ kν ≤ d − 1 for ν = 1, . . . , m

is a basis of the vector K-space L = K〈b〉, while for every r ∈ N, its subsetBr = αi1(b)k1 . . . αim(b)km |m ≥ 0; r ≤ i1 < i2 < · · · < im; and 0 ≤ kν ≤ d−1for ν = 1, . . . , m is a basis of the vector K-space K〈αr(b)〉.

If a ∈∞⋂

n=0

K〈αn(S)〉 then a ∈ K〈b〉 hence a = λ1b1 + · · · + λpbp for some

λ1, . . . , λp ∈ K and b1, . . . , bp ∈ B. Let αs(b) be the highest transform of b thatappears in b1, . . . , bp. Since a ∈ K〈αs+1(b)〉, a can be written as a = µ1b

′1 + · · ·+

µqb′q for some µ1, . . . , µq ∈ K and b′1, . . . , b

′q ∈ Bs+1. We arrive at the equality

λ1b1 + · · ·+ λpbp = µ1b′1 + · · ·+ µqb

′q that can hold only if the both expressions

in its sides belong to K. Therefore, a ∈ K, so that∞⋂

n=0

K〈αn(S)〉 = K.

(ii) If L : K is finite, then ld(L/K) = 1 (see Proposition 4.3.12), henceld(K〈a〉/K) = 1 for every a ∈ L, so that L = LK .

Conversely, if L = LK and S = a1, . . . , am, then ld(K〈a1〉/K) = 1 andld(K〈a1, . . . , ai〉/K〈a1, . . . , ai−1〉) = 1 for 1 < i ≤ m (by Proposition 4.3.12(iii), 1 ≤ ld(K〈a1, . . . , ai〉/K〈a1, . . . , ai−1〉) ≤ ld(K〈ai〉/K) = 1). ApplyingTheorem 4.3.4 we obtain that ld(L/K) = 1, and Proposition 4.3.12 shows thatL : K is finite.

Exercise 4.3.23 Let K be an ordinary difference field of arbitrary character-istic and let L be a finitely generated difference field extension of K. Prove thatL is purely inseparable over LK if and only if [L : K]s is finite.

Theorem 4.3.24 Let K be an ordinary difference field with a basic set σ = αand let L be a separably algebraic σ-field extension of K. Furthermore, let K∗

denote the inversive closure of K and let M be a σ-overfield of K∗〈L〉 containedin the inversive closure L∗ of L. Then MK∗ is the inversive closure of LK .

PROOF. If a ∈ MK∗ ⊆ L∗, then there exists r ∈ N such that αr(a) ∈ L.Furthermore, both a and αr(a) are separably algebraic over K (it follows from


the definition of a core). By Proposition 4.3.14, the σ-field K∗〈a〉 is inversive,so we can apply Corollary 4.3.6 and obtain that

ld(K〈αr(a)〉/K) = ld( (K〈αr(a)〉)∗/K∗) = ld(K∗〈a〉/K∗) = 1,

that is, αr(a) ∈ LK and a is in the inversive closure of LK . Furthermore, byTheorem 4.3.18, the σ-field MK∗ is inversive and contains LK (repeating theabove reasonings one can easily obtain that if a ∈ LK , then ld(K∗〈a〉/K∗) = 1).Therefore, MK∗ contains the inversive closure of LK , and hence coincides withthis closure.

The following result provides an important property of finitely generated al-gebraic difference field extensions of limit degree one.

Theorem 4.3.25 Let K be an inversive ordinary difference field with a basicset σ = α, let L be a σ-field extension of K such that L = K〈η1, . . . , ηm〉where elements η1, . . . , ηm are σ-algebraically independent over K (and thereforeσ-trdegKL = m). Let M be a σ-overfield of L such that the field extension M/Lis algebraic and ld(M/L) = 1. Then M is generated by adjoining to L a set ofelements which are algebraic over K.

PROOF. Without loss of generality we may assume that the field K isalgebraically closed in L. With this assumption we should show that M = L.

We proceed by induction on m. Let m = 1, so that L = K〈η〉 where ηis σ-transcendental over K. Let u be an element in M . Since ld(M/L) = 1,there is r ∈ N such that αr+k(u) ∈ K(u, . . . , αt(u)) for any k ∈ N. Let uschoose k so large that the set S of transforms αj(u) occurring in the polynomi-als Irr(αi(u),K), i = 1, . . . , r and the set T of transforms αj(u) occurring inIrr(αr+k(u),K) are disjoint. (It is possible, since Irr(αj(u),K) can be obtainedby applying αj to each coefficient of Irr(u,K). ) Let the rational expression forαr+k(u) in terms of u, . . . , αr(u) and transforms of η be arranged as a rationalfunction in members of T whose coefficients are rational combinations of othertransforms αi(η) and u, . . . , αr(u) with one coefficient unity. Let S′ be the setof transforms αi(u) appearing in these coefficients, and let U = S ∪ S′. ThenU ∩ T = ∅.

Let us adjoin to the field K new algebraically independent sets U (1), U (2), . . . ,such that CardU (i) = CardU for every i = 1, 2, . . . . Taking bijections U → U (i)

(where the image of an element a ∈ U is denoted by ai) and the identicalmapping T → T we can extend them to a K-isomorphism θi : K(U ∪ T ) →K(U (i) ∪ T ) which, in turn, can be extended to a field isomorphism

θi : K(U ∪ T )(u, . . . , αr(u), αr+k(u)) → K(U (i) ∪ T )(ui, . . . , αr(ui), αr+k(ui)).

If the rational expression for αr+k(u) contains a coefficient which is notalgebraic over K, then, since the sets U (i) are algebraically independent over K,this coefficient has distinct images under the isomorphisms θi. Then the αj(ui)are all distinct. Since they are the zeros of Irr(αr+k(u),K) which is unalteredby the isomorphisms, this is impossible. Therefore, every coefficient is algebraic


over K. But these coefficients lie in M , and since K is algebraically closed inM , the coefficients must be in K. Thus, αr+k(u) ∈ L, hence u ∈ L. It followsthat M = L.

To perform the induction step we set L = K〈η1, . . . , ηm〉 and assume thatour statement is true if the number of σ-generators of L over K which form aσ-transcendence basis of L/K is less than m. Then M is generated by adjoiningto L a set Σ of elements algerbaic over K〈η1〉. Since K〈η1〉〈Σ〉 and L are linearlydisjoint over K〈η1〉, ld(K〈η1〉〈Σ〉/K) = 1. Therefore, K〈η1〉〈Σ〉 is generatedby adjoining to L elements algebraic over K. This completes the proof of thetheorem.

As we have mentioned, the first generalization of the concept of limit degreeto the case of partial difference field extensions is due to P. Evanovich [61].Considering a difference field K with a basic set σ = α1, . . . , αn and itsfinitely generated σ-field extension M = K〈S〉, P. Evanovich inductively defineda characteristic ldn(M/K) of this extension as an element of the set N

⋃∞satisfying the following conditions (ld1) - (ld5).

(ld1) If M = L〈S〉 for a finite set S ⊆ M , then there exists a finitely gener-ated σ-overfield K ′ of K contained in L such that ldn(M/L) = ldn(K ′〈S〉/K ′).

(ld2) If S ⊆ M , then ldn(L〈S〉/K〈S〉) ≤ ldn(L/K) and ldn(L〈S〉/L) ≤ldn(K〈S〉/K). Equality will hold in both if S is σ-algebraically independentover L.

(ld3) If there is a σ-isomorphism φ of L onto a σ-field L′ and K ′ is aσ-subfield of L′ such that φ(K) = K ′, then ldn(L/K) = ldn(L′/K ′).

(ld4) If the σ-field extension L/K is finitely generated, then σ-trdegkL = 0if and only if ldn(L/K) < ∞.

(ld5) ldn(M/K) = ldn(M/L) · ldn(L/K).

If n = 1, then ld1 is defined to be the limit degree ld for ordinary differencefields. Suppose that ldn−1 is defined for difference (σ-) field extensions withCard σ = n − 1.

Let L/K be a finitely generated difference field extension with a basic setσ = α1, . . . , αn, L = K〈S〉 for a finite set S ⊆ L. For any m ∈ N, letLm = K〈⋃m

i=1 αin(S)〉σ′ where σ′ = α1, . . . , αn−1. Then Lm/Lm−1 is a fi-

nitely generated σ′-field extension. Applying (ld3) and (ld2) we obtain thatldn−1(Lm/Lm−1) ≥ ldn−1(Lm+1/Lm), hence there exists the limitlimm→∞ldn−1(Lm/Lm−1) = a where a ∈ N or a = ∞.

As in the case of ordinary difference fields (consider the proof of Lemma 4.3.2for n = 1), one can show that a is independent on the choice of σ-generators ofL/K. Now we define ldn(L/K) = a.

If L/K is not finitely generated, ldn(L/K) is defined to be the maximum ofldn(K ′/K) where K ′/K is a finitely generated σ-field subextension of L/K, ifthe maximum exists and ∞ if it does not.

Exercise 4.3.26 With the above notation, prove that ldn(M/K) coincides withthe limit degree of the extension M/K in the sense of Definition 4.3.


Exercise 4.3.27 Let K be an ordinary difference field with a basic set σ = αand let u, v, w be elements in some σ-overfield of K which are σ-algebraicallyindependent over K. Furthermore, let ξ, η and ζ be solutions of the σ-polynomialA = y3+uy2+vy+w ∈ K〈u, v, w〉y in one σ-indeterminate y in some difference(σ-) field containing K. Prove that if L = K〈u, v, w, α(ξ)〉, then A is irreduciblein Ly, ld(L〈ξ〉/L) = 1 and ld(L〈η〉/L) = 2.

4.4 The Fundamental Theorem on FinitelyGenerated Difference Field Extensions

In this section we prove the following result that plays the key role in the studyof finitely generated difference field extensions.

Theorem 4.4.1 Let K be a difference field with a basic set σ = α1, . . . , αn,M a finitely generated σ-field extension of K, and L an intermediate differencefield of M/K. Then the σ-field extension L/K is finitely generated.

PROOF. Let η1, . . . , ηr be a difference (σ-) transcendence basis of M/Land let L′ = L〈η1, . . . , ηr〉. First, we will show that if the σ-field extensionL′/K is finitely generated, then L/K is also finitely generated. Indeed, let L′ =K〈λ1, . . . , λm〉. Then each λi is a quotient of two polynomials in τ(ηk) (τ ∈T, 1 ≤ k ≤ r) with coefficients in L. Let U be the set of all these coefficients.We are going to show that L = K〈U〉. If a ∈ L, then a ∈ L′, so there existtwo polynomials P and Q in τ(ηk) with coefficients in K〈U〉 such that Qa = P .Since the set of all τ(ηk) (τ ∈ T, 1 ≤ k ≤ r) is algebraically independent overL, the coefficients of the corresponding products of τηk in Qa and P must beequal. Since a ∈ L and all the coefficients of P and Q belong to K〈U〉, we obtainthat a ∈ K〈U〉, so L = K〈U〉.

Let B be a σ-transcendence basis of L′/K. Since σ-trdeg(L′/K) ≤σ-trdeg(M/K) < ∞, the set B is finite. Therefore, in order to prove that L/Kis finitely generated, it is sufficient to prove that L′/K〈B〉 is finitely generated.Since σ-trdeg(M/K〈B〉) = σ-trdeg(M/L) + σ-trdeg(L′/K〈B〉) = 0 + 0 = 0,without loss of generality we can assume that σ-trdeg(M/K) = 0.

The equality σ-trdeg(M/K) = 0 implies that ld(M/K) < ∞ and henceld(L/K) < ∞. Therefore, there exists a finite set Φ ⊆ L such that ld(L/K) =ld(K〈Φ〉/K) and hence ld(L/K〈Φ〉) = 1. Clearly, in order to prove the theo-rem, it is sufficient to show that the σ-field extension L/K〈Φ〉 is finitely gener-ated, so without loss of generality we can assume that σ-trdeg(M/K) = 0 andld(L/K) = 1, so that ld(M/K) = ld(M/L).

Now, let M = K〈V 〉 where V is a finite set of σ-generators of M/K. Forevery j = 1, . . . , n, let σj = σ \ αj and for any set S ⊆ M , let K〈S〉σj

denote the σj-field extension of K (when K is treated as a difference field withthe basic set σj .) More general, for any distinct integers i1, . . . , ik ∈ 1, . . . , n(1 ≤ k ≤ n), σi1,...,ik

will denote the set σ \ αi1 , . . . , αik, and a σi1,...,ik

-fieldextension of K generated by a set S ⊆ M will be denoted by K〈S〉σi1,...,ik

.

4.4. FUNDAMENTAL THEOREM 293

Since ld(M/K) = ld(M/L), there exist p1, . . . , pn ∈ N such that

K(T(p1 +r1 +1, p2 +r2, . . . , pn +rn)V ) : K(Tp1 +r1, p2 +r2 . . . , pn +rn)(V )

= L(T(p1+r1+1, p2+r2, . . . , pn+rn)V ) : L(T(p1+r1, p2+r2 . . . , pn+rn)V )

= ldk(M/L) < ∞. (4.4.1)

for all r1, . . . , rn ∈ N.We are going to show that

L ⊆ K〈p1⋃

i=0

αi1(V )〉σ1

⋃· · ·

⋃K〈

pn⋃i=0

αin(V )〉σn

. (4.4.2)

Let us denote the set in the right-hand side of (4.4.2) by N and suppose thatL N , so that there exists η ∈ L \ N . Since η ∈ M = K〈⋃∞

i=0 αin(V )〉σn

andη /∈ K〈⋃pn

i=0 αin(V )〉σn

, there exists hn ∈ N such that

η ∈ K〈pn+hn+1⋃

i=0

αin(V )〉σn

\ K〈pn+hn⋃

i=0

αin(V )〉σn

. (4.4.3)

Since η ∈ K〈⋃pn+hn+1i=0 αi

n(V )〉σn= K〈⋃∞

j=0

⋃pn+hn+1i=0 αj

n−1αin(V )〉σn,n−1 and

η /∈K〈⋃pn−1j=0

⋃pn+hn+1i=0 αj

n−1αin(V )〉σn,n−1 (the last set is contained in N).

η ∈ K〈∞⋃

j=pn−1+hn−1+1

pn+hn+1⋃i=0

αjn−1α

in(V )〉σn,n−1\

K〈∞⋃

j=pn−1+hn−1

pn+hn+1⋃i=0

αjn−1α

in(V )〉σn,n−1 . (4.4.4)

The inclusions (4.4.3) and (4.4.4) show that

η ∈ K(T(∞, . . . ,∞, pn−1 + hn−1 + 1, pn + hn + 1)V )\K(T(∞, . . . ,∞, pn−1 + hn−1, pn + hn + 1)V ).

Proceeding in the same way we find h1, . . . , hn ∈ N such that

η ∈ K(T(p1 + h1 + 1, p2 + h2 + 1, . . . , pn + hn + 1)V )\K(T(p1 + h1, p2 + h2 + 1, . . . , pn + hn + 1)V ).

Since η ∈ L, we have

ld(M/K) = K(T(p1 + h1 + 1, p2 + h2 + 1, . . . , pn + hn + 1)V ) :

K(T(p1 + h1,

p2+h2+1, . . . , pn+hn+1)V ) > K(T(p1+h1+1, p2+h2+1, . . . , pn+hn+1)V ) :


K(T(p1+h1, p2+h2+1, . . . , pn+hn+1)V ∪η)) ≥ L(T(p1+h1+1, p2+h2+1,

. . . , pn + hn + 1)V ) : L(T(p1 + h1, p2 + h2 + 1, . . . , pn + hn + 1)V ∪ η))= L(T(p1+h1+1, p2+h2+1, . . . , pn+hn+1)V ) : L(T(p1+h1, p2+h2+1,

. . . , pn+hn+1)V ) = ld(M/K)

that contradicts (4.4.1). Thus, L ⊆n⋃

j=1

K〈pj⋃

i=0

αij(V )〉σj

.

Now we can complete the proof by induction on n = Card σ. If n = 0the statement of the theorem is a classical result of the field theory (see The-orem 1.6.1(ii)). Let n > 0. Since Card σj = n − 1 for j = 1, . . . , n, theinduction hypothesis implies that there exist finite sets S1, . . . , Sn such that

L⋂

K〈pj⋃

i=0

αij(V )〉σj

= K〈Sj〉σj (1 ≤ j ≤ n). Then L = K〈S1 ∪ · · · ∪ Sn〉, so

that the σ-field extension L/K is finitely generated. Because of the importance of Theorem 4.4.1 and in order to illustrate the

technique of autoreduced sets of difference polynomials, we present anotherproof of this theorem based on the results of Section 2.4.

ALTERNATIVE PROOF OF THEOREM 4.4.1. As before, let K be a dif-ference field with a basic set σ = α1, . . . , αn and let M = K〈η1, . . . , ηs〉 bea difference (σ-) field extension of K generated by a finite family η1, . . . , ηs.Furthermore, let L be an intermediate σ-field of the extension M/K. We aregoing to show that L/K is also a finitely generated σ-field extension.

Let P be the defining difference ideal of the s-tuple η = (η1, . . . , ηs) inthe algebra of σ-polynomials Ly1, . . . , ys over L, and let Σ = A1, . . . , Apbe a characteristic set of the σ-ideal P . Let ui, di and Ii denote the leaderof Ai, the degree degui

Ai, and the initial of the σ-polynomial Ai, respectively(i = 1, . . . , p), and let I(Σ) denote a subset of Ly1, . . . , ys consisting of 1 andall finite products of σ-polynomials of the form τ(IA) where A ∈ Σ, τ ∈ T .Furthermore, let Φ be the set of all coefficients of all σ-polynomials of Σ andlet K ′ = K〈Φ〉. Obviously, K ′ is a finitely generated σ-field extension of Kcontaining in L. We are going to prove the theorem by showing that K ′ = L.

Suppose that λ ∈ L \ K ′. Since λ ∈ M = K〈η1, . . . , ηs〉, there exist two

σ-polynomials A,B ∈ Ky1, . . . , ys such that λ =A(η)B(η)

. Then A(η)−λB(η) =

0 whence the σ-polynomial C = A−λB lies in P . It follows that C reduces to zeromodulo [Σ], that is, there exists J ∈ I(Σ) such that JC is a linear combinationof σ-polynomials of the form τ(Ai) (τ ∈ T, 1 ≤ i ≤ p) with coefficients Dτi ofthe form Dτi = D′

τi + λD′′τi where D′

τi, D′′τi ∈ K ′y1, . . . , ys.

Indeed, let v = τui be the Σ-leader of C (τ ∈ T, 1 ≤ i ≤ p), and let vq bethe highest power of v that appears in C. Then one can write C as

C = (C ′ + λC ′′)vq + C ′′′

4.5. ORDINARY DIFFERENCE FIELD EXTENSIONS 295

where the σ-polynomials C ′, C ′′ and C ′′′ lie in K ′y1, . . . , ys and do not con-tain ve with e > q. The first step of the reduction (described in the proof ofTheorem 2.4.1) sends C to the σ-polynomial

C1 = (τIi)C − vq−di(τAi)(C ′ + λC ′′) = [(τIi)A − vq−di(τAi)C ′]−λ[(τIi)B + vq−di(τAi)C ′′].

Let Ck be a σ-polynomial of the form Ck =Fk−λGk(Fk, Gk ∈ K ′y1, . . . , ys)obtained from C after the first k reduction steps, let vk = τuj be the Σ-leaderof Ck (τ ∈ T, 1 ≤ j ≤ p), and let vqk

k be the highest power of vk in Ck, so that

Ck = (C ′k + λC ′′

k )vqk

k + C ′′′k

where the σ-polynomials C ′k, C ′′

k and C ′′′k lie in K ′y1, . . . , ys and do not contain

vek with e > qk. Then the (k + 1)st reduction step transforms Ck into

Ck+1 = [(τIj)F − vqk−dj

k (τAj)C ′k] − λ[(τIj)G + v

qk−dj

k (τAj)C ′′k ].

Since this process reduces C to 0, we obtain that there exist J ∈ I(Σ) such thatJC = H1 + λH2 = 0 where H1 and H2 are σ-polynomials in K ′y1, . . . , ysthat can be written as linear combinations of elements of the form τ(Ai) (τ ∈T, 1 ≤ i ≤ p) with coefficients in K ′y1, . . . , ys. Moreover, as it is easy tosee from the description of the reduction process, H1 is the result of reductionof the σ-polynomial A with respect to Σ, hence H1 = 0 (otherwise A ∈ Pand λ = A(η)/B(η) = 0). Now the equality H1 = −λH2 and the fact thatH1,H2 ∈ K ′y1, . . . , ys imply that λ ∈ K ′. This completes the proof of thetheorem.

Corollary 4.5.2 Let K be an inversive difference field with a basic set σ, Ma finitely generated σ∗-field extension of K, and L and intermediate σ∗-field ofM/K. Then the σ∗-field extension L/K is finitely generated, that is, there existsa finite set V ⊆ L such that L = K〈V 〉∗.

PROOF. Let S be a finite set of σ∗-generators of M/K, so that M = K〈S〉∗.Then the chain K ⊆ L∩K〈S〉 ⊆ K〈S〉 can be considerd as a sequence of σ-fieldextensions. By Theorem 4.4.1, L∩K〈S〉 is a finitely generated σ-field extensionof K, so that there exists a finite set V ⊆ L such that L ∩ K〈S〉 = K〈V 〉. Letus show that L = K〈V 〉∗. Indeed, if a ∈ L, then a ∈ M = K〈S〉∗, hence thereexists τ ∈ T such that τ(a) ∈ K〈S〉. Therefore, τ(a) ∈ L∩K〈S〉 = K〈V 〉 hencea ∈ K〈V 〉∗. Since the inclusion K〈V 〉∗ ⊆ L is obvious, we obtained the desiredequality.

4.5 Some Results on Ordinary Difference FieldExtensions

Let K be an ordinary difference field with a basic set σ = α and let L be adifference overfield of K. In what follows, in accordance with the terminologyand notation introduced by R. Cohn, the transcendence degrees trdegKL and


trdegK∗L∗ will be also called the order and effective order of the σ-field exten-sion L/K; these characteristics will be denoted by ordG/F and EordG/F ,respectively. (As usual, K∗ denotes the inversive closure of a difference field K.)

It is easy to check that EordL/K ≤ ordL/K, EordL/K = ordL/K if K isinversive, and ordL/K = ∞ if σ-trdegKL > 0 (we leave the verification to thereader as an exercise). Furthermore, the properties of transcendence degree im-ply that for any chain K ⊆ L ⊆ M of ordinary difference field extensions we haveordM/K = ordM/L + ordL/K and EordM/K = EordM/L + EordL/K.

Recall that an ordinary difference field K with a basic set σ = α is calledaperiodic if there is no integer k ≥ 1 such that αk(a) = a for all a ∈ K. Wesay that K is completely aperiodic if either Char K = 0 and the σ-field K isaperiodic, or if Char K = p > 0 and the field K satisfies the following condition:if q and r are powers of p, and i, j ∈ N, then (αi(a))q = (αj(a))r for all a ∈ Kif and only if i = j and q = r.

Exercise 4.5.1 With the above notation, show that if an aperiodic σ-field Kof positive characteristic has infinitely many invariant elements, then K is com-pletely aperiodic.

[Hint: Show that if Char K = p > 0 and r, s ∈ N, r = s, then for any i, j ∈ Nthere are only finitely many invariant elements a ∈ K such that (αi(a))pr

=(αj(a))ps

.]

The complete aperiodicity plays an important role in R. Cohn’s theoremson simple difference field extensions which are the central results of this section.Note that the direct generalization of the theorem on primitive element of afinitely generated separable algebraic field extension (Theorem 1.6.17) to thedifference case is not true. The following counterexample is presented in [34].

Example 4.5.2 Let c1, . . . , cm (m > 1) be elements in some field extension of Qwhich are algebraically independent over Q. Let K denote the field Q(c1, . . . , cm)treated as an ordinary difference field whose basic set σ consists of the identityautomorphism α. Furthermore, let P denote the reflexive difference ideal gen-erated by the σ-polynomials αy − ci (1 ≤ i ≤ m) in the ring Ky (as usual,Ky denotes the ring of σ-polynomials in one σ-indeterminate y over K).

It follows from Proposition 2.4.9 that P is a reflexive prime σ-ideal, so it hassome generic zero (η1, . . . , ηm) with coordinates in some σ-overfield of K. LetL = K〈η1, . . . , ηm〉. Then α(ηi) = ci ∈ K for i = 1, . . . , m, hence every elementof L is a solution of some σ-polynomial of first order in Ky. Applying Corollary4.1.18 we obtain that if the σ-field extension L/K is generated by a singleelement, then trdegKL = 1. However, it is easy to see that trdegKL = m > 1.(If there is a polynomial f in m variables over K such that f(η1, . . . , ηm) = 0,then one can apply α to both sides of the last equality and obtain that theelements c1, . . . , cm are algebraically dependent over Q, contrary to the choiceof these elements.) Thus, L is a finitely generated σ-field extension of a difference(σ-) field K of zero characteristic, every element of L is σ-algebraic over K, butthere is no element ζ ∈ L such that L = K〈ζ〉.


Proposition 4.5.3 Let K be a completely aperiodic difference subfield of anordinary difference field L with a basic set σ = α. Let Ky1, . . . , ys be thealgebra of σ-polynomials in σ-indeterminates y1, . . . , ys over K and let A bea nonzero σ-polynomial in Ky1, . . . , ys. Then there exists and s-tuple η =(η1, . . . , ηs) with coordinates in L such that A(η) = 0.

PROOF. First of all, elementary induction arguments show that it is suf-ficient to consider the case s = 1. Thus, we assume that A is a nonzeroσ-polynomial in Ky (y is a σ-indeterminate over K).

If A is linear, but not homogeneous, then A(0) = 0. Suppose that A is alinear homogeneous σ-polynomial, so that A has a representation of the form

A = a1αi1y + · · · + arα

iry (4.5.5)

where ai are nonzero elements of K and i1, . . . , ir ∈ N. We are going to useinduction on r to show that there is η ∈ L such that A(η) = 0. If r = 1, thenA(1) = 0. Suppose that r > 1 and the statement is true for linear homoge-neous σ-polynomials with fewer than r terms in the representation of the form(4.5.5). Let u be an element of L such that αir (u) = αir−1(u), and let A′ is theσ-polynomial in Ky obtained by replacing y with uy everywhere in A. Thenthe nonzero σ-polynomial B = αir (u)A−A′ has fewer terms than A. Therefore,there exists ζ ∈ L such that B(ζ) = 0, hence ζ cannot annul both A and A′. Itfollows that either A(ζ) = 0 or A(uζ) = 0.

Now let A be arbitrary σ-polynomial in Ky of degree d > 1. We proceedby induction of d. The proposition has been proved for d = 1, and we assumethat it is true for σ-polynomials of degree less than d. Let Ky, z be the algebraof σ-polynomails in two σ-indeterminates y and z over K. In what follows, forany D ∈ Ky, z, the total degrees of D relative to the sets of variables αky |k ∈ N and αkz | k ∈ N will be denoted by degyD and degzD, respectively.Our induction hypothesis implies that if degyD < deg A(= degyA) anddegzD < deg A, then there exist elements η, ζ ∈ L such that D(η, ζ) = 0.

Let F and G be two σ-polynomials in Ky, z obtained from A by replacingy with y + z and z, respectively. Then F = A + G + C where either C = 0 orC = 0 and degyC < deg A, degzC < deg A. Note that if Char K = 0, thenC = 0. Indeed, in this case not every formal partial derivative ∂A/∂αky is 0, sothe formal Taylor expansion of F in powers of variables αkz (k ∈ N) contains anonzero summand of the form (∂A/∂αiy)αiz. This summand contains all termsof A which are of positive degree with respect to the set of transforms of y andlinear with respect to αiz. Therefore, such a term cannot be cancelled by anyterms from the other summands in the Taylor expansion. Also, such a termcannot be in A or G, so it must be in C whence C = 0.

Now we can complete the proof for the case C = 0. Let η and ζ be twoelements in L such that C(η, ζ) = 0 (such elements exists by the inductionhypothesis). If A(η) = 0 and A(ζ) = 0, then F (η, ζ) = C(η, ζ) = 0. Therefore,one of the elements η, ζ, or η + ζ does not annul A.


It remains to consider the case Char K = p > 0, C = 0. Let M be amonomial of A, that is, a power product of the form (αk1y)l1 . . . (αkqy)lq whichappears in A with a nonzero coefficient. If αiy and αjy, i = j, appear in M ,then M contributes to F a term with monomial M ′ obtained from M by thesubstitution of αiz for αiy. This term cannot be cancelled by any other termsand is in C. Thus, every monomial M of A is of the form M = (αiy)d. Supposethat d = kpr where k is not divisible by p. Then a term ((k(αiy)k−1)pr

appearsin (αiy + αiz)d and contributes a term to F . If k = 1, this term is in C. Since

C = 0, we have the only possibility: A =s∑

i=1

t∑j=1

aij(αiy)pj

where the coefficients

aij lie in K. Now one can apply the argument used in the first part of the proof(when we considered the case of a linear σ-polynomial A) and obtain (exploringthe complete aperiodicity instead of the aperiodicity) that there is an elementη ∈ L such that A(η) = 0.

Theorem 4.5.4 Let K be a completely aperiodic ordinary difference field witha basic set σ = α and let L be a σ-overfield of K such that σ-trdegKL = 0.If Char K = 0 or Char K = p > 0 and rld(L/K) = ld(L/K), then there existsan element θ ∈ L and a non-negative integer k such that αk(a) ∈ K〈θ〉 forany a ∈ L. Moreover, the element θ may be chosen as a linear combination ofthe members of any finite set of σ-generators of L/K with coefficients from anypre-assigned completely aperiodic σ-subfield of K.

PROOF. We start with the case Char K = 0 and L = K〈η, ζ〉 (obviously,it is sufficient to prove the result for the case of two σ-generators of L/K). Letu be an element in some σ-overfield of L which is σ-transcendental over L. Letv = η + ζu. Then v is σ-algebraic over K〈u〉, hence there exists s, k ∈ N suchthat αk(v) is algebraic over K(u, . . . , αs(u); v, . . . , αk−1(v)). Let us choose thesmallest possible k with this property. Then

k = trdegK〈u〉K〈u, v〉 ≤ trdegK〈u〉K〈u, η, ζ〉 ≤ trdegK〈u〉K〈η, ζ〉.Let A be a nonzero polynomial in s + k + 2 variables X1, . . . , Xs+k+2 withcoefficients in K such that A(u, . . . , αs(u); v, . . . , αk(v)) = 0 and A is of thelowest possible degree in Xs+k+2 (the variable which is replaced with αk(v) inthe last equality).

Since the elements u, . . . , αs(u) are algebraically independent over K〈η, ζ〉,the coefficients of the power products of the αi(u) in the expressions obtainedby replacing each αj(v), 0 ≤ j ≤ k, in A with αj(η) + αj(ζ)αj(u) and expand-ing formally are all equal to 0. Therefore, the formal partial derivative of thisexpansion with respect to αk(u) is 0, that is,

∂A/∂(αk(u)) + αk(ζ)∂A/∂(αk(v)) = 0.

Because of the minimum conditions on k and degree of A with respect to Xs+k+2,∂A/∂(αk(v)) = 0. It follows that αk(ζ) = −(∂A/∂(αk(u)))/(∂A/∂(αk(v))),hence αk(ζ) and, therefore, also αk(η) belong to K〈u, v〉. Thus, one can write


αk(η) = F/H and αk(ζ) = G/H where F , G, and H are σ-polynomials in u andv with coefficients in K and H = 0. (More precisely, there are σ-polynomialsF , G, and H in the algebra of σ-polynomials Ky, z in two σ-indeterminatesy and z over K such that the above expressions for αk(η) and αk(ζ) hold withF = F (u, v), G = G(u, v), and H = H(u, v).)

Let H be written as a σ-polynomial in u with coefficients in K〈η, ζ〉. ByProposition 4.5.3, there exists an element µ ∈ K such that H is not annulledwhen u is replaced by µ. Let θ = η = µζ. We are going to show that θ andk have the properties stated in the theorem. Let F, G, H become F ′, G′, H ′,respectively, when u is replaced by µ and v by θ. Then H ′ = 0.

Clearly, it is sufficient to show that H ′αk(η) = F ′ and H ′αk(ζ) = G′. (Theseequalities would imply that αk(η), αk(ζ) ∈ K〈θ〉 and, therefore, the inclusionαk(x) ∈ K〈θ〉 for any x ∈ L.) Considering the equality Hαk(η) = F one cansee that both its sides can be viewed as σ-polynomials in u with coefficientsin K〈η, ζ〉. Since u is σ-transcendental over K〈η, ζ〉, corresponding coefficientson both sides are equal. therefore, the equality is preserved if u is replaced byµ. Rewriting both sides of the resulting equality as appropriate combinationsof µ and η + µζ, we obtain that H ′αk(η) = F ′. Similarly H ′αk(ζ) = G′. Thiscompletes the proof in the case of characteristic 0.

Suppose that Char K = p > 0 and ld(L/K) = rld(L/K). Since the reducedlimit degree of a difference field extension cannot exceed its limit degree, onecan apply Theorem 4.3.4 and obtain that for every intermediate σ-field L′ of theextension L/K we have ld(L′/K) = rld(L′/K). Therefore, we may assume, asbefore, that L = K〈η, ζ〉 for some elements η, ζ ∈ L. Let u be a σ-transcendentalover L element in some σ-overfield of L, and let θ = η + uζ. If Φ and Ψ aresubsets of L such that Φ ⊆ Ψ then the statements of Theorem 1.6.23(viii)and Theorem 1.6.28(vi) imply that K〈u〉(Ψ) : K〈u〉(Φ) = K(Ψ) : K(Φ) and[K〈u〉(Ψ) : K〈u〉(Φ)]s = [K(Ψ) : K(Φ)]s.

Therefore, ld(L〈u〉/K〈u〉) = ld(L/K) and rld(L〈u〉/K〈u〉) = rld(L/K),hence ld(L〈u〉/K〈u〉) = rld(L〈u〉/K〈u〉). Furthermore, by the argument usedat the beginning of this paragraph, we obtain that ld(K〈u, v〉/K〈u〉) =rld(K〈u, v〉/K〈u〉). It follows that there exists s, k ∈ N such that αk(v) isalgebraic over the field K(u, . . . , αs(u); v, . . . , αk−1(v) ), and the degree andseparable factor of the degree of αk(v) over this field are equal. Then αk(v) isseparably algebraic over K(u, . . . , αs(u); v, . . . , αk−1(v)).

Let Z be an indeterminate over the field K(u, . . . , αs(u); v, . . . , αk−1(v) )and let f = f(Z) be an irreducible polynomial in the polynomial ringK(u, . . . , αs(u); v, . . . , αk−1(v) )[Z] such that f(αk(v)) = 0 and the coeffi-cients of f belong to K[u, . . . , αs(u); v, . . . , αk−1(v)] (the existence of such apolynomial follows from the conclusion made at the end of the previous para-graph). Since αk(v) is separably algebraic over K(u, . . . , αs(u); v, . . . , αk−1(v)),the formal partial derivative ∂C/∂Z is not 0, so that ∂C/∂Z is not annulled byαk(v). Let D be the polynomial expression in u, . . . , αs(u); v, . . . , αk(v) obtainedby replacing Z in f with αk(v). Then we can complete the proof of the firststatement of the theorem by repeating the argument of our proof in the case


Char K = 0 where A is replaced by D and the fact that (∂C/∂Z)(αk(v)) = 0is used to show that (∂D/∂(αk(v))) = 0.

It follows from Proposition 4.5.3 that the element µ in the first part of ourproof may be chosen from any completely aperiodic σ-subfield of K. This remarkproves the last statement of the theorem.

4.6 Difference Algebras

Let K be a difference (respectively, inversive difference) ring with a basic set σ.A K-algebra R is said to be a difference algebra over K or a σ-K-algebra (re-spectively, an inversive difference algebra over K or a σ∗-K-algebra) if elementsof σ (respectively, σ∗) act on R in such a way that R is a σ- (respectively, σ∗-)ring and α(au) = α(a)α(u) for any a ∈ K,u ∈ R,α ∈ σ (for any α ∈ σ∗ if R isa σ∗-K-algebra). A σ-K- (respectively, σ∗-K-) algebra R is said to be finitelygenerated if there exists a finite family η1, . . . , ηs of elements of R such thatR = Kη1, . . . , ηs (respectively, R = Kη1, . . . , ηs∗). If a σ-K- (or σ∗-K-)algebra R is an integral domain, then the σ-transcendence degree of R over Kis defined as the σ-transcendence degree of the corresponding σ- (respectively,σ∗-) field of quotients of R over K.

In what follows we consider some applications of the theorems on finitely gen-erated difference field extensions to difference algebras. We will formulate andprove some results about inversive difference algebras over inversive differencefields, leaving to the reader the formulations and proofs of the correspondingstatements for difference and “general difference” cases as exercises. (As before,by a “general difference” case we mean the study of a difference ring (field,module, algebra) R whose basic set is a union of a finite set σ of injective en-domorphisms of R and a finite set ε of automorphisms of R such that any twoelements of σ ∪ ε commute, see Section 3.6. for details.)

Let R be an inversive difference algebra over an inversive difference field Kwith a basic set σ = α1, . . . , αn and let U denote the set of all prime σ∗-ideals of R. As in Section 3.6 (see Proposition 3.6.16), one can consider the setBU = (P,Q) ∈ U × U|P ⊇ Q and the uniquely defined mapping µU : BU → Zsuch that

(i) µU (P,Q) ≥ −1 for every pair (P,Q) ∈ BU ;(ii) for any d ∈ N, the inequality µU (P,Q) ≥ d holds if and only if P = Q

and there exists an infinite chain

P = P0 ⊇ P1 ⊇ · · · ⊇ Q

such that Pi ∈ U and µU (Pi−1, Pi) ≥ d − 1 for i = 1, 2, . . . .

Definition 4.6.1 With the above notation, the least upper bound of the setµU |(P,Q) ∈ BU is called the type of the σ∗-K-algebra R. The least upperbound of the lengths k of chains P0 ⊇ P1 ⊇ · · · ⊇ Pk, such that P0, . . . , Pk ∈ Uand µU (Pi−1, Pi) = typeUR (i = 1, . . . , k), is called the dimension of R.

4.6. DIFFERENCE ALGEBRAS 301

The type and dimension of a σ∗-K-algebra R are denoted by typeUR anddimUR, respectively.

Theorem 4.6.2 Let K be an inversive difference field with a basic set σ =α1, . . . , αn, R = Kη1, . . . , ηs∗ a σ∗-K-algebra without zero divisors gener-ated by a finite set η = η1, . . . , ηs, and U the family of all prime σ∗-ideals ofR. Then:

(i) typeUR ≤ n.(ii) If σ-trdegKR = 0, then typeUR < n.(iii) If typeUR = n, then dimUR ≤ σ-trdegKR.(iv) If η1, . . . , ηs are σ-algebraically independent over K, then typeUR = n and

dimUR = s.

PROOF. Let Γ be the free commutative group generated by the set σ andfor any r ∈ N, let Γ(r) = γ ∈ Γ | ord γ ≤ r (recall that the order of anelement γ = αk1

1 . . . αknn ∈ Γ is defined as ord γ =

∑ni=1 |ki|). Furthermore, let

Rr (r ∈ N) denote the K-algebra K[γ(ηj) | γ ∈ Γ(r), 1 ≤ j ≤ s] and letRr = R for r ∈ Z, r < 0.

Let P be a prime σ∗-ideal of R and let ηi denote the canonical image of ηi

in the factor ring R/P (1 ≤ i ≤ s). It is easy to see that for every r ∈ N, P ∩Rr

is a prime ideal of the ring Rr and the quotient fields of the rings Rr/P ∩ Rr

and K[γ(ηj) | γ ∈ Γ(r), 1 ≤ j ≤ s] are isomorphic. By Theorem 4.2.5, thereexists a numerical polynomial ψ(t) in one variable t such that

ψ(t) = trdegKK[γ(ηj) | γ ∈ Γ(r), 1 ≤ j ≤ s] = trdegK(Rr/P ∩ Rr)

for all sufficiently large r ∈ Z, deg ψ(t) ≤ n and the polynomial ψ(t) can bewritten as

ψ(t) =2naP

n!tn + o(tn)

where aP = σ-trdegK(R/P ). It is easy to see that if P,Q ∈ U (as above, Udenote the set of all prime σ∗-ideals of R) and P ⊇ Q, then ψP (t) ψQ(t)where denotes the order on the set of all numerical polynomials introducedin Definition 1.5.10 (by Theorem 1.5.11, the set W of all Kolchin polynomialsis well-ordered with respect to ). Furthermore, if P,Q ∈ U and P Q, thenψP (t) ≺ ψQ(t). Indeed, in this case there exists r0 ∈ N such that P∩Rr Q∩Rr

for all r ≥ r0. Since trdegK(Rr/P∩Rr) and trdegK(Rr/Q∩Rr) are, respectively,the coheights of the ideals P ∩Rr and Q∩Rr in the ring Rr, Theorem 1.2.25(ii)shows that for all sufficiently large r ∈ Z,

ψP (r) = trdegK(Rr/P ∩ Rr) = coht(P ∩ Rr) < coht(Q ∩ Rr)

= trdegK(Rr/Q∩Rr) = ψQ(r).

It follows that if P,Q ∈ U , P ⊇ Q and ψP (t) = ψQ(t), then P = Q. Now thearguments of the proof of Theorem 3.6.24 show that for any d ∈ Z, d ≥ −1


and for any pair (P,Q) ∈ BU (that is, for any P,Q ∈ U such that P ⊇ Q), theinequality µU (P,Q) ≥ d implies the inequality deg(ψQ(t) − ψP (t)) ≥ d.

Since ψP (t) ≤ n for any prime σ∗-ideal P in R and ψP (t) < n if σ-trdegKR =0 (in this case aP = 0), we have deg(ψQ(t)−ψP (t)) ≤ n for any pair (P,Q) ∈ BU

and this inequality is strict if σ-trdegKR = 0. Therefore, for every (P,Q) ∈ BU ,we have µU (P,Q) ≤ n, and if σ-trdegKR = 0 then the last inequality is alwaysstrict. It follows that typeUR ≤ n and typeUR < n if σ-trdegKR = 0.

In order to prove statement (iii), suppose that typeUR = n and P0 P1 · · · Pd (d ∈ N) is a descending chain of prime σ∗-ideals of R suchthat µU (Pi−1, Pi) = typeUR = n for i = 1, . . . , d. Clearly, in order to prove theinequality dimUR ≤ σ-trdegKR, it is sufficient to show that d ≤ σ-trdegKR.

Since µU (Pi−1, Pi) = n (1 ≤ i ≤ d), the above considerations show thatdeg(ψPi

(t) − ψPi−1(t)) ≥ n hence aPi− aPi−1 ≥ 1 for i = 1, . . . , n. Now the

equalities

ψPd(t) − ψP0(t) =

d∑i=1

[ψPi(t) − ψPi−1(t)] =

d∑i=1

[2n

n!(aPi

− aPi−1)tn + o(tn)

]=

2n

n!(aPd

− aP0)tn + o(tn)

imply that

aPd− aP0 =

d∑i=1

(aPi− aPi−1) ≥ d.

On the other hand,

ψPd(t) − ψP0(t) ≤ ψ(0)(t) − ψP0(t) ≤ ψ(0)(t) = ψη|K(t) (4.6.1)

where ψη|K(t) is the σ∗-dimension polynomial of the extension K〈η1, . . . , ηs〉∗/Kassociated with the system of σ∗-generators η. Since

ψη|K(t) =2n

n!(σ∗-trdegKR)tn + o(tn)

(see Theorem 4.2.5), inequalities (4.6.1) show that d ≤ aPd−aP0 ≤ σ∗-trdegKR.

Thus, dimUR ≤ σ-trdegKR if typeUR = n.Suppose that η1, . . . , ηs are σ-algebraically independent over K. Let L denote

the set of all linear σ∗-ideals of R. (Recall that by a linear σ∗-ideal of R we meanan ideal I generated (as a σ∗-ideal) by homogeneous linear σ∗-polynomials of

the formm∑

i=1

aiγiykiwith ai ∈ K, γi ∈ Γσ (1 ≤ ki ≤ s for i = 1, . . . , m). ) Since

L ⊆ U (see Proposition 2.4.9), typeLR ≤ typeUR ≤ n. Therefore, in order toprove the equality typeUR = n, it is sufficient to show that typeLR ≥ n. Thisinequality, in turn, is a consequence of the inequality

µL([η1, . . . , ηp]∗, [η1, . . . , ηp−1]∗) ≥ n (p = 1, . . . , s) (4.6.2)

we are going to prove.


Let L(p) denote the set of all linear σ∗-ideals P ∈ L such that [η1, . . . , ηp−1]∗⊆P ⊆ [η1, . . . , ηp]∗. Then inequality (4.6.2) is equivalent to the inequality

µL(p)([η1, . . . , ηp], [η1, . . . , ηp−1]) ≥ n (p = 1, . . . , s). (4.6.3)

Furthermore, the canonical ring σ∗-isomorphism

Kη1, . . . , ηs∗/[η1, . . . , ηp−1]∗ ∼= Kηp, . . . , ηs∗ (p = 1, . . . , s)

induces a bijection of the family L(p) onto the family L(p)1 of all linear σ∗-

ideals of Kηp, . . . , ηs∗ whose σ∗-generators are of the formq∑

j=1

ajγj(ηp) with

aj ∈ K, γj ∈ Γ (q ≥ 1). Of course, this bijection preserves inclusions of σ∗-ideals.

Let L(p)2 (1 ≤ p ≤ s) be the family of all linear σ∗-ideals of the σ∗-K-

algebra Kηp∗. It is easy to see that if J ∈ L(p)2 , then JKηp, . . . , ηs∗ ∈ L(p)

1

and JKηp, . . . , ηs∗ ∩ Kηp∗ = J , so there is a one-to-one correspondencebetween the families L(p)

1 and L(p)2 that preserves inclusions.

Thus, in order to prove inequality (4.6.3) (and, therefore, inequality (4.6.2))it is sufficient to show that if Ky∗ is an algebra of σ∗-polynomials in oneσ∗-indeterminate y over K and B is the family of all linear σ∗-ideals of Ky∗,then typeBKy∗ ≥ n. We will prove this inequality by induction on n. If n = 0,the inequality is obvious. Let n > 0 and let σn = σ \ αn = α1, . . . , αn−1.Let us consider the descending chain

[y]∗ [(αn − 1)y]∗ · · · [(αn − 1)ry]∗ · · · (0)

of σ∗-ideals of Ky∗ and let Br (r ∈ N) be the set of all σ∗-ideals I ∈ Bsuch that [(αn − 1)r+1y]∗ ⊆ I ⊆ [(αn − 1)ry]∗. Denoting the canonical imageof the element (αn − 1)ry in the σ∗-ring K(αn − 1)ry∗/[(αn − 1)r+1y]∗ by zr

(r = 1, 2, . . . ), we can treat the last ring as a ring of σ∗n-polynomials Kzr∗σn

in one σ∗n-indeterminate zr over K (the index σn indicates that this ring, as

well as the field K, is considered with respect to the basic set σn). Notice thatKzr∗σn

is also a σ∗-ring where αn(zr) = zr.Let Γ′ be the subgroup of Γ generated by the set σ∗

n and let Ar denote thefamily of all σ∗

n-ideals of Kzr∗σngenerated by sets of elements of the form γ′zr

with γ′ ∈ Γ′. Then

µBr([(αn − 1)ry]∗, [(αn − 1)r+1y]∗) = µAr

([zr]∗σn, (0)),

so the inductive hypothesis leads to the inequality

µBr([(αn − 1)ry]∗, [(αn − 1)r+1y]∗) ≥ n − 1.


Therefore, µBr([y]∗, (0) ) ≥ n − 1 for every r ∈ N hence typeBKy∗ ≥ n. We

have proved inequality (4.6.2) and therefore the equality typeUR = n. Moreover,the above arguments show that if η1, . . . , ηs are σ-algebraically independent overK, then

µU ([η1, . . . , ηp]∗, [η1, . . . , ηp−1]∗ = typeUR = n

for p = 1, . . . , s, hence dimUR ≥ s. Combining this inequality with statement(iii) of our theorem we obtain that dimUR = s.

Exercise 4.6.3 Formulate and prove an analog of Theorems 4.6.2 for differencefield extensions.

We conclude this section with some results on local difference algebras. Inwhat follows, all fields are supposed to have zero characteristics.

Let K be an inversive difference field with a basic set σ = α1, . . . , αn andlet a σ∗-K-algebra A be a local ring with a maximal ideal m. Clearly, m is aσ∗-ideal of A, the residue field k = A/m can be naturally considered as a σ∗-fieldextension of K and m/m2 can be treated as a vector σ∗-k-space.

Proposition 4.6.4 Let K be an inversive difference field with a basic set σand let A be a local σ∗-K-algebra with a maximal ideal m. Furthermore, let kdenote the residue field A/m (treated as a σ∗-overfield of K). Then there existsan exact sequence of vector σ∗-k-spaces

0 → m/m2 ρ−→ ΩA|K⊗A

kν−→ Ωk|K → 0 (4.6.4)

where the σ-homomorphisms ρ and ν act as follows:

ρ(x + m2) = dA|Kx⊗

1, ν(dA|Ky⊗

1) = dk|K y (4.6.5)

for every x ∈ m, y ∈ A (y is the coset of y in the factor ring A/m).

PROOF. Since K is a field, every exact short sequence of K-modules splits.By Theorem 1.7.19(iii), there exists an exact sequence of vector σ∗-k-spaces

0 → m/m2 ρ−→ ΩA|K⊗A

kν−→ Ωk|K → 0

where the actions of σ-homomorphisms ρ and ν are defined by conditions(4.6.5). Let Ek and EA denote the rings of inversive difference (σ∗-) operatorsover k and A, respectively. Then m/m2 can be naturally considered as a leftEk-module, while Lemma 4.2.8 shows that Ωk|K and ΩA|K can be naturallytreated as left Ek- and EA- modules, respectively. As we have seen in Section 3.4,the vector k-space ΩA|K

⊗A k can be viewed as a left Ek-module where elements

of the free commutative group Γ generated by the set σ act in such a way thatγ(dA|Kx

⊗a) = γ(dA|Kx)

⊗γ(a) for every γ ∈ Γ, x ∈ A, a ∈ k. It remains


to show that the homomorphisms of vector k-spaces ρ and ν are actuallyσ-homomorphisms, that is,

ρ(γx) = γρ(x), ν(γ(dA|Kη⊗

1)) = γ(ν(dA|Kη⊗

1)) (4.6.6)

for any γ ∈ Γ, x = x + m2 ∈ m/m2, η ∈ A. Indeed, if γ ∈ Γ, then we have

ρ(γx) = ρ(γ(x) + m2) = dA|Kγ(x)⊗

1 = γdA|Kx⊗

1 = γρ(x)

and

ν(γ(dA|Kη⊗

1)) = ν(dA|Kγ(η)⊗

1) = dK|kγ(η) = γ(ν(dA|Kη⊗

1))

where η denotes the canonical image of η in the field k = A/m. Thus, themappings ρ and ν in (4.6.4) are σ-homomorphism of vector σ∗-k-spaces. Thiscompletes the proof.

In what follows, if A is a local algebra with a maximal ideal m over a fieldK and B a K-subalgebra of A, then Bm will denote the set of all elements of A

which can be written in the formf

gwhere f, g ∈ B and g /∈ m. (We write

f

gfor

fg−1 taking into account that elements of A \m are units of A.) Clearly, Bm isa local K-subalgebra of A with the maximal ideal m ∩ Bm.

Definition 4.6.5 Let K be an inversive difference field with a basic set σ andlet A be a local σ∗-K-algebra without zero divisors. We say that A is a local σ∗-K-algebra of finitely generated type if there exist elements η1, . . . , ηs ∈ A suchthat A = Kη1, . . . , ηs∗m where m is the maximal ideal of A.

Theorem 4.6.6 Let K be an inversive difference field with a basic set σ andlet an integral domain A be a local σ∗-K-algebra of finitely generated type witha maximal ideal m and the residue field k = A/m. Then

(i) m/m2 is a finitely generated vector σ∗-K-space.(ii) σ∗-dimkm/m2 ≥ σ∗-trdegKk.

PROOF. Let EA and Ek denote the rings of σ∗-operators over A and k,respectively, and let

0 → m/m2 ρ−→ ΩA|K⊗A

kν−→ Ωk|K → 0

be the exact sequence of Ek-modules considered in Proposition 4.6.4. We aregoing to show that ΩA|K

⊗A k is a finitely generated Ek-module. (Since the

ring Ek is left Noetherian (see Theorem 3.4.2), we shall obtain that m/m2 is alsoa finitely generated Ek-module.)

By the assumption of the theorem, there exist elements η1, . . . , ηs ∈ A suchthat A = Kη1, . . . , ηs∗m. Let us show that every element dA|Kξ (ξ ∈ A) can


be written as a linear combination of elements dA|Kη1, . . . , dA|Kηs with co-

efficients in EA. Indeed, any element ξ ∈ A can be written asf(η1, . . . , ηs)g(η1, . . . , ηs)

where f and g are σ∗-polynomials in σ∗-indeterminates y1, . . . , ys over K andg(η1, . . . , ηs) /∈ m. Since dA|Kf(η1, . . . , ηs) and dA|Kg(η1, . . . , ηs) can be writ-ten as linear combinations of elements dA|Kγ(ηi) = γdA|Kηi (γ ∈ Γ, 1 ≤i ≤ s), the element dA|Kξ =

1g2(η1, . . . , ηs)

[g(η1, . . . , ηs)dA|Kf(η1, . . . , ηs) −f(η1, . . . , ηs)dA|Kg(η1, . . . , ηs)] can be written as such a linear combination aswell.

It follows that elements dA|Kη1

⊗1,. . . , dA|Kηs

⊗1 generate ΩA|K

⊗A k as

a left Ek-module. Indeed, any generator dA|Kη⊗

A (η ∈ A) of the Ek-moduleΩA|K

⊗A k can be written in the form

s∑i=1

ri∑j=1

aijγijdA|Kηi

⊗1 =

s∑i=1

(ri∑

j=1

aijγij)(dA|Kηi

⊗1)

where aij ∈ A (1 ≤ i ≤ s, 1 ≤ j ≤ ri) and aij denotes the image of aij under

the natural epimorphism A → k = A/m. Sinceri∑

j=1

aijγij ∈ Ek (1 ≤ i ≤ s), the

elements dA|Kη1

⊗1, . . . , dA|Kηs

⊗1 generate the Ek-module ΩA|K

⊗A k. This

completes the proof of statement (i).Now let us prove the second statement of the theorem. As before, for any

r ∈ N let Γ(r) denote the set γ ∈ Γ | ord γ ≤ r, and for any η ∈ A let Γ(r)η =γ(η) | γ ∈ Γ(r). Furthermore, let Ar denote the local subring K[Γ(r)η1 ∪ · · · ∪Γ(r)ηs]m of Ar, let mr = m ∩ Ar, and let kr denote the residue field Ar/mr ofthe ring Ar. First of all, one can apply Theorem 1.2.25 (ii), (v) and obtain thefollowing inequality.

dimkr(mr/m2

r) ≥ trdegKAr − trdegKkr (4.6.7)

for every r ∈ N.Now, let Ky1, . . . , ys∗ denote the ring of σ∗-polynomials in σ∗-indeter-

minates y1, . . . , ys over K, let π : Ky1, . . . , ys∗ → Kη1, . . . , ηs∗ be thenatural σ-epimorphism of σ∗-K-algebras (π : yi → ηi for i = 1, . . . , s andπ(a) = a for every a ∈ K), and let P = Ker π. Furthermore, let B denotethe local σ∗-K-algebra Ky1, . . . , ys∗P and let n denote the maximal ideal ofB. It is easy to see that the homomorphism π can be naturally extended to aσ-homomorphism of local σ∗-K-algebras π′ : B → A such that π′(n) = m. LetQ = PB and let Br (r ∈ N) denote the local subring K[Γ(r)y1∪· · ·∪Γ(r)ys]n ofB. Also, for any r ∈ N and M ⊆ B, let Mr denote the set M ∩Br (in particular,we set nr = n ∩ Br). In what follows we identify the field Br/nr with the fieldkr via the σ-isomorphism naturally induced by π′).


By Theorem 1.7.19, for every r ∈ N, there exists an exact sequence

0 → nr/n2r → ΩBr|K

⊗Br

kr → Ωkr|K (4.6.8)

and a commutative diagram with exact rows

0 nr/n2r

⊗kr

k ΩBr|K⊗

Brk Ωkr|K

⊗kr

k 0

0 n/n2 ΩB|K⊗

B k Ωk|K 0

jr

ir hr (4.6.9)

where the first row is obtained by applying the functor⊗

krk to the exact se-

quence (4.6.8).We are going to show that the mapping jr in diagram (4.6.9) is injective.

To obtain this result we first notice that it is sufficient to show that for everyvector k-space V , the induced k-homomorphism of vector k-spaces

j∗r : Homk(ΩB|K⊗B

k, V ) → Homk(ΩBr|K⊗Br

k, V )

is surjective. This, however, is equivalent to asserting that the canonical mapDerK(B, V ) → DerK(Br, V ) is surjective, see Corollary 1.7.12 which impliesthe existence of sequences of natural isomorphisms

Homk(ΩB|K⊗B

k, V ) ∼= HomB(ΩB|K , V ) ∼= DerK(B, V )

and

Homk(ΩBr|K⊗Br

k, V ) ∼= HomB(ΩBr|K , V ) ∼= DerK(Br, V ).

Since B is the ring of quotients of a polynomial extension of the ring Br,every derivation from B to a vector k-space V can be extended to a derivationfrom B to V (see Proposition 1.7.9). It follows that the map j∗r is surjective hencejr is injective. Furthermore, the injectivity of jr in diagram (4.6.9) implies thatthe map ir in diagram (4.6.9) is also injective.

Since elements dB|K(γyi)⊗

1 (γ ∈ Γ, 1 ≤ i ≤ s) generate the vector k-spaceIm ir (r ∈ N), (Im ir)r∈N is an excellent filtration of the vector σ∗-k-spacen/n2 (we consider a filtration with r ∈ N as a filtration with r ∈ Z where allcomponents with negative indices are equal to 0).

Let Nr = Im ir (r ∈ N), N = n/n2, and let S denote the vector σ∗-k-space (Q + n2)/n2 ⊆ N . Furthermore, let Sr (r ∈ N) denote the image of theσ∗-k-space (Qr + n2

r)/n2r

⊗kr

k under the map ir. Then

Nr = ir( (nr/n2r)

⊗kr

k)/ir( (Qr + n2r)/n2

r)⊗kr

k)

= nr/(Qr + n2r)

⊗kr

k = (mr/m2r)

⊗kr

k,


the family (Sr)r∈N is an excellent filtration of the vector σ∗-k-space S, andm/m2 = N/S.

For every r ∈ N, let (m/m2)r denote the vector k-subspace of m/m2 gener-ated by the canonical image of mr in m/m2. Then ((m/m2)r)r∈N is an excellentfiltration of the vector σ∗-k-space m/m2 (this is the image of the excellent filtra-tion (Nr)r∈N under the canonical mapping N → m/m2). Setting S′

r = S ∩ Nr

we obtain (see Theorem 3.5.7) that (S′r)r∈N is an excellent filtration of S and,

obviously, Nr/S′r = (m/m2)r for every r ∈ N.

By Theorems 3.5.2 and 4.2.5, there exist numerical polynomials ψ1(t), ψ2(t),ψ3(t), and ψ4(t) in one variable t of degree at most n such that ψ1(r) = dimkNr,ψ2(r) = dimkSr, ψ3(r) = dimkS′

r and ψ4(r) = trdegKAr − trdegKkr for allsufficiently large r ∈ N. Since (Sr)r∈N and (S′

r)r∈N are two excellent filtra-

tions of the vector σ∗-k-space S,∆nψ2(t)

2n=

∆nψ3(t)2n

∈ Z (as before, for

any polynomial f(t), ∆nf(t) denote the nth finite difference of the polynomial:∆f(t) = f(t + 1)− f(t), ∆2f(t) = ∆(∆f(t)), . . . ). Furthermore, Theorem 4.2.5

shows that∆nψ4(t)

2n= σ-trdegKA − σ-trdegKk. Since ψ1(r) − ψ2(r) ≥ ψ3(r)

for all sufficiently large r ∈ N (see inequality (4.6.7)),

σ∗-dimk(m/m2) =∆n(ψ1(t) − ψ3(t))

2n=

∆n(ψ1(t) − ψ2(t))2n

≥ ∆nψ4(t)2n

= σ-trdegKA − σ-trdegKk. This completes the proof.

Suppose that K is a difference (but not necessarily inversive difference) fieldwith a basic set σ and A a local K-algebra with a maximal ideal m. The followingexample shows that even if m is a difference ideal, it is not necessarily a σ∗-idealof A.

Example 4.6.7 Let A be the ring of formal power series in one variable Xover Q. Treating Q as an ordinary difference ring whose basic set σ con-sists of the identical automorphism α, we can consider A as a σ-Q-algebraby setting α(X) = X2 (thus, α(

∑∞k=0 akXk) = α(

∑∞k=0 akX2k) for any series∑∞

k=0 akXk ∈ A). It is well-known that A is a local Q-algebra whose maximalideal m consists of all power series with zero free terms. Clearly, m is a differencebut not an inversive difference ideal of A.

Let K be a difference field with a basic set σ = α1, . . . , αn and letR = Kη1, . . . , ηs be a σ-K-algebra generated by a finite set η = η1, . . . , ηs.Furthermore, for any k ∈ N, let Rk denote the commutative K-algebraK[τ(ηi) | τ ∈ Tσ(k), 1 ≤ i ≤ s]. Since every Rk is a finitely generated al-gebra over a field, it has a finite Krull dimension. It would be interesting todescribe the function χ(k) = dimRk, in particular, to determine whether thisfunction is polynomial. Of course, if R is an integral domain, then χ(k) is apolynomial of k, since in this case R is isomorphic to the factor ring of the alge-bra of σ-polynomials Ky1, . . . , ys by a prime σ∗-ideal P . Then χ(k) = φP (k)where φP (t) is the σ-dimension polynomial of P (see Definition 4.2.21).


If R is not an integral domain, one can consider all essential prime divisorsQ1, . . . , Qm of the perfect ideal 0 of R and the corresponding σ-dimensionpolynomials φQi

(t) such that φQi(k) = trdegK(Rk/Qi

⋂Rk) (i = 1, . . . , m).

By the remark made after Theorem 4.2.15, the set of all difference di-mension polynomials is well-ordered with respect to the order ≺ such thatf(t) ≺ g(t) if and only if f(r) ≺ g(r) for all sufficiently large r ∈ Z. Letφ(t) = maxφQ1(t), . . . , φQm

(t) where the maximum is taken with respectto the order ≺. Let φ(t) = φQj

(t) for some j, 1 ≤ j ≤ m. It follows fromTheorem 1.2.25 that for any k ∈ N,

dimRk = maxtrdegK(Rk/Q) |Q is a minimal prime ideal ofRk.

Since Qj

⋂Rk is a prime ideal of Rk, it contains some minimal prime ideal p

of Rk (it follows from Proposition 1.2.1(vii) ). Then dimRk ≥ trdegK(Rk/p) ≥trdegK(Rk/Qj), hence

χ(k) = dimRk ≥ φ(k)

for all sufficiently large k ∈ N. On the other hand, for any prime ideal Q of Rk

(k ∈ N), trdegK(Rk/Q) does not exceed the number of generators of Rk over

K. Since this number is s

(k + n

n

)(see formula (1.4.16)), we obtain that

φ(k) ≤ χ(k) ≤ s

(k + n

n

)for all sufficiently large k ∈ N. Thus, χ(k) = dimRk is a function of polynomialgrowth. However, the conditions that imply the polynomial form of the functionχ(k) seem to be unknown. Of course, a similar observation can be made inthe case of a finitely generated inversive difference algebra over an inversivedifference field.

Chapter 5

Compatibility, Replicability,and Monadicity

5.1 Compatible and Incompatible DifferenceField Extensions

Definition 5.1.1 Let K be a difference field with a basic set σ and let L andM be two σ-overfields of F . The difference field extensions L/K and M/K aresaid to be compatible if there exists a σ-field extension N of K such that L/Kand M/K have σ-isomorphisms into N/K (i.e., there exist σ-isomorphisms ofL and M into N that leave the σ-field K fixed). Otherwise, the σ-field extensionsL/K and M/K are called incompatible.

The following example is due to R. Cohn.

Example 5.1.2 Let us consider Q as an ordinary difference field whose basicset σ consists of the identity automorphism α. If one adjoins to Q an elementi such that i2 = −1, then the resulting field Q(i) has two automorphisms thatextend α: one of them is the identical mapping (we denote it by the same letterα) and the other (denoted by β) sends an element a + bi ∈ Q(i) (a, b ∈ Q)to a − bi (complex conjugation). Then Q(i) can be treated as a difference fieldwith the basic set α, as well as a difference field with the basic set β.Denoting these two difference fields by L and M , respectively, we can naturallyconsider them as σ-field extensions of Q. Let us show that L/Q and M/Q areincompatible σ-field extensions. Indeed, suppose that there is a σ-field extensionE of Q and σ-isomorphisms φ and ψ, respectively, of L/Q and M/Q into E/Q.Let j = φ(i), k = ψ(i), and let γ be the translation of E that extends αand β. Then j2 = k2 = −1 whence either j = k or j = −k. Since γ(j) = jand γ(k) = −k, in both cases we obtain that j = −j, that is, j = 0. Thiscontradiction (j2 = −1) implies that the σ-field extensions L/Q and M/Q areincompatible.

311

312 CHAPTER 5. COMPATIBILITY, REPLICABILITY, MONADICITY

The existence of incompatible extensions plays an important part in thedevelopment of the theory of difference algebra. Of particular concern here isthe fact that the presence of incompatible extensions can inhibit the extensionof difference isomorphisms for difference field extensions. Notice that there is nosuch a phenomenon as incompatibility in the classical field theory: by Theorem1.6.40(ii), for every two field extensions L and M of a field K, there is a fieldextension of K which contains K-isomorphic copies of L and M .

Exercise 5.1.3 As in the preceding example, let us consider Q as an ordinarydifference field with a basic set σ = α where α is the identity automorphism.Let p(x) be an irreducible quadratic polynomial with rational coefficients and leta1 and a2 be its roots in C. Let K denote the ordinary difference field Q(a1, a2)with the identical translation, and let L be the difference field Q(a1, a2) withthe translation β that maps a1 to a2 and a2 to a1. Prove that the differencefield extensions K/Q and L/Q are incompatible.

In what follows we consider some results that play key roles in the study ofthe phenomenon of incompatibility. The first two theorems are natural gener-alizations of the results by R. Cohn [41, Chapter 7] to partial difference fieldextensions.

Theorem 5.1.4 Let K be an inversive difference field with a basic set σ =α1, . . . , αn and let L1 and L2 be two σ∗-field extensions of K. Let M bean overfield of K such that there exist field K-isomorphisms φi of Li into M(i = 1, 2) with the following properties:

a) The images φ1(L1) and φ2(L2) are quasi-linearly disjoint over K;b) The compositum of φ1(L1) and φ2(L2) coincides with M .Then there is a unique way of defining the action of elements of σ as injective

endomorphisms of M so that M becomes a σ-field extension of K and φi becomeσ-K-isomorphisms of Li into M (i = 1, 2).

PROOF. It is easy to see that for any αj ∈ σ (1 ≤ j ≤ n), φ1αj(K) andφ2αj(K) are quasi-linearly disjoint over K (see Theorem 1.6.44(i)). Further-more, the mapping ψij = φiαjφ

−1i (i = 1, 2) is an isomorphism of φi(Li) onto

φi(αj(Li)) which coincides with αj on K. By Theorem 1.6.43, there exists aunique extension of ψ1j and ψ2j to an isomorphism of M onto the compositumof φ1αj(K) and φ2αj(K). This extension defines an injective endomorphismof M that will be denoted by the same letter αj . Furthermore, our construc-tion of the extensions of the endomorphisms αj of K (1 ≤ j ≤ n) shows thatthe corresponding endomorphisms of M are pairwise commuting, so M can betreated as a σ-field extension of K. Clearly, φ1 and φ2 become, respectively,σ-K-isomorphisms of L1 and L2 into M .

The uniqueness of the desired σ-field structure on M follows from the factthat the action of any αj on M should extend ψ1j and ψ2j , and by Theorem1.6.3, such an extension is unique.

5.1. COMPATIBLE AND INCOMPATIBLE EXTENSIONS 313

Corollary 5.1.5 Let K be an inversive difference field with a basic set σ andlet L1 and L2 be two σ∗-field extensions of K. If the field extension L1/K isprimary, then there exist a σ-field extension M of K and σ-K-isomorphismsφi : Li → M (i = 1, 2) such that φ1(L1) and φ2(L2) are quasi-linearly disjointover K.

PROOF. By Theorems 1.6.40(ii) and 1.6.43, there exist an overfield F ofK and K-isomorphisms φi of Li into M (i = 1, 2) such that the fields φ1(L1)and φ2(L2) are quasi-linearly disjoint over K and M is the compositum of thesefields. By Theorem 5.1.4, M has a unique structure of a σ-field extension of Kwith respect to which φ1 and φ2 are σ-K-isomorphisms of L1 and L2 into M ,respectively.

Before stating the next theorem we would like to notice that if L/K is adifference field extension with a basic set σ and S(L/K) is the separable closureof K in L (that is, the set of all elements of L which are algebraic and separableover K), then S(L/K)) is an intermediate σ-filed of L/K.

Theorem 5.1.6 Let K be a difference field with a basic set σ and let L1 and L2

be two σ-field extensions of K. Furthermore, let S1 and S2 denote the separableclosures of K in L1 and L2, respectively. Then the difference field extensionsL1/K and L2/K are compaitible if and only if the σ-field extensions S1/K andS2/K are compaitible.

PROOF. Clearly, if L1/K and L2/K are compaitible, so are S1/K andS2/K. In order to prove the converse statement, assume first that the σ-fieldK is inversive and L1 and L2 are σ∗-field extensions of K. Then S1 and S2 arealso inversive, so they can be also treated as σ∗-field extensions of K. Let M bea σ-field extensions of K such that S1 and S2 have σ-K-isomorphisms into M .Without loss of generality we may assume that M actually contains S1 and S2.By Corollary 5.1.5, the field extensions L1/S1 and M/S1 are compatible.

Let N be a σ-overfield of M such that there exists a σ-S1-isomorphism φof L1 into N . Applying Corollary 5.1.5 once again we obtain that N/S2 andL2/S2 are compatible. Let Q be a σ-overfield of L2 such that there is a σ-S2-isomorphism ψ of N into Q. Then ψφ is a σ-K-isomorphism of L1 into Q. Thus,the σ-field extensions L1/K and L2/K are compatible.

Now suppose that L1/K and L2/K are difference (σ-) but not necessarilyinversive difference field extensions. As usual, for every difference (σ-) field F ,let F ∗ denote the inversive closure of F . Then the σ∗-field extensions S∗

1/K∗

and S∗2/K∗ are compatible. Furthermore, it is easy to see that S∗

1 and S∗2 are

separable closures of K∗ in L∗1 and L∗

2, respectively. It follows that the σ∗-fieldextensions L∗

1/K∗ and L∗2/K∗ are compatible hence the σ-field extensions L1/K

and L2/K are also compatible.

Corollary 5.1.7 Let K be an ordinary difference field with a basic set σ = αand let L1/K and L2/K be two compatible σ-field extensions of K. Then there


exist a σ-field extension M of L2 and a σ-K-isomorphism ρ of L1 into M withthe following property:

If F is any intermediate σ-field of L1/K, then σ-trdegL2〈ρ(F )〉L2〈ρ(L1)〉 =σ-trdegF L1 and EordL2〈ρ(L1)〉/L2〈ρ(F )〉 = EordL1/F .

PROOF. Since passing to inversive closures does not change the differencetranscendence degree and effective order of a σ-field extension, we can assumethat the σ-fields K, L1, L2, and F are inversive. Let S1 and S2 denote the sepa-rable closures of K in L1 and L2, respectively. As in the proof of Theorem 5.1.6,let M be a σ-overfield of K such that S1 and S2 have σ-K-isomorphisms intoM . Without loss of generality we may assume that the σ-field M is generatedby S1 and S2; in particular the field extension M/K is algebraic.

Let N be a σ-field extension of M such that there is a σ-S1-isomorphism φ ofL1 into N . Clearly, if a subset A of L1 is algebraically dependent or independentover K, then the set φ(A) is algebraically dependent or independent, respec-tively, over M and over S2. By Corollary 5.1.5, one can choose a σ-overfield Qof L2 such that there is a σ-S2-isomorphism ψ of L2 into Q, and L2 and ψ(N)are quasi-linearly disjoint over S2. Therefore, if a set B ⊆ N is algebraicallydependent or independent over S2, the set ψ(B) is algebraically dependent orindependent, respectively, over L2.

Let ρ = ψ φ. Then ρ is a σ-K-isomorphism of L1 into Q, and if a setC ⊆ L1 is algebraically dependent or independent over K, then the set ρ(C)is algebraically dependent or independent, respectively, over L1. Therefore, if asubset of L1 is algebraically dependent or independent over F , its image underρ is algebraically dependent or independent, respectively, over L2(ρ(F )). Sincethe order (which is also the effective order in the case of inversive fields) andthe σ-transcendence degree of a σ-field extension are completely determined byits algebraically independent sets, we obtain the conclusion of the corollary.

Theorem 5.1.8 Let K be an inversive difference field with a basic set σ and letL be a σ-overfield of K such that the field extension L/K is primary. Further-more, let φ be a σ-isomorphism of K into L. Then there exists an extension of φto a σ-isomorphism ψ of L into a σ-overfield of M of L such that M = L〈ψ(L)〉,L and ψ(L) are quasi-linearly disjoint over φ(K), and M/L is a primary fieldextension. Furthermore, if L is inversive, then M is also inversive.

If N is a σ-overfield of L such that φ has an extension to a σ-isomorphism χof L into N and N is the free join of L and χ(L) over φ(K), then there exists aunique σ-L-isomorphism ρ : M → N such that ρψ(a) = χρ(a) for every a ∈ L.

PROOF. It is easy to see that φ(K) is an inversive σ-subfield of L. Let usextend φ to a σ-isomorphism φ of L onto a σ-overfield H of φ(K). Then, bythe condition of our theorem, the field extension H/φ(K) is primary. ApplyingCorollary 5.1.5 we obtain that there exist a σ-field extension M of φ(K) andσ-φ(K)-isomorphisms µ : H → M and ν : L → M such that the field exten-sions µ(H)/φ(K) and ν(L)/φ(K) are quasi-linearly disjoint. Without loss ofgenerality we may assume that L is a σ-subfield of M and, with ψ denoting the


composition map µφ, M is the compositum of L and ψ(L). then L and ψ(L)are quasi-linearly disjoint over φ(K).

Since M is the free join of L and ψ(L) over φ(K) and the field extensionψ(L)/φ(K) is primary, we can apply Proposition 1.6.49 and obtain that theextension M/L is also primary. Furthermore, it follows from Proposition 2.1.8that if the σ-field L is inversive, then M = L〈ψ(L)〉 is inversive as well.

Applying Theorem 1.6.40 to the σ-φ(K)-isomorphism χψ−1 of ψ(L) ontoχ(L) and the identity automorphism of L, we obtain that there is a uniqueL-isomorphism ρ of M onto N which coincides with χψ−1 on ψ(L). Since theσ-field structures induced on N by ρ and by the action of σ on L and χ(L)coincide, and since L and χ(L) are quasi-linearly disjoint over φ(K) (by virtueof µ), it follows by Theorem 1.6.43 applied to each αi ∈ σ that the σ-fieldstructure on N is the same as the σ-field structure determined on N by M andρ. Thus, ρ is a σ-isomorphism of M onto N . The uniqueness of ρ in the senseof the statement of the theorem follows from the uniqueness of ρ in the abovesense (as an L-isomorphism M → N which coincides with χψ−1 on ψ(L)).

Exercise 5.1.9 Prove the statement obtained from Theorem 5.1.8 by replacingthe terms “primary” and “quasi-linearly disjoint” by “regular” and “linearlydisjoint”, respectively.

The following theorem is a version of the R. Cohn’s result for ordinary dif-ference field extensions, see [41, Chapter 9, Theorem I].

Theorem 5.1.10 Let K be a difference field with a basic set σ and let L be aσ-overfield of K such that L/K is compatible with every σ-field extension of K.Then every σ-isomorphism φ of K into a σ-overfield M of L can be extendedto a σ-isomorphism of L into a σ-overfield of M .

PROOF. First of all notice that one can extend φ to a σ-isomorphism of Lonto a σ-overfield of φ(K) (this extension will be still denoted by φ). Indeed,all we need to do for constructing such an extension is to choose a subset S ofM \ φ(K) whose cardinality is equal to Card(L \ K), fix a bijection ρ : L →φ(K) ∪ S that extends φ, and make L′ = φ(K) ∪ S a σ-overfield of φ(K) bydefining the operations and action of elements α ∈ σ as follows: if x, y ∈ L′, thenxy = ρ(ρ−1(x)ρ−1(y)), x + y = ρ(ρ−1(x) + ρ−1(y)), and α(x) = ρ(α(ρ−1(x))).

In what follow we fix such an extension of φ onto an overfield of φ(K) anddenote it by the same letter φ. (Of course, φ(L) need not lie in a σ-overfieldof M .) Since φ is a σ-isomorphism, the condition of the theorem imply thatφ(L)/φ(K) is compatible with every σ-field extension of φ(K). Therefore, thereexist a σ-overfield N of M and a σ-φ(K)-isomorphism ψ of φ(L) into N . Obvi-ously, ψφ maps L into N and coincides with φ on the field K.

The next example shows that the requirement of compatibility of L/K withevery σ-field extension of K is essential in the statement of Theorem 5.1.10.


Example 5.1.11 [41, Chapter 9, Example 1]) Let Q be the field of rationalnumbers treated as an ordinary difference field with the identity translation α.Let a and i denote the positive fourth root of 2 and the square root of −1,respectively (we assume a, i ∈ C). Furthermore, let us consider L = Q(i, a)as a difference overfield of Q where the action of α is defined by the equalitiesα(i) = −i and α(a) = −a, and let us treat the field K = Q(i) as an intermediatedifference field of L/Q.

Suppose that φ is a difference K-isomorphism of L into a difference overfieldof L. Since the field extension L/K is normal, φ is an automorphism of L(see Theorem 1.6.8). Also, the equalities a4 = 2 = φ(2) = (φ(a))4 imply thatφ(a) = aik for some positive integer k. Since α(a) = a, α(aik) = −aik. On theother hand, α(aik) = (−i)k(−a) = (−1)k+1(aik). It follows that k is even, henceφ(a2) = a2. Thus, every difference K-isomorphism of L leaves fixed elements ofthe field K〈a2〉. At the same time, it is easy to see that there is a difference K-automorphism ψ of K〈a2〉 such that ψ(a2) = −a2. Obviously, ψ has no extensionto a difference K-isomorphism of L.

The following statement describes a particular class of ordinary differencefield extensions L/K that are compatible with every difference field extensionof K.

Proposition 5.1.12 Let K be an ordinary difference field with a basic set σ =α, Ky1, . . . , ys the algebra of σ-polynomials in σ-indetermiantes y1, . . . , ys

over K and Φ a family of linear homogeneous σ-polynomials in Ky1, . . . , ys.Then

(i) The reflexive closure P of the σ-ideal [Φ] is a prime reflexive σ-ideal.(ii) If η = (η1, . . . , ηs) is a generic zero of P , then the field L = K〈η1, . . . , ηs〉

is purely transcendental over K (in the usual sense of the field theory).(iii) The difference field extension L/K is compatible with every σ-field ex-

tension of K.

PROOF. Let K∗ denote the inversive closure of K and let R denote the al-gebra of σ∗-polynomials in σ∗-indeterminates y1, . . . , ys over K∗ (thus, R can betreated as an inversive closure of the σ-ring Ky1, . . . , ys). Let Φ∗ = αif | i ∈Z, f ∈ Φ ⊆ R and let Ψ denote the set of all linear combinations of ele-ments of Φ∗ with coefficients in K∗. Furthermore, for every k = 0, 1, . . . , letYk = αiyj | 0 ≤ i ≤ k, 1 ≤ j ≤ s and Ψk = Ψ

⋂K[Yk]. Then Ψk is a system of

linear polynomials in the polynomial ring K[Yk]. Clearly, this system is closedunder linear combinations with coefficients in K and generates a prime ideal inK[Yk]. Furthermore, if m,n ∈ N and n > m, then Ψn

⋂K[Ym] = Ψm and every

solution of Ψm as a set of linear polynomials in K[Ym] extends to a solutionof Ψn. Then a generic zero of the ideal Pm = P

⋂K[Ym] in K[Ym] extends to

a solution of Φn and hence of Pn. Therefore, Pm = Pn

⋂K[Ym]. Furthermore,

denoting the set αiyj | 1 ≤ i ≤ k, 1 ≤ j ≤ s (k ≥ 1) by Y ′k and using the

above arguments one obtains that Pn ∩ K[Y ′n] is the ideal generated in K[Y ′

n]by Ψ

⋂K[Y ′

n]. Since for any σ-polynomials f , the inclusion α(f) ∈ Ψ impliesf ∈ Ψ, we obtain that the inclusion α(f) ∈ Pn implies f ∈ Pn.


It follows from the above discussion that P =∞⋃

k=0

Pk is a reflexive prime

σ-ideal in Ky1, . . . , ys and P⋂

K[Yk] = Pk for all k ∈ N. Therefore, P is thereflexive closure of [Φ].

In order to prove (ii), we notice that if η = (η1, . . . , ηs) is a generic zero of P ,then the sk-tuple (η1, . . . , ηs, αη1, . . . , α1ηs, . . . , α

kηs) is a generic zero of Pk. Itfollows that K(η1, . . . , ηs, αη1, . . . , α1ηs, . . . , α

kηs) is purely transcendental overK for k = 0, 1. . . . whence L = K〈η1, . . . , ηs〉 is purely transcendental over K.

Statement (iii) is a direct consequence of Theorem 5.1.6. Let K be a difference field with a basic set σ = α1, . . . , αn. In what follows,

for any αi1 , . . . , αik∈ σ (1 ≤ k ≤ n), (K;αi1 , . . . , αik

) will denote the field Ktreated as a difference field with a basic set αi1 , . . . , αik

.Definition 5.1.13 With the above notation, we say that K satisfies the uni-versal compatibility condition if every two σ-extensions of K are compatible. Kis said to satisfy the stepwise compatibility condition if there exists a permu-tation (i1, . . . , in) of (1, . . . , n) such that all difference fields (K;αi1 , . . . , αik

),1 ≤ k ≤ n satisfy the universal compatibility condition.

Remark 5.1.14 It follows from Theorem 5.1.6 that any algebraically closed orseparably algebraically closed difference field, as well as the inversive closureof such a field (see Proposition 2.1.9(i) ), satisfies the stepwise compatibilitycondition.

Theorem 5.1.15 Let K be a difference field with a basic set σ, α ∈ σ, and Kα

denote the field K treated as a difference field with the basic set σ \ α. If forevery α ∈ σ, Kα satisfies the stepwise compatibility condition, then there existsa σ-overfield L of K such that L is an algebraic closure of K.

PROOF. We proceed by induction on n = Card σ. Suppose first that n = 1,so that K is an ordinary difference field with a basic set σ = α. ApplyingTheorem 1.6.6(v) with L = K, M = K (an algebraic closure of K) and φ = iα,where i is the embedding of K into K, we obtain that there exists an extensionof α to an endomorphism of K. Thus, K can be treated as a σ-overfield of K.

Assume the statement of the theorem to be true for n = q where q is apositive integer. Let K be a difference field with a basic set σ = α1, . . . , αq+1consisting of q + 1 translations such that, for some ordering αi1 , . . . , αiq+1 ofelements of σ, the difference field K ′ = (K;αi1 , . . . , αiq

) satisfies the stepwisecompatibility condition. Of course, K ′ satisfies the condition on K of the state-ment of the theorem, so we can apply the induction hypothesis and, settingσ′ = αi1 , . . . , αiq

, obtain that there exists an algebraic closure H of K thathas a structure of a σ′-overfield of K ′. Since any two σ′-field extensions of K ′

are compatible, we may, by Theorem 5.1.10, extend αiq+1 , considered as a σ′-isomorphism of K ′ into K ′, to a σ′-isomorphism αiq+1 of H into a σ′-overfieldof H. For each element a ∈ H, αiq+1(a) is algebraic over the field αiq+1(K

′)and hence also over K. Therefore (since K admits at most one algebraic closure


in any given overfield of K), αiq+1(a) ∈ H. It follows that the structure of Has a σ′-field, together with αiq+1(K

′), determines a σ-overfield structure on analgebraic closure of K (the action of αq+1 is defined as the action of αiq+1).Thus, the proof is completed by induction.

The first part of the proof of Theorem 5.1.15 leads to the following statementabout ordinary difference fields.

Corollary 5.1.16 Let K be an ordinary difference field with a basic set σ =α. Then there exists an algebraic closure K of K that has a structure of aσ-overfield of K.

Although any ordinary difference field K has a difference overfield which isan algebraic closure of K, the following example provided by I. Bentsen [10]shows that this result cannot be generalized to partial difference fields.

Example 5.1.17 Let us consider Q as a difference field with a basic set σ =α1, α2 where α1 and α2 are the identity automorphisms. Let b and i denotethe positive square root of 2 and the square root of −1, respectively (we assumeb, i ∈ C), and let F = Q(b, i). Let us extend α1 and α2 to automorphisms of thefield F setting α1(b) = −b, α1(i) = i, α2(b) = b, α2(i) = −i, and consider F asa σ-overfield of Q. Then there exists no σ-overfield of F which is an algebraicclosure of F . Indeed, if it is not so, then there exists a σ-overfield G of F whichcontains an element a such that a2 = b. Since (α1(a))2 = α1(a2) = −b and(α2(a))2 = α2(a2) = b, we have α1(a) = λai and α2(a) = µa where λ and µdenote plus or minus 1. Then α1α2(a) = λµia and α2α1(a) = −λµia. Thus,α1 and α2 do not commute at a, which contradicts the assumption that G is aσ-overfield of F .

The following example, also due to I. Bensten [10], shows that the converseof Theorem 5.1.15 is not true.

Example 5.1.18 Let Q be the field of rational numbers treated as a differ-ence field with two translations α1 and α2 each of which acts as an identityautomorphism on Q. α1 and α2 extend trivially to identity automorphisms ofthe algebraic closure H of Q contained in C. Let K denote Q treated as anordinary difference field with a basic set σ′ = α where α is either α1 or α2,and let F = Q(i) (i2 = −1). Let L1 and L2 denote the σ′-field extensions of Kconstructed on F via the continuations of α such that α(i) = i and α(i) = −i,respectively. (As usual, we denote the continuation by the same letter α.) Thenthe σ′-field extensions L1/K and L2/K are incompatible (see Example 5.1.2),so for either definition of K, K does not satisfy the universal compatibilitycondition.

5.2. DIFFERENCE KERNELS 319

5.2 Difference Kernels over Ordinary DifferenceFields

In this section we consider R. Cohn’s theory of difference kernels over ordinarydifference fields. This theory plays an important role in the theory of ordinaryalgebraic difference equations; in particular, the technique of difference kernels,as we shall see in Chapter 7, allows one to prove the fundamental existencetheorem for ordinary difference polynomials. Below we basically follow [41]; thecorresponding theory for partial difference fields is presented in Chapter 6.

Definition 5.2.1 Let K be an ordinary inversive difference field with a basicset σ = α. A difference (or σ-) kernel of length r ≥ 1 over K is an orderedpair R = (K(a0, . . . , ar), τ) where each ai is itself an s-tuple (a(1)

i , . . . , a(s)i ) over

K (a positive integer s is fixed) and τ is an extension of α to an isomorphismof K(a0, . . . , ar−1) onto K(a1, . . . , ar) such that τai = ai+1 for i = 0, . . . , r − 1.(In other words, τ(aj)

i ) = a(j)i+1 for 0 ≤ i ≤ r − 1, 1 ≤ j ≤ s.) If r = 0, then

R = (K(a0), τ) where τ = α : K → K.Two kernels R = (K(a0, . . . , ar), τ) and R′ = (K(b0, . . . , bq), τ ′) (where

b0 has the same number of coordinates as a0) are said to be equivalent (orisomorphic) if q = r and there exists a K-isomorphism φ : K(a0, . . . , ar) →K(b0, . . . , br) such that φ(ai) = bi (i = 0, . . . , r) φτ = τ ′φ on K(a0, . . . , ar−1).

Note, that in the last definition and below we use the following convention:if elements x1, . . . , xm belong to the domain of a mapping φ and x denotes them-tuple (x1, . . . , xm), then φ(x) is the m-tuple (φ(x1), . . . , φ(xm)).

Definition 5.2.2 With the above notation, the transcendence degree of the dif-ference kernel R is defined to be trdegK(a0,...,ar−1)K(a0, . . . , ar); it is denotedby δR.

In what follows, while considering difference kernels over an ordinary differ-ence field K, we always assume that K is inversive.

Definition 5.2.3 Let R = (K(a0, . . . , ar), τ) be a difference kernel of length rover an ordinary inversive difference field K. A prolongation R′ of R is definedas a difference kernel of length r+1 consisting of an overfield K(a0, . . . , ar, ar+1)of K(a0, . . . , ar) and an extension τ ′ of τ to an isomorphism of K(a0, . . . , ar)onto K(a1, . . . , ar+1).

As before, a set of the form a = a(i)|i ∈ I will be referred to as an indexinga (with the index set I), and if J ⊆ I, then the set a(i)|i ∈ J will be calleda subidexing of a. If R = (K(a0, . . . , ar), τ) is a difference kernel and a0 is asubindexing of a0 (that is, a0 = (a(i1)

0 , . . . , a(iq)0 ), 1 ≤ i1 < · · · < iq ≤ s), then

ak (k = 1, . . . ) will denote the corresponding subindexing of ak.

Theorem 5.2.4 Every difference kernel R = (K(a0, . . . , ar), τ) over an or-dinary difference field K with a basic set σ = α has a prolongation R′ =


(K(a0, . . . , ar, ar+1), τ ′). Moreover, one can chose a prolongation R′ with thefollowing properties.

(i) If a0 is a subindexing of a0 such that ar is algebraically independent overK(a0, . . . , ar−1), then ar+1 is algebraically independent over K(a0, . . . , ar) (sothat δR = δR′).

(ii) The setr+1⋃i=0

ai is algebraically independent over K.

PROOF. Let P be the prime ideal with generic zero ar of the polyno-mial ring K(a0, . . . , ar−1)[X1, . . . , Xs] in indeterminates X1, . . . , Xs. Let P ′ bethe set obtained from P by replacing the coefficients of the polynomials ofP by their images under τ . Clearly, P ′ is a prime ideal of the polynomialring K(a1, . . . , ar)[X1, . . . , Xs]. Let J denote the ideal of the polynomial ringK(a0, a1, . . . , ar)[X1, . . . , Xs] generated by P ′, let Q be an essential prime di-visor of the ideal J in , and let ar+1 be a generic zero of Q. Since ar+1 isalso a generic zero of P ′, Theorem 1.2.42(iv) implies that there is an iso-morphism τ ′ of K(a0, . . . , ar) onto K(a1, . . . , ar+1) which extends τ . Thus,R′ = (K(a0, . . . , ar+1), τ ′) is a prolongation of R.

Let a0 be a subindexing of a0 such that ar is algebraically independent overK(a0, . . . , ar−1). Then ar is contained in a transcendence basis br of ar overK(a0, . . . , ar−1). Then the existence of the isomorphism τ ′ implies that br+1 isa transcendence basis of ar+1 over K(a1, . . . , ar). By Theorem 1.2.42(viii), everycomplete set of parameters of P ′ is a complete set of parameters of Q, hencebr+1 is also a transcendence basis of ar+1 over K(a0, . . . , ar). It follows that br+1

and therefore its subset ar+1 are algebraically independent over K(a0, . . . , ar).To prove statement (ii) note that for every i = 0, . . . , r+1, ai is algebraically

independent over K(a0, . . . , ai−1). Indeed, we have proved this for i = r+1, it istrue for i = r by the condition of the theorem, and for every i = 0, . . . , r− 1 thestatement follows from the fact that τ r−i is an isomorphism of K(a0, . . . , ai) ontoK(ar−i, . . . , ar). Therefore, ai is algebraically independent over K(a0, . . . , ai−1)

whence the setr+1⋃i=0

ai is algebraically independent over K.

Definition 5.2.5 Let R = (K(a0, . . . , ar), τ) be a difference kernel of length rover an ordinary inversive difference field K. A prolongation R′ of the kernelR is called generic if δR′ = δR.

A generic prolongation of a difference kernel R = (K(a0, . . . , ar), τ) over aninversive ordinary difference field can be constructed as in the proof of Theorem5.1.4. Indeed, let P be the prime ideal with generic zero ar of a polynomial ringK(a0, . . . , ar−1)[X1, . . . , Xs] in s indeterminates X1, . . . , Xs. Let P ′ be obtainedfrom P by replacing the coefficients of the polynomials of P by their imagesunder τ . Then P ′ is a prime ideal of K(a1, . . . , ar)[X1, . . . , Xs] and generates anideal P in K(a0, . . . , ar)[X1, . . . , Xs]. Let Q be an essential prime divisor of Pin the last ring and let ar+1 be a generic zero of Q. Then ar+1 is also a generic


zero of P ′ and there is an isomorphism τ ′ : K(a0, . . . , ar) → F (a1, . . . , ar+1)that extends τ . We obtain the desired generic prolongation.

Conversely, if R′ = (K(a0, . . . , ar+1), τ ′) is any generic prolongation of R,then ar+1 is a solution of the ideal P and hence of one of its essential primedivisors Q. It follows that cohtQ = coht P = trdegK(a0,...,ar−1)K(a0, . . . , ar) =trdegK(a0,...,ar)K(a0, . . . , ar+1), so that ar+1 is a generic zero of Q. Thus, everygeneric prolongation of R can be constructed by the foregoing procedure.

Example 5.2.6 (see [41, Chapter 6, section 2]). Let K be an inversive ordi-nary difference field of zero characteristic with a basic set σ = α, Ky thering of σ-polynomials in one σ-indeterminate y over K, and A an irreducibleσ-polynomial in Ky (that is, A is irreducible as a polynomial in variablesy, αy, α2y, . . . over K). Assuming that A contains y and αmy, m > 0, is thehighest transform of y in A, we shall use prolongations of difference kernels toconstruct a solution of A. First, let us consider an m-tuple a = (a(1), . . . , a(m))whose coordinates constitute an algebraically independent set over K. Now wedefine an m-tuple a1 as follows: we set a

(i)1 = a(i+1) for i = 1, . . . , m− 1, replace

yi−1 by a(i) in A (1 ≤ i ≤ m), find a solution of the resulting polynomial in oneunknown ym and take it as a(m). Since A involves y, a(1) will be algebraically de-pendent on a(2), . . . , a(m), a

(m)1 over K. Therefore, a

(1)1 , . . . , a

(m)1 are algebraically

independent over K whence there is a difference kernel R1 defined over K by theextension of the translation α to an isomorphism τ0 : K(a) → K(a1). By succes-sive application of Theorem 5.2.4 we find a sequence a0 = a, a1, . . . such that akernel Rk+1 is defined by an isomorphism τk : K(a0, . . . , ak) → K(a1, . . . , ak+1)(k = 0, 1, . . . ) and Rk+1 is a prolongation of Rk. Then K(a0, a1, . . . ) becomes adifference overfield of K where the extension of α (denoted by the same letter)is defined by α(b) = τk(b) whenever b ∈ K(a0, . . . , ak). It is clear that this fieldcoincides with K〈a(1)〉 and the element a(1) is a solution of A.

Theorem 5.2.7 Let R = (K(a0, . . . , ar), τ) be a difference kernel over an in-versive ordinary difference field K with a basic set σ = α.

(i) There are only finitely many distinct (that is, pairwise non-isomorphic)generic prolongations of R.

(ii) Let R′ = (K(a0, . . . , ar+1), τ ′) be a generic prolongation of a R and leta0 be a subindexing of a0 which is algebraically independent over K(a1, . . . , ar).Then a0 is algebraically independent over K(a1, . . . , ar+1).

PROOF. Let Q1, . . . , Qk be all essential prime divisors of the ideal J de-fined in the proof of Theorem 5.2.4. As we have mentioned in the discussion afterDefinition 5.2.5, every Qi yields a generic prolongation R′ of R and all genericprolongation can be obtained in this way. To prove the second part of the theo-rem, one may assume that a0 is a σ-transcendence basis of a0 over K(a1, . . . , ar)(that is, a transcendence basis of K(a0, . . . , ar) over K(a1, . . . , ar)). Since τ ′ is anisomorphism of K(a0, . . . , ar) onto K(a1, . . . , ar+1), trdegKK(a0, . . . , ar−1) =trdegKK(a1, . . . , ar) and trdegKK(a0, . . . , ar) = trdegKK(a1, . . . , ar+1). Ap-plying the additive property of transcendence degree (see Theorem 1.6.30(iv))


we obtain trdegK(a1,...,ar+1)K(a0, . . . , ar+1) = trdegK(a0,...,ar)K(a0, . . . , ar+1) =trdegK(a0,...,ar−1)K(a0, . . . , ar) = trdegK(a1,...,ar)K(a0, . . . , ar).

Since a0 contains a transcendence basis of a0 over K(a1, . . . , ar+1), the lastequalities show that a0 is such a basis, hence a0 is algebraically independentover K(a1, . . . , ar+1).

Let R = (K(a0, . . . , ar), τ) be a difference kernel over an inversive ordinarydifference field K with a basic set σ = α and let b0 be a subindexing of a0. Ifbr is a transcendence basis of ar over K(a0, . . . , ar−1), then b0 is called a specialset. Clearly, such a set consists of δR elements and it is also a special set for anygeneric prolongation R′ of R. If b0 is a subindexing of a0 such that b0 containsa special set and trdegK(b0,...,br−1)K(b0, . . . , br) = δR, then the order of R withrespect to b0 is defined as ordb0R = trdegK(b0,...,br)K(a0, . . . , ar). (In this case,b0 is said to be a subindexing of a0 for which ordb0R is defined.)

Suppose that b0 itself is a special set. Then it is easy to see that ordb0R isdefined and ordb0R = trdegK(b0,...,br)K(a0, . . . , ar) = trdegK(b0,...,br)K(a0, . . . ,ar−1, br) = trdegK(b0,...,br−1)K(a0, . . . , ar−1), see Theorem 1.6.30(iv) . If δR = 0,we consider b0 = ∅ and define the order ordR of the kernel R as ordR =trdegKK(a0, . . . , ar) = trdegKK(a0, . . . , ar−1). If r = 0, then ordb0R is definedif b0 contains a special set which is a transcendence basis of K(a0) over K. Inthis case ordb0R = 0.

If a subindexing b0 of a0 is a special set, we define the degree db0R and reduceddegree rdb0R of R with respect to b0 to be K(a0, . . . , ar) : K(a0, . . . , ar−1; br)and [K(a0, . . . , ar) : K(a0, . . . , ar−1; br)]s, respectively. (As usual, if L/K is afield extension, then [L : K]s denotes the separable degree of L over K.)

Theorem 5.2.8 With the above notation, let b0 be a subindexing of a0 for whichordb0R is defined.

(i) If R′ = (K(a0, . . . , ar+1), τ ′) is a generic prolongation of a kernel R =(K(a0, . . . , ar), τ), then ordb0R′ is defined and ordb0R = ordb0R′.

(ii) Suppose that b0 itself is a special set. Then db0R and rdb0R are finite.Furthermore, if R′

1, . . . ,R′h are all distinct (pairwise non-isomorphic) prolonga-

tions of R, then∑h

i=1 rdb0R′i = rdb0R and

∑hi=1 db0R′

i ≤ db0R. (Of course, inthe case of characteristic 0 the last inequality becomes an equality.)

PROOF. (i) Since ordb0R is defined, b0 contains a special set which, as wehave noticed, is also a special set for the generic prolongation R′. Since everytranscendence basis of br+1 over K(a0, . . . , ar) is algebraically independent overK(b0, . . . , br), we obtain that

trdegK(b0,...,br)K(b0, . . . , br+1) ≥ trdegK(a0,...,ar)K(a0, . . . , ar, br+1)= trdegK(a0,...,ar)K(a0, . . . , ar+1) = δR′. (5.2.1)

Applying the isomorphism τ ′ we obtain that

trdegK(b0,...,br)K(b0, . . . , br+1) ≤ trdegK(b1,...,br)K(b1, . . . , br+1)= trdegK(b0,...,br−1)K(b0, . . . , br) = δR . (5.2.2)


Since δR = δR′ (R′ is a generic prolongation of R), the equalities (5.2.1) and(5.2.2) imply that

trdegK(b0,...,br)K(b0, . . . , br+1) = δR′ . (5.2.3)

Thus, ordb0R′ is defined and, moreover, equations (5.2.1) and (5.2.3) show that

trdegK(b0,...,br)K(b0, . . . , br+1) = trdegK(a0,...,ar)K(a0, . . . , ar+1) .

whence

trdegKK(a0, . . . , ar) = trdegKK(b0, . . . , br) + trdegK(b0,...,br)K(b0, . . . , br+1)= trdegKK(b0, . . . , br) + ordb0R.

The last two equalities imply that

trdegKK(a0, . . . , ar+1) = trdegK(a0,...,ar)K(a0, . . . , ar+1)+trdegKK(a0, . . . , ar)= trdegK(b0,...,br)K(b0, . . . , br+1)+trdegKK(b0, . . . , br)

+ordb0R = trdegKK(b0, . . . , br+1) + ordb0R.

On the other hand, by the definition of the order we have

trdegKK(a0, . . . , ar+1) = trdegKK(b0, . . . , br+1) + ordb0R′

whence ordb0R = ordb0R′.(ii) In the prove the second part of the theorem we use the notation of the

proof of Theorem 5.2.4: P denotes the prime ideal of the polynomial ringK(a0, . . . , ar−1)[X1, . . . , Xs] with generic zero ar, P ′ is the prime ideal ofthe ring K(a1, . . . , ar)[X1, . . . , Xs] obtained from P by replacing the coeffi-cients of the polynomials in P by their images under τ , and J is the ideal ofK(a0, a1, . . . , ar)[X1, . . . , Xs] generated by P ′. Let Q1, . . . , Qm be the essentialprime divisors of the ideal J . Then each Qi produces a generic prolongationR′

i of R, and every generic prolongation of R is isomorphic to one of R′i (see

the proof of Theorem 5.2.4 and the remark after Definition 5.2.5). Since b0 is aspecial set for each R′

i, db0R′i and rdb0R′

i (1 ≤ i ≤ m) are defined.Let u0 be a generic zero of the ideal P ′ of K(a1, . . . , ar)[X1, . . . , Xs] and

let v0 be a the subindexing of u0 with the same superscripts as the coor-dinates of b0. Let z1, . . . , zk be the complete set of conjugates of u0 overK(a1, . . . , ar, v0) (see the corresponding definition after Theorem 1.6.23). Theneach Qj (1 ≤ j ≤ m) has some zij

as a generic zero. Let c1, . . . , cm be allsuch generic zeros. Of course, they form a maximal subset of z1, . . . , zkpairwise not equivalent under a K(a0, . . . , ar, v0)-isomorphism. Since db0R =K(a0, . . . , ar) : K(a0, . . . , ar−1, br) = K(a1, . . . , ar, u0) : K(a1, . . . , ar−1, v0),db0R′ = K(a0, . . . , ar, ci) : K(a0, . . . , ar, v0) (1 ≤ i ≤ m) and the correspondingequalities hold for the reduced degrees, statement (ii) of the theorem followsfrom equalities (1.6.1) - (1.6.3).


Let K be an ordinary difference field with a basic set σ = α and L =K〈η1, . . . , ηs〉 a σ-overfield of K generated by an s-tuple η = (η1, . . . , ηs). Thenthe contraction of α to an isomorphism

τr : K(η, α(η), . . . , αr−1(η)) → K(α(η), . . . , αr(η))

(r = 1, 2, . . . ) defines a difference kernel R = (K(a0, . . . , ar), τr) of length rover K.

Conversely, starting with a difference kernel one can introduce the conceptof its realization as follows.

Definition 5.2.9 With the above notation, let R = (K(a0, . . . , ar), τ) be a dif-ference kernel with a0 = (a(1)

0 , . . . , a(s)0 ). An s-tuple η = (η1, . . . , ηs) with coor-

dinates from a σ-overfield of K is called a realization of R if η, α(η), . . . , αr(η)is a specialization of a0, . . . , ar over K. If this specialization is generic, the re-alization is called regular.

If there exists a sequence R(0) = R,R(1),R(2), . . . of kernels, each a genericprolongation of the preceding, such that η is a regular realization of each R(i),then η is called a principal realization of R.

Theorem 5.2.10 Let R = (K(a0, . . . , ar), τ) be a difference kernel over aninversive ordinary difference field K with a basic set σ = α.

(i) There exists a principal realization of the kernel R. If η is such a real-ization, then σ-trdegKK〈η〉 = δR.

(ii) Let b0 be a subindexing of a0 such that ordb0R is defined and letζ is the corresponding subindexing of a principal realization η of R. ThentrdegK〈ζ〉K〈η〉 = ordb0R. Furthermore, if b0 is a special set, then ζ is aσ-transcendence basis of K〈η〉 over K.

(iii) The number of distinct principal realizations of the kernel R is finite. Letus denote them by (1)η, . . .(m) η. If b0 is a special set and (i)ζ is the corresponding

subset of the components of (i)η (1 ≤ i ≤ m), thenm∑

i=1

rld(K〈(i)η〉/K〈(i)ζ〉) =

rdb0R andm∑

i=1

ld(K〈(i)η〉/K〈(i)ζ〉) ≤ db0R.

(iv) If a subindexing b0 of a0 is a special set for the kernel R and ζ is the cor-responding subindexing of a principal realization η of R, then trdegK〈ζ〉K〈η〉 =trdeg(K〈ζ〉)∗(K〈η〉)∗ if and only if b0 is algebraically independent over the fieldK(a0, . . . , ar) or b0 = ∅.

(v) If η is a regular realization of the kernel R, but not a principal realization,then σ-trdegKK〈η〉 < δR.

(vi) A realization of the kernel R which specializes over K to a principalrealization is a principal realization, and the specialization is generic.

PROOF. Repeating the proof of Theorem 5.2.4 we find a sequence of differ-ence kernels R = R0,R1,R2, . . . such that each kernel Rm = (K(a0, . . . , ar+m),


τm) is a generic prolongation of Rm−1 (m = 1, 2, . . . ). The isomorphisms τm ofK(a0, . . . , ar+m−1) onto K(a1, . . . , ar+m) define the automorphism of the fieldK(a0, a1, . . . ) which coincides with τm on every its subfield K(a0, . . . , ar+m)(m ∈ N). Denoting this automorphism by α, we obtain a structure of thedifference field extension K〈a0〉 on the field K(a0, a1, . . . ), and a0 is a principalrealization of R. Furthermore, as it follows from the remark after Definition5.2.5, every distinct principal realizations of R are obtained in this way (bychoosing all distinct principal prolongations Rm of Rm−1 (m = 1, 2, . . . ) in theabove sequence).

Let b0 be a subindexing of a0 such that ordb0R is defined. Since b0 con-tains a special set, trdegK(a0,...,ar+m,br+m+1,...,br+m+k)K(a0, . . . , ar+m+k) = 0 forall m, k ∈ N. By Theorem 5.2.8(i), trdegK(b0,...,br+m+k)K(a0, . . . , ar+m+k) =ordb0R whence trdegK(b0,...,br + m + k)K(a0, . . . , ar + m, br + m + 1, . . . , br + m + k) =ordb0R. It follows that a transcendence basis of the coordinates of a0, . . . , ar+m

over K(b0, . . . , br+m) remains algebraically independent over K(b0, . . . , br+m+k)for every k ∈ N, hence it remains algebraically independent over K(b0, b1, b2, . . . ).Thus, trdegK(b0,b1,b2,... )K(a0, . . . , ar+m+k, b0, b1, b2, . . . ) = ordb0R.

Since the last equality is true for every m ∈ N, we obtain the equalitytrdegK〈b0〉K〈a0〉 = ordb0R which implies that b0 contains a σ-transcendencebasis of K〈a0〉 over K. If b0 is a special set, such a basis is a σ-transcendence basisof a over K. Indeed, Theorem 5.2.4(ii) shows that the set

⋃mi=1 bi is algebraically

independent over K for every m ∈ N, so that b0 is σ-algebraically independentover K. Since a special set contains δR elements, σ-trdegKK〈a〉 = δR. Sinceall principal realizations can be obtained in the process used to obtain a0, thiscompletes the proof of the first two statements of the theorem.

In order to prove statement (iii) notice that for any special set b0 and any m ∈N we have db0Rm+1 ≤ db0Rm and rdb0Rm+1 ≤ rdb0Rm (see theorem 6.1.8(ii)).Therefore, there exists p ∈ N such that db0Rm = db0Rp and rdb0Rm = rdb0Rp

for all integer m ≥ p. It follows that for all such m we have K(a0, . . . , ar+m) :K(a0, . . . , ar+m−1, br+m) = db0Rp.

By Theorem 5.2.4, each set br+m+i (i = 1, 2, . . . ) is algebraically independentover K(a0, . . . , ar+m+i−1), hence the set

⋃∞i=1 br+m+i is algebraically indepen-

dent over K(a0, . . . , ar+m).Let U =

⋃∞j=0 bj . Applying Theorem 1.6.28(v) we obtain that K(a0, . . . ,

ar+m, U) : K(a0, . . . , ar+m−1, U) = db0Rp for all m ≥ p, whence ld(K〈a0〉/K〈b0〉) = db0Rp. Similarly one can obtain that rld(K〈a0〉/K〈b0〉) = rdb0Rp.

Theorem 5.2.8(ii) shows that the sum of all reduced degrees with respect toa special set b0 of all distinct kernels obtained from R by m successive prolon-gations is rdb0R and the sum of their degrees is at most db0R. This implies theinequality and equality in part (iii) of the theorem.

Let us prove statement (iv). If trdegK〈ζ〉K〈η〉 = trdegK〈ζ〉∗K〈η〉∗ and b0 = ∅,then one has trdegK(ζ,α(ζ),α2(ζ),... )K(ζ, α(η), α2(η), . . . ) = trdegK(ζ,α(ζ),α2(ζ),... )

K(η, α(η), α2(η), . . . ) = trdegK(α(ζ),α2(ζ),... )K(α(η), α2(η), . . . ).


(The last equality is obtained from the second one by applying α to the fieldsK(ζ, α(ζ), α2(ζ), . . . ) and K(η, α(η), α2(η), . . . )).

It follows that

trdegK(α(η),α2(η),... )K(ζ, α(η), α2(η), . . . )

= trdegK(α(ζ),α2(ζ),... )K(ζ, α(ζ), α2(ζ), . . . ).

Since ζ is algebraically independent over the field K(α(ζ), α2(ζ), . . . ), ζ is alsoalgebraically independent over this field, hence b0 is algebraically independentover K(a0, . . . , ar).

Conversely, suppose that b0 is algebraically independent over K(a1, . . . , ar)or b0 = ∅. Since b0 is a special set, we have trdegK(b0,...,br)K(a0, . . . , ar) =trdegK(b0,...,br−1)K(a0, . . . , ar−1) = trdegK(b1,...,br)K(a1, . . . , ar) hence

trdegK(a1,...,ar)K(a0, . . . , ar) = trdegK(b1,...,br)K(b0, . . . , br).

At the same time, our assumptions about b0 imply thattrdegK(a1,...,ar)K(b0, a1, . . . , ar) = trdegK(b1,...,br)K(b0, . . . , br) hence

trdegK(a1,...,ar)K(b0, a1, . . . , ar) = trdegK(a1,...,ar)K(a0, . . . , ar).

The last equality shows that every coordinate of a0 is algebraic over the fieldK(b0, a1, . . . , ar). Furthermore, using induction on k ∈ N, one can easily obtainthat every coordinate of η, α(η), . . . , αk(η) is algebraic over the field K(ζ, α(ζ),. . . , αk(ζ), αk+1(η), . . . , αk+r(η)) and hence over the field K(ζ, α(ζ), . . . , αk(ζ),αk+1(η), αk+2(η), . . . ). We arrive at the equality trdegK(ζ,α(ζ),... )K(η, α(η), . . . )= trdegK(ζ,α(ζ),... )K(ζ, α(ζ), . . . , αk(ζ), αk+1(η), αk+2(η), . . . ) which, togetherwith Proposition 4.1.19, implies statement (iv) of our theorem.

Let η = (η1, . . . , ηs) be a regular realization of R but not a principal real-ization of this kernel. Let p = σ-trdegKK〈η〉 and let ηi1 , . . . , ηip

(1 ≤ i1 < · · · <ip ≤ s) be a σ-transcendence basis of K〈η〉 over K. Then for every q ∈ N, theset αj(ηiν

) | 0 ≤ j ≤ q, 1 ≤ ν ≤ p is algebraically independent over K hence

trdegKK(η, α(η), . . . , αq(η)) ≥ p(q + 1). (5.2.4)

Since η is not a principal realization of R, there exists the smallest integer d,d ≥ r, such that the kernel K(η, α(η), . . . , αd+1(η)), τd+1) is not a generic pro-longation of the kernel K(η, α(η), . . . , αd(η)), τd)(we use the notation introducedbefore Definition 5.2.9). Then αd+1(η) is a zero but not a generic zero of an idealof dimension δR similar to the ideal Q in the proof of Theorem 5.2.4. Therefore,

trdegK(η,α(η),...,αd(η))K(η, α(η), . . . , αd+1(η)) ≤ δR− 1.

It is easy to see that for every m ∈ N, trdegK(η,α(η),...,αd+m(η))

K(η, α(η), . . . , αd+m+1(η)) ≤ trdegK(α(η),...,αd+m(η))K(α(η), . . . , αd+m+1(η)) =trdegK(η,α(η),...,αd+m−1(η))K(η, α(η), . . . , αd+m(η)). By induction on m we obtain


that trdegK(η,α(η),...,αd+m(η))K(η, α(η), . . . , αd+m+1(η)) ≤ δR − 1 for everym ∈ N hence

trdegKK(η, α(η), . . . , αq(η)) ≤ trdegKK(η, α(η), . . . , αd(η)) + q(δR− 1).(5.2.5)

Considering equalities (5.2.4) and (5.2.5) for large q we obtain that p = σ-trdegKK〈η〉 ≤ δR− 1 < δR.

In order to prove the last statement of the theorem, consider a realizationη of the kernel R and a principal realization ξ of R such that η σ-specializesto ξ over K. Since the defining difference ideal of η is contained in the definingdifference ideal of ξ, σ-trdegKK〈η〉 ≥ σ-trdegKK〈ξ〉 = δR. The last inequality,together with part (v) of our theorem, implies that η is a principal realizationof the kernel R and σ-trdegKK〈η〉 = σ-trdegKK〈ξ〉. Let a subindexing b0 ofa0 be a special set, and let ζ and λ be subindexing of η and ξ, respectively,that correspond to the subindexing b0. Part (ii) of our theorem shows that ζand λ are σ-transcendence bases of K〈η〉 and K〈ξ〉, respectively, over K andσ-trdegK〈ζ〉K〈η〉 = σ-trdegK〈λ〉K〈ξ〉. It follows that the specialization of η to ξis generic.

Let K be an inversive ordinary difference field with a basic set σ = α andlet η = (η1, . . . , ηs) be an s-tuple with coordinates in some σ-overfield of K.Denoting the extension of α to K〈η〉 by the same letter α, we see that for anyr ∈ N, η is a regular realization of the kernel R = (K(η, α(η), . . . , αr(η)), τ)where τ is the restriction of α on K(η, α(η), . . . , αr−1(η)). If r is sufficientlylarge, then η is the only principal realization of R.

To prove this, let us consider the ring of σ-polynomials Ky1, . . . , ys in σ-indeterminates y1, . . . , ys over K. Let P be the prime σ∗-ideal of Ky1, . . . , yswith generic zero η and let Σ = g1, . . . , gd be a basis of P (that is, Σ is a setof generators of P as a perfect σ-ideal in Ky1, . . . , ys; by Theorem 2.5.11, onecan always choose a finite basis of P ). Furthermore, let r denote the maximalnumber i such that some term of the form αiyj (1 ≤ j ≤ s) appears in one of theσ-polynomials g1, . . . , gd. It is easy to see that if ζ = (ζ1, . . . .ζs) is a realizationof the kernel R = (K(η, α(η), . . . , αr(η)), then ζ annuls every gj (1 ≤ j ≤ d)and therefore every σ-polynomial in P . It follows that η σ-specializes to ζ overK. If ζ is a principal realization of R, then Theorem 5.2.10(vi) shows that ηis also a principal realization of this kernel and that η is the only (up to aσ-K-isomorphism) principal realization of R.

We have arrived at the following statement.

Proposition 5.2.11 Let K be an inversive ordinary difference field with a basicset σ and let η = (η1, . . . , ηs) be an s-tuple with coordinates in some σ-overfieldof K. Then η is the unique principal realization of a kernel R over K such thatevery realization of R is a specialization of η over K.

Remark 5.2.12 Note that every kernel is equivalent to a kernel of length 0 or 1in the sense that its realizations generate the same extensions. Indeed, with the


notation of Definition 5.2.1 (with a kernel R = (K(a0, . . . , ar), τ), r > 1), onecan define two rs-tuples b0 and b1 with coordinates of the s-tuples a0, . . . , ar−1

and a1, . . . , ar, respectively. Then (K(b0, b1), τ) is a kernel of length 1 equivalentto R.

5.3 Difference Specializations

Recall that by a difference (σ-) specialization of an indexing a = a(i)|i ∈ Iover a difference field K with a basic set σ we mean a σ-K-homomorphismsφ of Ka into a σ-overfield of K (the indexing φa = φa(i)|i ∈ I is alsocalled a σ-specialization of a over K). Obviously, an indexing b = b(i)|i ∈ Iof elements of some σ-overfield of K is a σ-specialization of a if and only if(τ(bi) | i ∈ I, τ ∈ Tσ is a specialization of the indexing (τ(ai) | i ∈ I, τ ∈ Tσin the sense of Definition 1.6.55. Also, every σ-specialization over K naturallydefines a σ-specialization of every indexing in Ka (but not necessarily ofevery indexing in K〈a〉). If no confusion can result, we shall omit reference toa σ-field K and/or basic set σ while talking about difference specializations(so “specializations” will mean σ-specialization over a difference field underconsideration). Also, we shall often use notation a for a specialization of anindexing a or its coordinates (so that a specialization of an indexing a = a(i)|i ∈I will be denoted by a = a(i)|i ∈ I). If b = a(j)|j ∈ J is a subindexing ofa (J ⊆ I) and a is a specialization of a, then b will denote the correspondingspecialization of b. In particular, if the index set I is finite, I = 1, . . . , s,b = (a(i1), . . . , a(ik)) is a subindexing of a = (a(1), . . . , a(s)) (1 ≤ i1 < · · · <ik ≤ s) and a = (a(1), . . . , a(s)) is a specialization of a, then b will denote thespecialization (a(i1), . . . , a(ik)) of b. Recall that a specialization φ is called genericif it is a σ-isomorphism. Otherwise it is called proper. Note that if φ : a → b isa generic specialization over K, then φ has a unique extension to a differenceisomorphism of K〈a〉 onto K〈a〉.

Proposition 5.3.1 Let K be a difference field with a basic set σ and let a =(a(1), . . . , a(s)) be a σ-specialization of an s-tuple a = (a(1), . . . , a(s)) over K.Furthermore, let b = (a(i1), . . . , a(ik)) be a subindexing of a and b the correspond-ing σ-specialization of b. Then, if b is a σ-algebraically independent set over K,so is b, and hence σ-trdegKK〈a(1), . . . , a(s)〉 ≤ σ-trdegK〈a(1), . . . , a(s)〉.

PROOF. For every r ∈ N, our σ-specialization of b naturally induces aspecialization of the indexing Σr = τ(a(ij) | (τ, j) ∈ Tσ(r) × 1, . . . , k (weshall denote this specialization by Σr). By Theorem 1.6.58, trdegKK(Σr) ≤trdegKK(Σr) for all r ∈ N, hence the leading coefficient of the difference di-mension polynomial of the σ-field extension K〈b〉/K associated with the setof σ-generators a(i1), . . . , a(ik) is less than or equal to the leading coefficientof the difference dimension polynomial of the σ-field extension K〈b〉/K asso-ciated with the set of σ-generators a(i1), . . . , a(ik). Applying Theorem 4.2.1(iii)we obtain the inequality σ-trdegKK〈b〉 ≤ σ-trdegKK〈b〉 which implies the first

5.3. DIFFERENCE SPECIALIZATIONS 329

statement of the theorem and (after setting b = a) the desired inequality σ-trdegKK〈a(1), . . . , a(s)〉 ≤ σ-trdegKK〈a(1), . . . , a(s)〉.

Theorem 5.3.2 (R. Cohn) Let K be an ordinary difference field with a basicset σ = α and let η and ζ be two finite indexings in a difference overfield ofK such that the field extension K〈η, ζ〉/K〈η〉 is primary. Furthermore, let c bea nonzero element in Kη, ζ and let B be a σ-transcendence basis of ζ overK〈η〉. Then almost every σ-specialization η of η over K can be extended to aσ-specialization η, ζ of η, ζ over K such that

(i) c = 0.(ii) Every σ-transcendence basis of ζ over K〈η〉 specializes to a σ-transcen-

dence basis of ζ over K〈η〉.(iii) If B denotes the σ-specialization of B, then trdeg(K〈η,B〉)∗(K〈η, ζ〉)∗ =

trdeg(K〈η,B〉)∗(K〈η, ζ〉)∗.PROOF. In order to satisfy (i), we extend ζ to the indexing ζ∪c−1 (consid-

ering c−1 as the last element of the extended indexing). Then, if a specializationη of η is extended to ζ ∪c−1, the specialization of c is not zero (otherwise, wewould arrive at a contradiction 1 = cc−1 = 0). Also, if conditions (ii) and (iii)are valid for the indexing η and ζ ∪ c−1 (with the last indexing instead of ζ),these conditions are also valid for η and ζ. (For condition (ii) this statement isobvious. Furthermore, since c ∈ Kη, ζ, B is a σ-transcendence basis of ζ∪c−1

over K〈η〉, so if (iii) holds for η, ζ ∪ c−1, it also holds for η, ζ.)Let η = (η(1), . . . , η(s)), ζ = (ζ(1), . . . , ζ(m)), and let Ky, z denote the ring

of σ-polynomials in σ-indeterminates y1, . . . , ys, z1, . . . , zm over K (y denotesthe s-tuple (y1, . . . , ys) and z denotes the m-tuple (z1, . . . , zm)). Let P be thereflexive prime σ-ideal of Ky, z with generic zero (η, ζ).

Let t be a positive integer satisfying the following condition: there is a basisof the σ-ideal P which contains no σ-polynomial f with ordzj

f > t for j =1, . . . , m. (Since P has finite bases, such an integer t exists.) Let L denote theinversive closure of K〈η〉. Then σ-trdegLL〈ζ〉 = σ-trdegK〈η〉K〈η, ζ〉 (one caneasily check this inequality taking into account that L = (K〈η〉)∗). Let R denotethe difference kernel (L(ζ, α(ζ), . . . , αt(ζ)), α).

By Proposition 2.1.9(iii) and Theorem 1.6.48(i), the field extension L(ζ, α(ζ),. . . , αt(ζ) /L is primary. Applying Proposition 1.6.64 we obtain that there is anelement c = 0 in (Kη)∗ with the following property: if φ is a specialization ofthe coordinates of αi(η) (i ∈ N) over K∗ and φ(c) = 0, then φ has an extensionto a nondegenerate specialization of ζ, α(ζ), . . . , αt(ζ) and a unique nondegener-ate extension to a specialization of ζ, α(ζ), . . . , αt−1(ζ). Let u = αj(c) be sometransform of c (j ∈ N).

Let η → η be a σ-specialization of η over K such that u = 0. This spe-cialization extends to a unique specialization of the inverse transforms α−k(η)(k > 0) such that u = 0. Let ζ, . . . , αt(ζ) be a nondegenerate extension ofthis specialization (in the sense of Definition 1.6.62) to ζ, α(ζ), . . . , αt(ζ). Thenζ, α(ζ), . . . , αt−1(ζ) and α(ζ), . . . , αt(ζ) are nondegenerate extensions of the spe-cialization of the αi(η) to ζ, α(ζ), . . . , αt−1(ζ) and α(ζ), . . . , αt(ζ), respectively.


Since u = 0 and α(u) = 0, these nondegenerate extensions of the specializationof the αi(η) are unique (see Proposition 1.6.64).

To complete the proof of the theorem we need the following lemma.

Lemma 5.3.3 With the above notation, let L denote the inversive closure ofK〈η〉 (the translation of this field will be denoted by the same letter α as thetranslation of K). Then α can be extended from L to an isomorphism τ ofL(ζ, α(ζ), . . . , αt−1(ζ)) onto L(α(ζ), . . . , αt(ζ)) such that τ(αi(ζ)) = αi+1(ζ)(0 ≤ i ≤ t − 1).

PROOF. Let β denote the translations of L〈ζ〉 (which is an extension ofα), let α′ be an extension of α to an isomorphism of L(ζ, α(ζ), . . . , αt−1(ζ)),and let φ denote our specialization (indicated also as a bar over the cor-responding letter). Then the mapping γ = α′φβ−1 coincides with φ on(Kη)∗ and provides a nondegenerate extension of φ to a specialization ofα(ζ), . . . , αt(ζ). By Proposition 1.6.64, such an extension is unique, so there isan L-isomorphism θ of L(γα(ζ), . . . , γαt(ζ)) onto L(α(ζ), . . . , αt(ζ)) such thatθγαi(ζ) = αi(ζ), 1 ≤ i ≤ t. Then τ = θα′ is an isomorphism which is definedon φβ−1( (Kη)∗[α(ζ), . . . , αt(ζ)]) = (Kη)∗[ζ, α(ζ), . . . , αt−1(ζ)], coincideswith α on (Kη)∗η, and maps αi(ζ) onto the field αi+1(ζ) (0 ≤ i ≤ t − 1).Extending τ to an isomorphism of the quotient field L(ζ, α(ζ), . . . , αt−1(ζ))onto L(α(ζ), . . . , αt(ζ)) we obtain the desired extension of α.

COMPLETION OF THE PROOF OF THE THEOREM

Using the above notation and the extension τ , whose existence is establishedby the last lemma, we can consider a kernel R = (L(ζ, α(ζ), . . . , αt(ζ)), τ).Denoting a principal realization of this kernel by ζ (this should not lead to anyconfusion), we obtain that η, ζ is a specialization of η, ζ (since η, ζ is a solutionof a basis of P ). Thus, almost every specialization of η has an extension.

Let λ be a σ-transcendence basis of ζ = (ζ(1), . . . , ζ(m)) over K〈η〉 andfor any r ∈ N, let λr denote the set αr(ζ(k)) | ζ(k) ∈ λ. Then for allsufficiently large r ∈ N, say for all r ≥ r0 (r0 is a fixed positive inte-ger) and for every i = 1, . . . , m, the elements αr(ζ(i)), . . . , αr(ζ(i)) are alge-braically dependent over L(λ, α(λ), . . . , αr(λ)). Let us choose t in the abovepart of the proof sufficiently large, so that for any σ-transcendence basisλ of ζ over K〈η〉 (there are only finitely many such bases) and for everyi = 1, . . . , m, the elements αt(ζ(i)), . . . , αt(ζ(i)) are algebraically dependentover L(λ, α(λ), . . . , αt(λ)). Since the extension of the specialization of αi(η)is nondegenerate, (ζ(i), α(ζ(i)), . . . , αt(ζ(i))) is algebraically dependent overL(λ, α(λ), . . . , αt(λ)) for i = 1, . . . , m (αi(λ) denotes the specialization ofαi(λ)). We obtain that each αi(λ) is σ-algebraic over L〈λ〉, hence λ contains aσ-transcendence basis of ζ over L. Since ζ is a principal realization, a special setω for the kernel R is a σ-transcendence basis of ζ. Because of the nondegeneracyof the extension, ω is a special set of R and hence a σ-transcendence basis of ζover L. It follows that if λ is a σ-transcendence basis of ζ over K〈η〉, then λ is

5.3. DIFFERENCE SPECIALIZATIONS 331

a σ-transcendence basis of ζ over K〈η〉, that is, condition (ii) of the theorem issatisfied.

In order to satisfy (iii) suppose, first, that B = ∅. Then the nondegen-eracy implies that ordR = ordR, hence trdegK∗(K〈η, ζ〉)∗ = trdegLL〈ζ〉 =trdegLL〈ζ〉 = trdeg(K〈η〉)∗(K〈η, ζ〉)∗, so (iii) is satisfied.

Now let B = ∅. By Theorem 4.4.1. one can choose a finite set S such thatK〈η,B, S〉 is the algebraic closure of K〈η,B〉 in K〈η, ζ〉. As we just proved,there exists a nonzero element a ∈ Kη,B, S such that every specializa-tion η, B, S of η,B, S over K with a = 0 can be extended to a special-ization ζ of ζ with trdeg(K〈η,B,S〉)∗(K〈η, ζ〉)∗ = trdeg(K〈η,B,S〉)∗(K〈η, ζ〉)∗ =trdeg(K〈η,B〉)∗(K〈η, ζ〉)∗.

Since the field extension K〈η,B, S〉/K〈η〉 is primary (see Theorem 1.6.48),the preceding part of the proof shows that

(I) Almost every specialization η of η over K has an extension to a spe-cialization η, B, S of η,B, S with B σ-algebraically independent over K anda = 0.

Now, notice that any element u ∈ S is algebraic over K〈η,B〉, hence it is azero of some polynomial fu in one variable with coefficients in Kη,B. If c is anonzero coefficient of fu, then c can be written as a polynomial gc in elementsτ(b) (τ ∈ Tσ, b ∈ B) with coefficients in Kη. Let d be such a coefficient ofgc, d = 0. If η is a specialization of η such that d = 0, a = 0, and B is σ-algebraically independent over K (so that the specialization satisfies (I)), thenc = 0. It follows that

(II) If B and S are as in (I), then every element of S is algebraic overK〈η,B〉.

Let η be a specialization of η with the property described in (I) and letB and S be as in (I) (so that (II) also holds). In this case, as we have seen,ζ exists such that η, ζ is a specialization of η, ζ and trdeg(K〈η,B,S〉)∗(K〈η, ζ〉)∗ =trdeg(K〈η,B〉)∗(K〈η, ζ〉)∗. This equality, together with (II), implies thattrdeg(K〈η,B,S〉)∗(K〈η, ζ〉)∗ = trdeg(K〈η,B〉)∗(K〈η, ζ〉)∗, so that our specializationsatisfies condition (iii). This completes the proof of the theorem.

The following example is due to R. Cohn ([41, Chapter 7, Example 3]). Itshows that one cannot drop the requirement that the difference field exten-sion K〈η, ζ〉/K〈η〉 in the hypothesis of Theorem 5.3.2 is primary. (Without thisassumption one cannot even conclude that extensions of almost every special-ization of η exist.)

Example 5.3.4 Let us consider Q as an ordinary difference field whose basicset σ consists of the identity translation α. Let a be an element transcendentalover Q and let Q(a) be treated as a σ-overfield of Q such that α(a) = −a.Furthermore, let b = a2. Then α(b) = b, and b is transcendental over Q. Ifη ∈ Q, then η is a specialization of b over Q, since it lies in the irreduciblevariety of the σ-polynomial αy − y ∈ Qy and b is a generic zero of thisvariety. We are going to show that it is not true that almost every specializationof b can be extended to a specialization of b, a.


Let η ∈ Q, η = 0. Then the specialization of b to η2 cannot be extendedto b, a. Indeed, if such an extension exists, then a would specialize to η or −η.but neither of these is a solution of the σ-polynomial αy + y annulled by a.Suppose that there is c ∈ Qb, c = 0, such that every specialization of b whichdoes not specialize c to 0 can be extended to a. Then c ∈ Q[b], hence at mostfinite number of specializations of b specialize c to 0. At the same time, we justdescribed an infinite set of specializations of b which cannot be extended to b, a.

5.4 Babbitt’s Decomposition. Criterionof Compatibility

Throughout this section K will denote an ordinary difference field with a basicset σ = α. An element a in some σ-overfield of K is said to be normal overK if K(a) is a normal field extension of K in the usual sense (see Section 1.6,in particular, Theorem 1.6.8).

It is easy to check that if a is normal over K, then the field extension K〈a〉/Kis normal. Indeed, in this case K(a) is a splitting field over K of the minimalpolynomial f(X) = Irr(a,K). Therefore, K〈a〉 is a splitting field over K of thefamily of polynomials (f (i))i∈N where f (i) is a polynomial in K[X] obtained byapplying αi to every coefficient of f .

Let L be an algebraic σ-field extension of K. By Corollary 5.1.16, there existsan algebraic closure L of L which has a structure of a σ-overfield of L. ApplyingTheorem 1.6.13(v) we obtain that L contains a normal closure N of the field Lover K. Let us show that N is a σ-overfield of L.

Indeed, since N is a splitting field the family of polynomials Irr(a,K) | a ∈L over K (see Theorem 1.6.13(i)), every element u ∈ N can be written

in the form u =f(a1, . . . , am)g(a1, . . . , am)

where f and g are polynomials in m vari-

ables with coefficients in K, and a1, . . . , am are roots of some polynomialsIrr(b1,K), . . . , Irr(bm,K), respectively, with b1, . . . , bm ∈ L. Then α(u) =f1(α(a1), . . . , α(am))g1(α(a1), . . . , α(am))

where the polynomials f1 and g1 are obtained by apply-

ing α to every coefficient of f and g, respectively, and α(a1), . . . , α(am) areroots of the polynomials Irr(α(b1),K), . . . , Irr(α(bm),K), respectively. There-fore, α(u) ∈ N , so that N is a σ-field extension of L.

Proposition 5.4.1 Let K be an ordinary difference field with a basic set σ =α, L an algebraic σ-field extension of K, and N a normal closure of L overK (treated as a σ-field extension of L). Let a be an element of L such thatld(K〈a〉/K) = 1 and let b be an element of N conjugate to a (that is, b is a rootof Irr(a,K)). Then ld(K〈b〉/K) = 1.

PROOF. Let pi(x) denote the polynomial obtained by applying αi to everycoefficient of Irr(a,K) (i = 0, 1, . . . ). Since K〈a〉 = K(a, α(a), α2(a), . . . ), Ncontains a normal closure Na of K〈a〉 over K which is a splitting field of the

5.4. BABBITT’S DECOMPOSITION 333

family of polynomials pi(x) | i = 0, 1, . . . over K (see Theorem 1.6.13(iii)). ByProposition 4.3.12(ii), if ld(K〈a〉/K) = 1, then K〈a〉 : K < ∞. It follows thatNa : K < ∞ (see Theorem 1.6.13(iv)). Since Irr(b,K) = Irr(a,K), Na is alsoa normal closure of K〈b〉 over K. Applying Theorem 1.6.13(iv) once again weobtain that K〈b〉 : K < ∞ whence ld(K〈b〉/K) = 1.

Corollary 5.4.2 Let K be an ordinary difference field with a basic set σ = αand let N be a σ-overfield of K such that the field extension N/K is normal.Then the core NK is a normal field extension of K.

Exercise 5.4.3 Let K be an inversive ordinary difference field and let L be adifference overfield of K such that the field extension L/K is algebraic. Provethat if N is a normal closure of L over K, then the core NK contains a normalclosure of LK over K.

The following example shows that, with the notation of the last exercise,NK needs not to coincide with the normal closure of LK over K.

Example 5.4.4 Let K = Q(X) be the field of rational fractions in one inde-terminate X over Q. Then K can be treated as an inversive ordinary differencefield whose basic set σ consists of a translation α such that α(f(X)) = f(X +1)for any f(X) ∈ K. Let L be a field extension of K obtained by adjoining toK elements 4

√X + i (i ∈ N). (Formally speaking, one should adjoin to K a

zero of the polynomial Y 4 − X ∈ K[Y ] and obtain a field K1. Then we ad-join to K1 a zero of the polynomial Y 4 − (X + 1) ∈ K1[Y ] and obtain a fieldK2, then K2 is extended to a field K3 by adjoining a root of the polynomialY 4 − (X + 2) ∈ K2[Y ], etc. Finally, we define L as the union of all fields Ki.)

Treating L as a σ-field extension of K with α( 4√

X + j) = 4√

X + j + 1 (j ∈N) and applying Theorem 4.3.20 we obtain that LK =

⋂∞n=0 K〈αn( 4

√X)〉 =⋂∞

n=0 Q(X, 4√

X + n, 4√

X + n + 1, . . . ) = K, so the normal closure of LK

over K coincides with K. On the other hand, the normal closure of Lover K is the field N = Q(i,X, 4

√X, 4

√X + 1, . . . ) (i =

√−1). This fieldis a σ-overfield of K where the extension of α from L to N is defined bythe condition α(i) = i. Using Theorem 4.3.20 once again we obtain thatNK =

⋂∞n=0 Q(i,X, 4

√X + n, 4

√X + n + 1, . . . ) = Q(i,X) K.

Definition 5.4.5 Let L be a σ-overfield of K such that L = K〈a〉 for someelement a ∈ L and the field extension L/K is algebraic. If K(a, α(a)) : K(a) =ld(L/K), then a is said to be a standard generator of L over K. If a is a standardgenerator of L over K such that K(a) : K is as small as possible, a is called aminimal standard generator of L over K.

If L is a normal field extension of K, L = K〈u〉 for some u ∈ L, andK(u, α(u)) : K(u) = ld(L/K), then u is said to be a normal standard generatorof L over K. If an element u ∈ L is a normal standard generator of L over Ksuch that K(u) : K is as small as possible, u is said to be a minimal normalstandard generator of L over K.


In what follows we assume that L is a finitely generated σ-field extension ofK such that the extension L/K is separably algebraic. Then one can always finda standard generator of L over K as follows. Suppose that L = K〈S〉 for a finiteset S ⊆ L and ld(L/K) = K(S, α(S), . . . , αi+1(S)) : K(S, α(S), . . . , αi(S))for some i ∈ N. By Theorem 1.6.17, one can choose c ∈ L such that c isa linear combination of elements of S

⋃α(S)

⋃ · · ·⋃αi(S) with coefficientsin K and K(c) = K(S, α(S), . . . , αi(S)). Then L = K〈c〉 and K(c, α(c)) =K(S, α(S), . . . , αi+1(S)), so that c is a standard generator of L over K. If, in ad-dition, L/K is normal, then the set S′ of all roots of all polynomials Irr(a,K),a ∈ S, is a finite set of σ-generators of L over K such that the field exten-sion K(S′)/K is normal. As before, if ld(L/K) = K(S′, α(S′), . . . , αj+1(S′)) :K(S′, α(S′), . . . , αj(S′)) for some j ∈ N, we can find an element u ∈ L suchthat K(u) = K(S′, α(S′), . . . , αj(S′)). Then u is normal over K, L = K〈u〉 andld(L/K) = K(u, α(u)) : K(u), so that u is a normal standard generator of Lover K.

In what follows, if L is a difference field extension of an ordinary differencefield K, then LK denotes the core of this extension (see Definition 4.3.17).

Lemma 5.4.6 Let K be an inversive ordinary difference field with a basic setσ = α. Let L be a finitely generated σ-field extension of K such that thefield extension L/K is separably algebraic and normal. Then LK is a simplenormal inversive σ-field extension of K such that ld(LK/K) = 1. Furthermore,(LK)K = LK .

PROOF. The fact that LK is an inversive σ-field extension of K withld(LK/K) = 1 was proved in Section 4.3 (see the remark after Definition 4.3.17).The normality of the extension LK/K follows from Corollary 5.4.2. Furthermore,Theorem 4.4.1 implies that LK is a finitely generated σ-field extension of K.Applying Proposition 4.3.12 we obtain that LK : K < ∞ hence LK = K〈u〉 forsome element u ∈ LK . Without loss of generality we can assume that u is a stan-dard generator of LK over K, that is K(u, α(u)) : K(u) = ld(LK/K) = 1. ThenK(α(u)) ⊆ K(u). Since K is inversive, we obtain that K(u, α(u)) : K = K(u) :K = K(α(u)) : K = K(α(u)) : K hence K(α(u)) = K(u) and LK = K(u).Finally, since ld(K〈u〉/K) = 1, u ∈ (LK)K hence (LK)K = LK .

Definition 5.4.7 A finitely generated ordinary difference (σ-) field extensionL of a σ-field K is called benign if

(i) L/K is an algebraic, normal and separable field extension.(ii) There exists an element u ∈ L such that L = K〈u〉, u is normal over K

and K(u) : K = ld(L/K).

It follows from Theorem 4.3.22 that if L is a benign extension of an ordinarydifference field K of zero characteristic, then LK = K. One of the main resultsof this section, Theorem 5.4.13, shows the existence of decomposition of a fi-nitely generated ordinary difference field extension L/K, where L is separablyalgebraic and normal over K, into a finite sequence of benign extensions. Wewill follow [4] in the proof of this theorem.


Definition 5.4.8 Let K be an ordinary difference field with a basic set σ = αand let L be a simple σ-field extension of K (that is, the extension L/K can begenerated by a single σ-generator). Let u ∈ L be a σ-generator of L over K suchthat K(u) : K is minimal. Then u is called a minimal generator of L over K,and K(u) : K is said to be the minimal degree of L over K, written md(L/K).

If md(L/K) = ld(L/K), then L is called a mild difference (σ-) field extensionof K.

Definition 5.4.9 Two difference fields K and L with the same basic set σ arecalled equivalent, written K L, if there exist identical inversive closures of Kand L.

Let F ∗ be an inversive difference field with a basic set σ and let F1 andF2 be σ-subfields of F ∗ with the inversive closure F ∗. Let K/F1 and L/F2 beσ-field extensions and let K∗ and L∗ denote the inversive closures of K andL, respectively. Then we say that the σ-field extensions K/F1 and L/F2 areequivalent if there is a σ-F ∗-isomorphism of K∗ onto L∗. (In most cases of suchan equivalence we shall have K∗ = L∗.)

Lemma 5.4.10 Let K and L be ordinary difference fields with basic set σ = αsuch that K L. Let M denote the common inversive closure of K and L,and let u1, . . . , un be elements in some σ-overfield of M such that for everyi = 0, 1, . . . , n − 1, K〈u1, . . . , ui+1〉 is a mild σ-filed extension of K〈u1, . . . , ui〉with minimal generator ui+1. Then there exist m1, . . . , mn ∈ N such thatL〈αr1(u1), . . . , αri+1(ui+1)〉 is a mild σ-filed extension of L〈αr1(u1), . . . , αri(ui)〉for all integers ri ≥ mi (0 ≤ i ≤ n − 1).

PROOF. We proceed by induction on n. Since K〈u1〉/K is a mild differ-ence field extension and u1 is a minimal generator of K〈u1〉 over K, we obtainthat K(u1) : K = K(u1, α(u1), . . . , αi+1(u1)) : K(u1, α(u1), . . . , αi(u1)) for i =1, . . . , n−1. Therefore, if f(X) is the minimal polynomial of u1 over K, f(X) =Irr(u1,K), then the polynomial fi+1(X), obtained by applying αi+1 to everycoefficient of f(X) (i = 0, . . . , n−1), is irreducible over K(u1, α(u1), . . . , αi(u1)).

Since the coefficients of f(X) lie in K ⊆ M , fm1(X) ∈ L[X] for suf-ficiently large m1 ∈ N. Then for every r1 ∈ N, r1 ≥ m1, the polyno-mial fr1+i(X) is irreducible over L(αr1(u1), . . . , αr1+i−1(u1)) (i = 0, 1, . . . ).Indeed, if it is not so, then for all sufficiently large j ∈ N, fr1+i+j(X)would be reducible over K(αr1+j(u1), . . . , αr1+j+i−1(u1)) and hence overK(u1, α(u1), . . . , αr1+j+i−1(u1)) which is impossible. This completes the proofof the case n = 1.

Suppose that the statement of the lemma is true for the case n − 1. SinceK〈u1, . . . , un−1〉 L(αr1(u1), . . . , αrn−1(un−1)) for any r1, . . . , rn−1 ∈ N, thestatement for u1, . . . , un follows from the first part of the proof (the case n = 1)with un playing role of u1, the field K〈u1, . . . , un−1〉 playing role of K, andL(αr1(u1), . . . , αrn−1(un−1)) playing role of L.

Lemma 5.4.11 Let K and L be ordinary difference fields with basic set σ = αsuch that K L. Let u1, . . . , un be elements in some σ-field extension of thecommon inversive closure of K and L such that for every i=0, 1, . . . , n−1,


the element ui+1 is normal over the field K〈u1, . . . , ui〉. Then there existm1, . . . ,mn ∈ N such that αri+1(ui+1) is normal over L〈αr1(u1), . . . , αri(ui)〉for all ri ∈ N, ri ≥ mi (0 ≤ i ≤ n − 1).

PROOF. Let f(X) = Irr(u1,K). Since u1 is normal over K, the zeros off(X) can be written as polynomials in u1 with coefficients in K. Therefore, thereexists m1 ∈ N such that for any r1 ∈ N, r1 ≥ m1, we have fr1(X) ∈ L[X] andevery zero of fr1(X) can be written as a polynomial in αr1(u1) with coefficientsin L. (As before, for any d ∈ N, fd(X) denotes the polynomial obtained byapplying αd to every coefficient of f(X).) Hence αr1(u1) is normal over L. Nowthe statement of the lemma follows by induction on n.

The last two lemmas immediately imply the following result.

Theorem 5.4.12 Let K and L be ordinary difference fields with basic setσ = α, let K L, and let u1, . . . , un be elements in some σ-field extension ofthe common inversive closure of K and L such that for every i = 0, 1, . . . , n−1,K〈u1, . . . , ui+1〉 is a benign extension of K〈u1, . . . , ui〉 with minimal gen-erator ui+1. Then there exist m1, . . . ,mn ∈ N such that for all ri ∈ N,ri ≥ mi (0 ≤ i ≤ n − 1), L〈αr1u1, . . . , α

ri+1(ui+1)〉 is a benign extension ofL〈αr1(u1), . . . , αri(ui)〉 with normal minimal generator αri+1(ui+1).

Theorem 5.4.13 (Babbitt’s Decomposition Theorem). Let K be an ordinaryinversive difference field with a basic set σ = α and let L be a finitelygenerated σ-field extension of K such that the field extension L/K is sepa-rably algebraic and normal. Then there exist elements u1, . . . , un ∈ L such thatL LK〈u1, . . . , un〉 and for every i = 0, . . . , n−1, LK〈u1, . . . , ui+1〉 is a benignextension of LK〈u1, . . . , ui〉 with normal minimal generator ui+1.

PROOF. Without loss of generality we can assume that LK = K. Let us setn = ld(L/K) and consider the following two possibilities: 1) L does not containa proper normal field extension of K whose limit degree over K is less than n;2) L contains such an extension of K.

Case 1. Suppose that there is no intermediate difference field F between Kand L such that F = K, F/K is normal and ld(F/K) < n. Let η be a normalstandard generator of L over K which minimizes K(η) : K. Since K is inversive,K(α(η)) : K = K(η) : K. Furthermore, it follows from Theorem 1.6.15(x) thatK(α(η)) : K(η)

⋂K(α(η)) = K(η, α(η)) : K(η) = n. Let ζ be a primitive

element of the extension K(η)⋂

K(α(η))/K, so that K(ζ) = K(η)⋂

K(α(η)).Then α(ζ) ∈ K(α(η)) and K(ζ, α(ζ)) ⊆ K(α(η)).

Since the field extension K〈ζ〉/K is normal, it follows from our assumptionthat either K〈ζ〉 = K or ld(K〈ζ〉/K) = n. If K〈ζ〉 = K, then K(α(η)) : K =K(η) : K = n, so that L is a benign extension of K with normal minimalgenerator η. If ld(K〈ζ〉/K) = n, then n ≤ [K(ζ, α(ζ)) : K(ζ)] ≤ [K(α(η)) :K(η)

⋂K(α(η))] = n, so that K(ζ, α(ζ)) = K(α(η)) and K〈ζ〉 = K〈α(η)〉. In

this case ζ is a normal standard generator of K〈α(η)〉 over K and K(ζ) : K =[K(ζ, α(ζ)) : K]

[K(ζ, α(ζ)) : K(ζ)]=

[K(α(η)) : K]n

=[K(η) : K]

n.


Since L = K〈η〉 does not have a normal standard generator over K of degreeless than K(η) : K, the same can be said about the isomorphic field K〈α(η)〉,so that n = 1. Thus, ld(K〈η〉/K) = 1 hence η ∈ LK = K and L = K.

Case 2. Suppose that there is a normal σ-field extension F of K such thatK F ⊆ L and the limit degree k = ld(F/K) satisfies the condition 1 < k < n.As before we also suppose that LK = K. Furthermore, proceeding by inductionon n, we assume that the theorem holds for extensions of limit degree less than n.

Let L∗ and F ∗ denote inversive closures of L and F , respectively (F ∗ ⊆ L∗).Since the field extension F/K is normal, so is F ∗/K. Therefore, the composi-tum LF ∗ is a normal finitely generated σ-field extension of F ∗. By Lemma5.4.6, the σ-field (LF ∗)F∗ is inversive and (LF ∗)F∗ = F ∗〈w〉 for some ele-ment w ∈ LF ∗. Since the fields F ∗〈w〉 and L∗ are inversive, any transformof w generates the σ-field extension F ∗〈w〉/F and without loss of generalitywe can assume that w ∈ L and w is a standard generator of F 〈w〉 over F .

Then α(w) ∈ F (w), so that α(w) can be written as α(w) =m∑

i=0

ciwi for some

c0, . . . , cm ∈ F . Clearly, the restrictions of all K-automorphisms of L to thefield F (w) form the set of all relative isomorphisms of F (w) over K. Therefore,

if α(v) is a conjugate of α(w) over K, then α(v) =m∑

i=0

ρ(ci)ρ(w)i where ρ is

the restriction of some K-automorphism of L to F (w). Since the field exten-sion F/K is normal, ρ(ci) ∈ F for i = 0, . . . , m. Therefore, every conjugateof α(w) over K lies in the field F (W ) where W denotes the set of all conju-gates of w over K. It follows that F (W ) = F 〈W 〉 whence ld(F 〈W 〉/F ) = 1.Therefore, ld(F 〈W 〉/K) = [ld(F 〈W 〉/F )][ld(F/K)] = ld(F 〈W 〉/F ) = k. Sincethe field extension F/K is normal, so is F (W )/K. We obtain that F 〈W 〉 isa normal σ-field extension of K, F 〈W 〉K = K, and ld(F 〈W 〉/K) = k < n.By the induction hypothesis, there exist elements u1, . . . , ul ∈ F 〈W 〉 such thatF 〈W 〉 F 〈u1, . . . , ul〉 and for every i = 1, . . . , l, F 〈u1, . . . , ui〉 is a benign ex-tension of F 〈u1, . . . , ui−1〉 with normal minimal generator ui.

It follows from our previous considerations that LF ∗ is a finitely generatednormal σ-field extension of F ∗〈W 〉. Since ld(F ∗〈W 〉/F ∗) = 1, we have F ∗〈W 〉 =(LF ∗)F∗ , so that the σ-field F ∗〈W 〉 is inversive. Furthermore, since

ld(LF ∗/F ∗〈W 〉) ≤ ld(L/F 〈W 〉) =ld(L/K)

ld(F 〈W 〉/K)=

n

k< n,

we can apply the induction hypothesis and obtain that there exist elementsvl+1, . . . , vs ∈ LF ∗ such that LF ∗ F ∗〈W 〉〈vl+1, . . . , vs〉 and for every j =l+1, . . . , s, F ∗〈W 〉〈vl+1, . . . , vj〉 is a benign extension of F ∗〈W 〉〈vl+1, . . . , vj−1〉with normal minimal generator vj . Since F ∗〈W 〉 F 〈W 〉 K〈u1, . . . , ul〉,it follows from Theorem 5.4.12 that if we set uj = αm(vj) (l + 1 ≤ j ≤ s)with m ∈ N sufficiently large, then K〈u1, . . . , ui+1〉 is a benign extension ofK〈u1, . . . , ui〉 with normal minimal generator ui+1 (0 ≤ i ≤ s − 1). Obviously,


L LF ∗ F ∗〈W 〉〈vl+1, . . . , vs〉 K〈u1, . . . , us〉.

This completes the proof of the theorem.

Let L be a finitely generated difference field extension of an inversive ordinarydifference field K such that L/K is algebraic and normal. Then a finite sequenceu1, . . . , us of elements of L whose existence is established by Theorem 5.4.13 issaid to define a benign decomposition of L over K.

In what follows we are going to apply Babbitt Decomposition Theorem tothe study of compatibility of ordinary difference field extensions. We begin witha statement which is a direct consequence of Proposition 2.1.7(v). As usual theinversive closure of a difference field F is denoted by F ∗ and the core of F oversome its difference subfield K is denoted by FK . Furthermore, we assume thatthe field extensions considered below are separable.

Proposition 5.4.14 Let K be an inversive ordinary difference field with a basicset σ = α and let L and M be σ-field extensions of K. Then

(i) The σ-field extensions L/K and M/K are incompatible if and only if theextensions L∗/K and M∗/K are incompatible.

(ii) Suppose that the field extensions L/K and M/K are algebraic and equiv-alent (as difference field extensions of K). If F is another σ-overfield of K,then the σ-field extensions L/K and F/K are incompatible if and only if theextensions M/K and F/K are incompatible.

Our next result shows that the study of compatibility of difference fieldextensions can be reduced to the study of compatibility of finitely generatedextensions.

Theorem 5.4.15 Let K be an ordinary difference field with a basic set σ = αand let L and M be σ-overfields of K. Then the difference field extensions L/Kand M/K are incompatible if and only if there exist intermediate σ-fields L′ andM ′ of L/K and M/K, respectively, such that the σ-field extensions L′/K andM ′/K are finitely generated and incompatible.

PROOF. It follows from Theorem 5.1.6 that we may assume that the fieldextensions L/K and M/K are algebraic. Let S be a set of σ-generators of L overK and let B be a set of elements in some σ-overfield of M which is σ-algebraicallyindependent over M and has the same cardinality as S. We suppose that a one-to-one correspondence between the sets S and B is fixed and consider KB asa ring of σ-polynomials in the set of σ-indeterminates B. Let P be the reflexiveprime σ-ideal of KB consisting of σ-polynomials which are annulled whenmembers of B are replaced by the corresponding members of S. Let Q be theperfect σ-ideal of MB generated by P .

We are going to prove the following criterion:The σ-field extensions L/K and M/K are incompatible if and only if 1 ∈ Q.


Indeed, if L/K and M/K are compatible, then there are elements in a σ-overfield of M which when substituted for the corresponding elements of Bannul every σ-polynomial of P and, therefore, every σ-polynomial of Q. Then1 /∈ Q. Conversely, suppose that 1 /∈ Q. By Proposition 2.3.4 there exists areflexive prime σ-ideal Q′ of MB such that Q ⊆ Q′ and 1 /∈ Q′. Then thedifference factor ring MB/Q′ is an integral domain whose quotient σ-field Ncontains M . Let S′ denote the image of the set B under the natural epimorphismMB → MB/Q′. Since S′ is a solution of the set of σ-polynomials of P ,there is a σ-K-homomorphism of KS onto KS′. Since every element of Sis algebraic over K (the extension L/K is algebraic), this mapping is actually aσ-isomorphism. Therefore, there is a σ-K-isomorphism of L into N whence theσ-field extensions L/K and M/K are compatible.

Applying the criterion just proved, we obtain that P generates the unit idealin MB. Then there exists a finite subset B1 of B such that P ∩KB1 gener-ates the unit ideal in MB1. Let S1 be the finite subset of S that correspondsto B1 under the bijection between S and B, and let L′ = K〈S1〉. Applying theabove criterion to L′ (instead of L) and P ∩ KB1 (instead of P ) we obtainthat the σ-field extensions L′/K and M/K are incompatible. Application of thesame procedure to M produces a finitely generated σ-field extension M ′ of Ksuch that M ′ ⊆ M and the extensions L′/K and M ′/K are incompatible.

Proposition 5.4.16 Let F and N be difference overfields of an inversive or-dinary difference field K with a basic set σ = α. Let G be a benign σ-fieldextension of F and let φ be a σ-K-isomorphism of F onto a σ-subfield F ′ ofN . Then φ can be extended to a σ-K-isomorphism of G into N . In particular(if F = K), a benign extension G/K is compatible with every σ-field extensionof K.

PROOF. Let η be a minimal generator of G over F , let f(X) = Irr(η, F ),and let g(X) be a polynomial in F ′[X] obtained from f(X) by replacing thecoefficients of f by their images under φ. If ζ is any zero of g(X), then φ canbe extended to a field isomorphism φ0 of F (η) onto F ′(ζ) such that φ(η) = ζ.

Let f1(X) denote the polynomial obtained from f(X) by applying α toevery coefficient of f . Since η is a minimal generator of a benign extension ofF , the polynomial f1(X) is irreducible over F (η). Let g1(X) be a polynomialobtained from f1(X) by replacing the coefficients of f by their images underφ0. Since f1(X) ∈ F [X] and φ is a σ-isomorphism, α(ζ) is a zero of g1(X),so we can extend φ0 to an isomorphism φ1 of F (η, α(η)) onto F ′(ζ, α(ζ)) suchthat φ1(α(η)) = α(ζ). A straightforward induction argument now yields thestatement of the proposition.

The following statement gives a criterion of the compatibility for separablyalgebraic normal difference field extensions.

Proposition 5.4.17 Let K be an ordinary difference field with a basic set σ =α and let L and M be difference (σ-) field extensions of K such that thefield extensions L/K and M/K are normal. Then a necessary and sufficient


condition for the compatibility of the σ-field extensions L/K and M/K is thecompatibility of the difference field extensions LK/K and MK/K.

PROOF. Clearly, the incompatibility of LK/K and MK/K implies the theincompatibility of L/K and M/K. Therefore, in order to prove the theorem,one should show that if L/K and M/K are incompatible, so are LK/K andMK/K.

Suppose that LK and MK have σ-K-isomorphisms into a σ-field extension Hof K. Without loss of generality we can assume that the field H is algebraicallyclosed (see Corollary 5.1.16). Furthermore, it follows from Theorem 5.4.15 andTheorem 5.1.6 that we can suppose that the σ-field extensions L/K and M/Kare finitely generated and separably algebraic. Finally, we can assume that theσ-field K is inversive. (Indeed, if it is not so, we may replace K with its inversiveclosure K∗ and replace L and M with K∗〈L〉 and K∗〈M〉, respectively. Thenthe cores LK and MK will be replaced by their inversive closure (see Theorem4.3.24). Clearly, these replacements do not affect compatibility.)

Under all these assumptions, let u1, . . . , um and v1, . . . , vn define benigndecompositions of the extensions L/K and M/K, respectively. By Proposi-tion 5.4.14(ii), in order to complete the proof, it is sufficient to show thatLK〈u1, . . . , um〉 and MK〈v1, . . . , vn〉 have σ-K-isomorphisms into H. This fact,however, is a direct consequence of Proposition 5.4.16, so our proposition isproved.

The following considerations are aimed at removing the condition of normal-ity in Proposition 5.4.17 and obtaining a criterion of compatibility of arbitrarydifference field extensions. Note that the proof of the corresponding statement in[41, Chapter 7] assumed that the normal closure of the core of a difference fieldextension L/K coincides with the core of the extension N/K where N is thenormal closure of L over K. Unfortunately, this is not always so, see Example5.4.4. In his recent paper on compatibility theorem (to appear in the PacificJournal of Mathematics) R. Cohn has presented a correct proof of the generalform of this theorem, which can be considered as a criterion of compatibility.Our proof is a slight modification of R. Cohn’s arguments.

In what follows we treat some fields as ordinary difference fields with respectto different translations. In order to avoid confuses and make our arguments insuch considerations clear, it is sometimes convenient to use different notationfor a difference field and its underlying field, that is, the same field treated as aregular (non-difference) field. We adopt the following notation and terminology.If K is an ordinary difference field with a basic set σ = α, then K will denotethe underlying field of K (i. e., if a capital letter denotes some difference field,then the same letter with a “hat” denotes the corresponding underlying field).The field K will be also denoted by (K, α). Furthermore, if an overfield L of Kcan be treated as a difference overfield of K with respect to several extensions ofα to endomorphisms of L, β is one of such extensions, and S is a set of differencegenerators of L/K with respect to β, we write L = K〈S〉β .


Lemma 5.4.18 Let K be an ordinary difference field with a basic set σ = α,let N be a normal field extension of the field K and let β and γ be two extensionsof α to endomorphisms of N . Let Nβ and Nγ denote the difference fields (N , β)and (N , γ), respectively, treated as difference overfields of K. Furthermore, letFβ and Fγ denote the cores of Nβ and Nγ over K, respectively. Then Fβ = Fγ .

PROOF. Obviously, it is sufficient to show that Fβ ⊆ Fγ (the oppositeinclusion would follow by symmetry). Let a ∈ Fβ . Then K〈a〉β is a finite ex-tension of K (see Proposition 4.3.12). Let p(X) = Irr(a,K) and let pi(X) bethe polynomial obtained by applying αi to every coefficient of p(X) (i ∈ N).Since the field extension N/K is normal, N contains the set A of all roots of allpolynomials pi(X) and K(A) is the normal closure of K〈a〉β contained in N .Since K〈a〉β : K < ∞, K(A) : K < ∞ (see Theorem 1.6.13(iv)). Furthermore,since a is a root of p(X), γi(a) is a root of pi(X) for every i ∈ N. It follows thatK(A) is also the normal closure of K〈a〉γ over K contained in N . Therefore,K〈a〉γ : K < ∞ hence a ∈ Fγ . Thus, Fβ ⊆ Fγ .

Lemma 5.4.19 If L is a difference overfield of an ordinary difference field K,then the core of L over LK coincides with LK .

PROOF. Let a be an element of the core of L over LK . Then ld(LK〈a〉/LK) =1. Therefore, ld(K〈a〉/K) ≤ ld(LK〈a〉/K) = ld(LK〈a〉/LK)ld(LK/K) = 1. Itfollows that ld(K〈a〉/K) = 1, hence a ∈ LK .

Lemma 5.4.20 Let K be an ordinary difference field with a basic set σ = αand let L be a σ-overfield of K. Let L∗ be the inversive closure of L, K∗ theinversive closure of K contained in L∗, and L a σ-overfield of K∗〈L〉 containedin L∗. Then LK∗ = (LK)∗ where (LK)∗ denotes the inversive closure of LK .

PROOF. Let a ∈ LK∗ . Then a ∈ L∗ and a is separably algebraic over K∗,hence there exists r ∈ N such that αr(a) ∈ L and αr(a) is separably algebraicover K. Applying Corollary 4.3.6 we obtain that

ld(K〈αr(a)〉/K) = ld((K〈αr(a)〉)∗/K∗) = ld((K〈a〉)∗/K∗) = ld(K∗〈a〉/K∗) = 1.

Therefore, αr(a) ∈ LK whence a ∈ (LK)∗. On the other hand, Theorem 4.3.19shows that the σ-field LK∗ is inversive. Furthermore, LK ⊆ LK∗ . Indeed, ifa ∈ LK , then

1 ≤ ld(K∗〈a〉/K∗) ≤ ld((K〈a〉)∗/K∗) = ld(K〈a〉/K) = 1,

hence a ∈ LK∗ . It follows that (LK)∗ ⊆ LK∗ hence (LK)∗ = LK∗

Lemma 5.4.21 The following statements are equivalent.(i) Let K be an ordinary difference field and let L and M be two algebraic

difference field extensions of K such that the extensions LK/K and MK/K arecompatible. Then the extensions L/K and M/K are also compatible.


(ii) If L is an algebraic difference field extension of an ordinary differencefield K such that LK = K, then L/K is compatible with every difference fieldextension of K.

(iii) If L is an algebraic difference field extension of an ordinary differencefield K such that LK = K, then L/K is compatible with every normal differencefield extension of K.

PROOF. Clearly, (i) implies (ii), and (ii) implies (iii). Also, it is easy to seethat (iii) implies (ii). Indeed, let L/K be an algebraic ordinary difference fieldextension such that LK = K and let M be any difference overfield of K. Thenthe normal closure N of the field M over K can be equipped with a structureof a difference overfield of M (see the remark at the beginning of this section).Applying statement (iii) we obtain that difference field extensions L/K andN/K are compatible. Therefore, L/K and M/K are compatible as well.

It remains to prove statement (i) assuming that statement (ii) is true. LetL/K and M/K be algebraic ordinary difference field extensions such that LK/Kand MK/K are compatible. Then there are difference K-isomorphisms of LK

and MK , respectively, into a difference overfield F of K. Without loss of gener-ality we can assume that LK and MK are contained in F .

Since the core of L over LK is LK (see Lemma 5.4.19), statement (ii) impliesthat the difference field extensions L/LK and F/LK are compatible, so thereare difference LK- (and therefore K-) isomorphisms of L and F , respectively,into a difference overfield G of K. Without loss of generality we can assume thatG contains F as its difference subfield. Since the core of M over MK is MK ,one can apply statement (ii) and obtain that the extensions M/MK and G/MK

are compatible, so there are difference MK- (and therefore K-) isomorphisms ofG and M into a difference overfield H of K. It follows that there are differenceK-isomorphisms of L and M into H, so L/K and M/K are compatible.

Theorem 5.4.22 (Criterion of Compatibility). Let K be an ordinary differencefield with a basic set σ = α and let L and M be difference (σ-) field extensionsof K. Then a necessary and sufficient condition for the compatibility of theσ-field extensions L/K and M/K is the compatibility of the difference fieldextensions LK/K and MK/K.

PROOF. Because of Theorems 5.1.6 and 5.4.15 it is sufficient to prove thetheorem under the assumption that the difference field extensions L/K andM/K are finitely generated and separably algebraic. Furthermore, without lossof generality we may assume that the difference field K is inversive. Indeed, letK∗ denote the inversive closure of K and let L = K∗〈L〉 and M = K∗〈M〉.Suppose that the σ-field extensions LK/K and MK/K are compatible, thatis, there are σ-K-isomorphisms of LK and MK into some σ-overfield Ω of K.Then these isomorphisms can be naturally extended to σ-K∗-isomorphisms ofthe inversive closures (LK)∗ and (MK)∗ of LK and MK , respectively, into theinversive closure Ω∗ of Ω. By Lemma 5.4.20, LK∗ = (LK)∗ and MK∗ = (MK)∗,


hence there are σ-K∗-isomorphisms LK∗ → Ω∗ and MK∗ → Ω∗. If our theoremis true for K∗ (instead of K), then the extensions L/K∗ and M/K∗ are com-patible, hence L/K and M/K are compatible.

Finally, Lemma 5.4.21 shows that it is sufficient to prove the following result:

(∗) Let K be an inversive ordinary difference field with a basic set σ = α, letL and M be finitely generated separably algebraic difference field extensions ofK, and let LK = K. Then the difference field extensions L/K and M/K arecompatible.

In order to prove this statement, let us denote the basic translation ofL by αL (this is an extension of α to an endomorphism of L) and consider anormal closure N of L over K as an underlying field of a difference overfield Nof L with a translation αN (the existence of an extension of αL to an endomor-phism of N is established at the beginning of this section). Thus,we obtain achain of difference field extensions K = (K, α) ⊆ L = (L, αL) ⊆ N = (N , αN ).

By Proposition 4.3.12, the core F = NK is a finite extension of K, soF = K(a) for some element a ∈ F . Let p(y) be the minimum polynomial of aover K which will be also treated as a difference polynomial in the ring Ky ofdifference polynomials in one difference indeterminate y over K. Furthermore,for every i ∈ Z, let pi(y) denote the polynomial in K[y] obtained by applyingαi to every coefficient of p(y). Since K ⊆ M , p(y) ∈ My. By Theorem 7.2.1,whose proof is independent of the results of this section and their consequences,there is a solution η of the difference polynomial p(y) in some difference fieldextension Ω = (Ω, αΩ) of M . Without loss of generality we can assume thatthe extension Ω/K is normal (otherwise one can replace Ω by a difference fieldwhose underlying field is the normal closure of Ω over K).

Let S = p(y), p1(y), p−1(y), . . . ⊆ Ky. Since the difference field exten-sion F/K is normal and for every i ∈ Z, αi

N (a) is a root of pi(y) contained in F ,F is a splitting field of the family S over K. On the other hand, for every i ∈ Z,αi

Ω(η) is a root of pi(y) contained in Ω. Since the extension Ω/K is normal, Ωcontains a splitting field G of the family S over K. Furthermore, since for everyzero b of a polynomial pi(y), αΩ(b) is a zero of pi+1(y), the restriction αG ofαΩ to G is an isomorphism of the field G onto itself. Thus, G = (G, αG) is adifference subfield of Ω = (Ω, αΩ).

It follows from Theorem 1.6.5(iii) that there is a K-isomorphism µ of Fonto G (both fields are splitting fields of the same set of polynomials over K).Then β = µ−1αGµ is an endomorphism of F whose restriction on K coincideswith α. Thus, Fβ = (F , β) is a difference overfield of K.

The following diagram illustrates the arrangement of the fields under con-sideration.


L

K

N

F

K

MG

Ω

M〈η〉

µ

Since F is a finite normal separable extension of K, β(F )/K is also suchan extension. Furthermore, F

⋂L = LK = K and β(F )

⋂αL(L) = K = α(K)

(since K = α(K) ⊆ β(F )⋂

αL(L) ⊆ F⋂

L = K). It follows that the fieldextensions F /K and L/K are linearly disjoint (see the last part of Theorem1.6.24). Applying Theorems 1.6.43 and 1.6.5(ii) to the diagram

L

K

N

F

α(K) = Kα

N

α

αL

β

we obtain that there exists an isomorphism α from N into itself (it is shown bya dotted line) such that the restrictions of α on F and L coincide with β andαL, respectively. (Theorem 1.6.43 gives an isomorphism φ of LF into N whichextends αL and β, and Theorem 1.6.5 (ii) (see also Theorem 1.6.8(v)) impliesthe existence of an extension of φ to the desired isomorphism α : N → N , sinceN is a splitting field of the family Irr(a,K) | a ∈ L over LF .)

Let N ′ = (N , α). Then N ′ is a normal difference extension of K with the coreFβ = (F , β) (see Lemma 5.4.18). Since µ is a difference K-isomorphism of Fβ

onto G ⊆ Ω, the difference field extensions Fβ/K and ΩK/K are compatible. (Asusual, ΩK denotes the core of the difference field Ω over K.) Taking into accountthat both N ′/K and Ω/K are normal, one can apply Proposition 5.4.17 andobtain that these difference field extensions are compatible. Since L = (L, αL)and M are intermediate difference fields of the extensions N ′/K and Ω/K,respectively, the extensions L/K and M/K are compatible as well.

Note that Theorem 5.4.22 can be proved without referring to the existencetheorem for ordinary difference polynomials (Theorem 7.2.1), if one uses thealgebraic closure of M as the difference field Ω in our proof of Theorem 5.4.22.Indeed, by Corollary 5.1.16, Ω can be treated as a difference overfield of M


whose translation αΩ is an extension of the translation αM of M (which, inturn, extends the translation α of K). Clearly, Ω contains a splitting field Gof the set of polynomials S = p(y), p1(y), p−1(y), . . . ⊆ Ky (we use thenotation of the proof of Theorem 5.4.22) and αΩ(G) ⊆ G. Then we can considerthe isomorphism µ of the field F onto G and complete the prove as above.

Theorems 5.4.15 and 5.4.22 imply the following statement.

Corollary 5.4.23 Let K be an ordinary difference field with a basic set σ andlet L and M be two σ-field extensions of K. Then the following statements areequivalent.

(i) L/K and M/K are incompatible.(ii) There exist finitely generated σ-field extensions L′ and M ′ of K such that

L′ ⊆ L, M ′ ⊆ M , and L′/K and M ′/K are incompatible.(iii) LK/K and MK/K are incompatible.(iv) LK/K and M/K are incompatible.

Corollary 5.4.24 Let K be an ordinary difference field with a basic set σ = αand let L be a σ-overfield of K such that the field extension L/K is primary.Furthermore, let M and N be two σ-overfields of L such that ML/L is equivalentto L〈MK〉/L, and NL/L is equivalent to L〈NK〉/L. Then M/L and N/L arecompatible if and only if M/K and N/K are compatible.

PROOF. Let L′ be a σ-overfield of L such that there is a σ-K-isomorphismφ of MK into L′ and L′ = L〈φ(MK)〉. Since the field extension φ(MK)/K is al-gebraic, L′ is the free join of φ(MK) and L. By Proposition 1.6.49(i), φ(MK)/Kand L/K are quasi-linearly disjoint.

Let L′′ be another σ-overfield of L such that there is a σ-K-isomorphismψ of MK into L′′ and L′′ = L〈φ(MK)〉. By Theorem 1.6.43(ii), there is anisomorphism ρ of L′′ onto L′ which is identical on L and coincides with φψ−1 onψ(MK). Obviously, ρ is a σ-L-isomorphism of L′′ onto L′, so that the describedextension L′ of L is unique up to a σ-L-isomorphism. Of course, the similarstatement holds for the corresponding construction for NK .

Since the compatibility of M/L and N/L trivially implies the compatibilityof M/K and N/K, we assume that M/K and N/K are compatible and showthat the same is true for M/L and N/L. First, we observe that in our case theextensions MK/K and NK/K are also compatible. Let F be a σ-overfield of Ksuch that there are σ-K-isomorphisms µ and ν of MK and NK , respectively,into F . Without loss of generality, we can also assume that F is the compositumof µ(MK) and ν(NK). Then F/K is an algebraic extension and the σ-fieldextensions F/K and L/K are compatible by Theorem 5.1.6. let G be a σ-overfield of L such that F has a σ-K-isomorphism into G. Then MK and NK

have σ-K-isomorphisms into G. Let M ′ and N ′ denote the images of MK andNK , respectively, under these isomorphisms. By the first part of the proof,L〈M ′〉 is σ-L-isomorphic to L〈MK〉 and L〈N ′〉 is σ-L-isomorphic to L〈NK〉,therefore the extensions L〈MK〉/L and L〈NK〉/L are compatible. It follows that


the equivalent extensions ML/L and NL/L are compatible. Applying Theorem5.4.22 we obtain that the extensions M/L and N/L are also compatible.

Theorem 5.4.25 Let K be an ordinary difference field with a basic set σ =α and let L = K〈S〉 where the set S is σ-algebraically independent over K.Furthermore, let M be a σ-overfield of L. Then the σ-field extensions ML/Land L〈MK〉/L are equivalent.

PROOF. If a ∈ MK , then K〈a〉 is a finite field extension of K. Therefore,K〈S∪a〉 is a finite extension of K〈S〉 (see Theorem 1.6.2) and ld(L〈a〉/L) = 1.Furthermore, a is algebraic and separable over L (this element is algebraic andseparable over K), so that a ∈ ML. Since L ⊆ ML, we obtain that L〈MK〉 ⊆ ML.

Let b ∈ ML and let K ′ denote the algebraic closure of the field K in itsoverfield L〈b〉 = K〈S ⋃b〉 which is a σ-subfield of ML. If u ∈ K ′, thenK〈S ∪ u〉 is a finite field extension of K〈S〉. Since the set

⋃∞i=0 αi(S) is al-

gebraically independent over K ′, K〈u〉 is a finite extension of K (see Theorem1.6.28(v)). Furthermore, even if Char K > 0, the element u is separable overK. Indeed, by Theorem 1.6.28(vii), the fields K〈u〉 and L are linearly disjointwhence Irr(u,K) = Irr(u,L). Since u ∈ ML and therefore u is separable overL, we obtain that u is separable over K.

Since b ∈ ML, there exists k ∈ N such that every αi(b) (i ≥ 0) lies inK〈S〉(b, α(b), . . . , αk(b)). Furthermore, there is a positive integer r such thatthe elements b, α(b), . . . , αk(b) are algebraic over the field K(

⋃ri=0 αi(S)). Let

h be a positive integer such that k + h > r. Then αk+h(b) is algebraic overK(

⋃∞i=r+1 αi(S)).

Let U =⋃∞

i=r+1 αi(S), V =⋃r

i=0 αi(S), E = K ′(U⋃αk+h(b)), F =

K ′(V⋃b, α(b), . . . , αk(b)), and G = K ′(U

⋃V

⋃b, α(b), . . . , αk(b)) =K〈S ⋃b〉. Since U , V and U

⋃V are transcendence bases of E, F and

G, respectively, over the field K ′, we obtain that G is the free join of E andF over K ′ (see Theorem 1.6.40(ii) and Theorem 1.6.37(v)). Since the field K ′

is algebraically closed in E, it follows from Proposition 1.6.49(i) that the fieldextensions E/K ′ and F/K ′ are quasi-linearly disjoint.

In order to complete the proof, suppose first that Char K = 0. Then E and Fare linearly disjoint over K ′. Since αk+h(b) ∈ K〈S〉(b, α(b), . . . , αk(b)) ⊆ F (U),

αk+h(b) =f

gwhere f, g ∈ F [U ]. Let wi | i ∈ J be a basis of F as a vector

space over K ′. Then

f =∑i,j

λijwiuj , g =∑i,j

µijwiuj (5.4.1)

where uj are distinct products of nonnegative powers of elements of U , theelements λij and µij lie in K ′, and each sum is finite. Since g = 0, there arenonzero coefficients among µij . Without loss of generality we can assume thatµ11 = 0. Since the set wi | i ∈ J is linearly independent over E by lineardisjointness, the equation αk+h(b)g = f implies αk+h(b)

∑j

µ1juj =∑

j

λ1juj


where the summation is extended over all indices j such that at least one of theelements λ1j , µ1j in the sums in (5.4.1) is different from zero.

Since the power products uj are linearly independent over K ′ and µ11 = 0,we have

∑j

µ1juj = 0 whence αk+h(b) ∈ K ′(U) ⊆ K ′〈S〉 ⊆ L〈MK〉. Thus, if b is

any element of ML, then some transform of b lies in L〈MK〉. Since L〈MK〉 ⊆ ML,the extensions ML/L and L〈MK〉/L are equivalent.

Now suppose that Char K = p > 0. then the perfect closures E and F of Eand F , respectively, are linearly disjoint over the perfect closure K of K ′. Setting

αk+h(b) =f

gas before, and letting the wi now denote a basis of F as a vector

space over K, we obtain expressions similar to those in (5.4.1): f =∑

i,j λijwiuj

and g =∑

i,j µijwiuj where λij and µij lie in K and µ11 = 0. Then

αk+h(b)∑

j

µ1juj =∑

j

λ1juj . (5.4.2)

Since the power products uj are linearly independent over any algebraic exten-sion of K ′ and, in particular, over K, the sum in the left-hand side of (5.4.2) isnot 0, hence αk+h(b) ∈ K(U). Then there exists a positive integer m such that(αk+h(b))pm ∈ L〈MK〉, so that the element αk+h(b) is purely inseparable overL〈MK〉. since this element is in the core ML, it is separable over L and there-fore over L〈MK〉. By Theorem 1.6.19(i), αk+h(b) ∈ L〈MK〉. This completes theproof of the theorem.

Theorem 5.43.25 and Corollary 5.4.24 imply the following statement.

Corollary 5.4.26 Let K be an ordinary difference field with a basic set σ = αand let L = K〈S〉 where the set S is σ-algebraically independent over K. LetM be a σ-field extension of K such that the extensions L/K and M/K areequivalent. Furthermore, let F and G be two σ-field extensions of M such thatF/K and G/K are separably algebraic and normal. Then F/M and G/M arecompatible if and only if F/K and G/K are compatible.

We conclude this section with some applications of the preceding results tothe study of difference specializations. The correspondent results are due to R.Cohn [38], [41, Chapter 7].

Theorem 5.4.27 Let K be an ordinary difference field with a basic set σ = αand let L be a σ-overfield of K such that the σ-field extension L/K is finitelygenerated and primary. Let M be a finitely generated σ-field extension of L suchthat the extensions ML/L and L〈MK〉/L are equivalent. Furthermore, let η andζ be finite indexings such that L = K〈η〉 and M = L〈ζ〉. Finally, let B be a σ-transcendence basis of ζ over L, and let c be a nonzero element of Kη, ζ. Thenalmost every σ-specialization η of η over K such that M/K and K〈η〉/K arecompatible can be extended to a σ-specialization η, ζ of η, ζ with the properties(i), (ii), and (iii) of Theorem 5.3.2.


PROOF. Let ζ∗ denote a finite set of σ-generators of ML over L. Then,in order to prove the theorem, it is sufficient to show that almost every σ-specialization η of η over K such that K〈η, ζ〉/K and K〈η〉/K are compatiblecan be extended to a σ-specialization of η, ζ∗ over K which does not annul apre-assigned nonzero element u ∈ Kη, ζ∗.

Let ζ ′ be a finite set of σ-generators of MK over K. Since the extensionsML/L and L〈MK〉/L are equivalent, there is m ∈ N such that αm(u) ∈ K〈η, ζ ′〉and αm(a) ∈ K〈η, ζ ′〉 for every element a in ζ∗. Since the elements of ζ ′ arealgebraic over K〈η〉, there exists an element v ∈ Kη with the following prop-erties:

(a) If a is an element of ζ∗, then αm(a) can be represented as a fractionza

vwhere za ∈ Kη, ζ ′.

(b) αm(u) =w

vfor some w ∈ Kη, ζ ′.

Since w is algebraic over K〈η〉, there is a polynomial f(X) in one variableX with coefficients in K〈η〉 such that f(w) = 0. Let e denote the term of f(X)free of X. Since u = 0, we can assume that e = 0.

Clearly, if a σ-specialization of η over K does not annul ve and extends to aσ-specialization of η, ζ ′ over K, then it also extends to a σ-specialization of η, ζ∗

over K not annulling u. To complete the proof of the theorem, it is sufficientto show that every (and therefore almost every) σ-specialization η of η over Ksuch that K〈η, ζ〉/K and K〈η〉/K are compatible extends to a σ-specializationof η, ζ′ over K.

Assuming that the conditions stated in the last paragraph hold, we observefirst that the compatibility of K〈η, ζ〉/K and K〈η〉/K imply the compatibilityof K〈ζ′〉 and K〈η〉/K. Let K〈η, ζ ′〉/ be a σ-overfield of K〈η〉 such that K〈ζ ′〉has a σ-K-isomorphism into K〈η, ζ ′〉/, the image of ζ ′ being ζ ′. Let us showthat η, ζ ′ is a σ-specialization of η, ζ ′ over K, that is, the αi(η), αi(ζ ′) (i ∈ N)form a specialization of αi(η), αi(ζ ′) over the field K. Indeed, it is clear thatthe αi(η) form a specialization of the αi(η) and the αi(ζ ′) of the αi(ζ ′). Also,the αi(ζ ′) are algebraic over K and the field extension K(η, α(η), α2(η), . . . )/Kis primary by our hypothesis. Applying Proposition 1.6.49(i) we obtain thatthe fields K(ζ ′, α(ζ ′), α2(ζ ′), . . . ) and K(η, α(η), α2(η), . . . )/K are quasi-linearlydisjoint over K. Now Proposition 1.6.65 shows that the αi(η) and αi(ζ ′) form aspecialization of the αi(η) and αi(ζ ′). This completes the proof.

Theorem 5.4.28 Let K be an ordinary difference field of zero characteris-tic with a basic set σ = α. Let L = K〈η, ζ〉 be a finitely generated σ-fieldextension of K where η = (η1, . . . , ηk) and ζ = (ζ1, . . . , ζl) are finite index-ings with elements η1, . . . , ηk σ-algebraically independent over K. Furthermore,let B be a σ-transcendence basis of ζ over K〈η〉 and let c be a nonzero el-ement of Kη, ζ. Finally, let Ky1, . . . , yk be the ring of σ-polynomials inσ-indeterminates y1, . . . , yk over K.

Then there exists a nonzero σ-polynomial A ∈ Ky1, . . . , yk such that if anindexing η = (η1, . . . , ηk) is not a solution of A and K〈η〉/K is compatible withL/K, then there is a σ-specialization (η, ζ) of (η, ζ) with the properties (i), (ii),and (iii) of Theorem 5.3.2.


PROOF. It follows from Theorems 5.4.25 and 5.4.27 that there exists anonzero element µ ∈ Kη such that every σ-specialization of η which does notspecializes µ to zero and satisfies the stated property of compatibility can beextended to a σ-specialization of η, ζ with the properties (i), (ii), and (iii) ofTheorem 5.3.2.

Since the components of η are σ-algebraically independent over K, any in-dexing of k elements in a σ-overfield of K is a specialization of η. If µ is expressedas a σ-polynomial in η1, . . . , ηk with coefficients in K, then one can choose thedesired σ-polynomial A as the result of replacing each ηi in this expression of µby the corresponding σ-indeterminate yi (1 ≤ i ≤ k).

The following example shows that the condition of σ-algebraic independenceof η1, . . . , ηk in the formulation of Theorem 5.4.28 is essential.

Example 5.4.29 Let G and H denote ordinary difference fields constructedon Q(i) (i2 = −1) with the use of the identical automorphism and complexconjugation, respectively. (We consider Q as a difference field whose basic setσ consists of the identical translation α and denote the translations of G andH by the same symbol α.) To avoid confusion, we shall denote i, treated as anelement of the field H, by j, so that G = Q〈i〉 and H = Q〈j〉.

Let us adjoin to H a σ-algebraically independent over H element η and letζ = jη. If η, ζ is a σ-specialization of η, ζ over Q with η = 0, then H is σ-Q-

isomorphic to Q〈 ζ

η〉, since

(ζ

η

)2

= −1 and the translation sends the element

ζ

ηto − ζ

η.

It is easy to see that the variety of a linear difference polynomial αky − iy(k = 1, 2, . . . ) of the difference polynomial ring Gy is irreducible. Let λk be ageneric zero of this variety. Then for every k = 1, 2, . . . , λk = 0, G ⊆ Q〈λk〉, andλk is a σ-specialization of η over Q, since η is σ-algebraically independent overQ. This σ-specialization cannot be extended to a σ-specialization λk, ζ of η, ζover Q. Indeed, if such an extension exists, then Q〈λk, ζ〉/K would contain theintermediate σ-subfield G and a σ-subfield σ-Q-isomorphic to H. This wouldcontradict the incompatibility of G/Q and H/Q established in Example 5.1.2.

The following theorem shows that “in most cases” an indexing and its spe-cialization generate compatible difference field extensions.

Theorem 5.4.30 Let K be an ordinary difference fields with a basic set σ =α and let η be a finite indexing whose coordinates lie in some σ-overfield ofK. Then almost every σ-specialization of η over K generates a σ-field extensionof K compatible with K〈η〉/K.

PROOF. Let S be the separable part of K〈η〉 over K. By Theorem 4.4.1,there exists a finite set ζ such that S = K〈ζ〉. Furthermore, there is a nonzeroelement µ ∈ Kη such that every member of ζ is a quotient of some element


of Kη by µ. Then every σ-specialization η of η over K which does not spe-cialize µ to 0 can be extended to a σ-specialization η, ζ of η, ζ over K. Thenζ, α(ζ), α2(ζ), . . . is a specialization of ζ, α(ζ), α2(ζ), . . . over the field K (in thesense of the Definition 1.6.55). Since all elements of all αi(ζ) (i ∈ N) are alge-braic over K, they have only generic specializations. It follows that there is aK-isomorphism from K(ζ, α(ζ), α2(ζ), . . . ) to K(ζ), . . . with αi(ζ) correspond-ing to αi(ζ), i = 0, 1, . . . . Therefore, K〈ζ〉 is σ-K-isomorphic to K〈ζ〉, and sinceK〈ζ〉 ⊆ K〈η〉, the σ-field extensions K〈ζ〉/K and K〈η〉/K are compatible. Ap-plying Theorem 5.1.6 we obtain that K〈η〉/K and K〈η〉/K are also compatible.This completes the proof.

Theorems 5.4.28 and 5.4.30 lead to a generalization if the difference versionof the Nullstellensatz (Theorem 2.6.5). In what follows, K denotes an ordinarydifference field with a basic set σ = α and Ky1, . . . , ys denotes the ring ofσ-polynomials in σ-indeterminates y1, . . . , ys over K. As before, we say that anindexing (a set) is σ-algebraic over K if every its coordinate (respectively, everyelement of the set) is σ-algebraic over K.

Lemma 5.4.31 With the above notation, let M and N be distinct varietiesover Ky1, . . . , ys. Then their subsets consisting of solutions σ-algebraic overK are distinct.

PROOF. Suppose, first, that M is a nonempty irreducible variety overKy1, . . . , ys and N is a variety over Ky1, . . . , ys, which does not containM. Then there exists a σ-polynomial A ∈ Φ(N )\Φ(M) (we use the notation ofSection 2.6). Let (η1, . . . , ηs) be a generic zero of M and let K ′ be the algebraicclosure of K in K〈η1, . . . , ηs〉. Let L be a completely aperiodic σ-overfield ofK ′, which is σ-algebraic over K ′. (For example, L is the σ-field obtained byadjoining to K ′ a generic zero of the σ-polynomial α(z) − z − 1 in the ring ofσ-polynomials in one σ-indeterminate z over K ′.) Then Theorem 5.1.6 impliesthat the σ-field extensions L/K and K〈η1, . . . , ηs〉/K are compatible.

Let ζ be a σ-transcendental basis of the set η = η1, . . . , ηs over K and letλ = η \ ζ. Without loss of generality we can assume that the elements of ζ andλ form a k-tuple (η1, . . . , ηk) (1 ≤ k ≤ s) and the (s − k)-tuple (ηk+1, . . . , ηs)denoted by the same letters ζ and λ. Let u denotes the indexing (y1, . . . , yk)of our σ-indeterminates and let µ be the element of Kη1, . . . , ηs obtained bysubstituting ηi for yi in A (1 ≤ i ≤ s). Since A /∈ Φ(M), µ = 0.

By Theorem 5.4.28, there exists a nonzero σ-polynomial B ∈ Ku with thefollowing property: if ζ is an indexing of k elements in a σ-overfield of K, ζ is nota solution of B and K〈ζ〉/K is compatible with K〈ζ, λ〉/K, then there exists aσ-specialization ζ, λ of ζ, λ such that every coordinate of λ is σ-algebraic overK and the σ-specialization of µ is not 0. Then ζ, λ is an s-tuple in M whichdoes not annul B and therefore this s-tuple does not lie in N .

Since L/K and K〈η1, . . . , ηs〉/K are compatible, it follows from Proposition4.5.3 that ζ can be chosen as a set of elements of L. Then every coordinate ofζ is σ-algebraic over K hence the s-tuple ζ, λ is σ-algebraic over K. We have


obtained that M contains a solution, which is σ-algebraic over K and does notlie in N . this completes the proof.

Theorem 5.4.32 Let K be an ordinary difference field with a basic set σ andKy1, . . . , ys an algebra of σ-polynomials in σ-indeterminates y1, . . . , ys overK. Let S ⊆ Ky1, . . . , ys and let A be a σ-polynomial in Ky1, . . . , ys. Thenthe following conditions are equivalent.

(i) A is annulled by every solution of the set S which is σ-algebraic over K.

(ii) A lies in the perfect ideal S of the ring Ky1, . . . , ys.

PROOF. The implication (ii) =⇒ (i) is obvious, so we just need to showthat (i) implies (ii). Suppose that (i) holds, but A /∈ S. By Theorem 2.6.5, thevariety M(S) strictly contains M(S ∪ A). Applying Lemma 5.4.31 we obtainthat there is an s-tuple η ∈ M(S) \M(S ∪ A), which is σ-algebraic over K.This contradicts condition (i).

We complete this section with a result on the existence of simple (that is,generated by one element) incompatible difference field extensions obtained withthe use of the criterion of compatibility.

Theorem 5.4.33 Let K be an inversive ordinary difference field with a basicset σ = α, let L = K〈η1, . . . , ηm〉 be a σ-field extension of K such that σ-trdegKL = m (so that η1, . . . , ηm is a σ-transcendence basis of L over K). LetF and G be finitely generated incompatible σ-field extensions of the inversiveclosure L∗ of L. Then there exist elements a ∈ F and b ∈ G such that the σ-fields K〈a〉 and K〈b〉 are algebraic extensions of K and the σ-field extensionsK〈a〉/K and K〈b〉/K are incompatible.

PROOF. Since the σ-field extensions F/L∗ and G/L∗ are incompatible, thesame is true for the extensions N(F )/L∗ and N(G)/L∗ where N(F ) and N(G)denote the normal closures of F and G, respectively, over L∗. Therefore, withoutloss of generality we can assume that F and G are normal extensions of L∗. ByTheorem 5.4.22, the σ-field extensions FL∗/L∗ and GL∗/L∗ are incompatible.It follows (see Proposition 2.1.7(v)) that there exist σ-field extensions F ′ andG′ of L such that F ′ ⊆ F , G′ ⊆ G, the extensions F ′/L and G′/L are algebraicand incompatible, and ld(F ′/L) = ld(G′/L) = 1. Then the σ-fields F ′ and G′

are generated by the adjunction to L some elements a and b, respectively, whichare algebraic over K.

If K〈a〉/K and K〈b〉/K are compatible, then one can find a σ-field exten-sion H of K containing σ-subfields σ-isomorphic to K〈a〉 and K〈b〉. Adjoin-ing m elements annulling no nonzero σ-polynomial with coefficients in H (andhence annulling no nonzero σ-polynomial with coefficients in K), we obtain aσ-overfield of H which contains a σ-subfield σ-isomorphic to L. We obtain a σ-field extension of L which contains σ-subfields σ-isomorphic to L〈a〉 and L〈b〉.Thus, the extensions L〈a〉/L and L〈b〉/L are compatible, contrary to the factthat F ′/L and G′/L are incompatible. This completes the proof.


5.5 Replicability

Definition 5.5.1 Let K be a difference field with a basic set σ and L/K, M/Ktwo σ-field extensions of K. Then the number of σ-K-isomorphisms of L into Mis called the replicability of L/K in M/K. The replicability of a σ-field extensionL/K is defined as the maximum of replicabilities of L/K in all σ-field extensionsof K, if this maximum exists, or ∞ if it does not.

It is easy to see that if a difference field extension L/K is finitely generated(in the difference sense), then it is sufficient to define the replicability of L/Kconsidering only difference fields in the universal system over K (see Definition2.6.1). Furthermore, if L/K is a difference field extension and K∗ and L∗ arethe inversive closures of K and L, respectively, then the replicability of L∗/K∗

is the same as the replicability of L/K. (We leave the proof of this fact to thereader as an exercise.) The following statement is a version of the R. Cohn’s“fundamental replicability theorem” (see [41, Chapter 7, Theorem II]) in thecase of partial difference fields.

Theorem 5.5.2 Let K be difference field with a basic set σ and L a σ-fieldextension of K. Then a necessary condition for finite replicability of L/K isthat every element of L have a transform of some order algebraic over K. Thiscondition is sufficient if L is finitely generated over K.

PROOF. Suppose, first, that the σ-fields K and L are inversive and thereplicability of L/K is finite. Let L′ be the algebraic closure of K in L, andsuppose that L contains elements no transforms of which are algebraic over K.We shall show that for every i ∈ N, there exists an inversive σ-overfield Li ofK and 2i σ-K-isomorphisms of L into Li with the property that if a ∈ L \ K ′,then the images of a under these isomorphisms are all distinct. Clearly, theseimages are not in the σ-field L′

i of all elements of Li algebraic over K.We construct the sequence of σ-fields Li by induction starting with L0 = L.

Suppose that the fields L0, . . . , Li − 1 (i ≥ 1) has been found. By Corollary5.1.5 (with L′

i−1 for K and Li−1 for both L1 and L2 in the conditions of theCorollary), we find a σ-overfield M of Li−1 such that Li−1/L′

i−1 has two σ-Li−1-isomorphisms φ and ψ into M with φ(Li−1) and ψ(Li−1) quasi-linearly disjointover L′

i−1 and M their compositum. Clearly, the σ-field M is inversive, and forany distinct elements a, b ∈ Li−1 \L′

i−1, the elements φ(a), φ(b), ψ(a), ψ(b) areall distinct (see Theorem 1.6.43). Thus, we can set Li = M . Since the replicabil-ity of L/K in Li/K is at least 2i, the existence of the sequence L0, L1, . . . impliesthat the replicability of L/K is ∞. This completes the proof of the necessity inthe case of inversive difference field extensions. The necessity in the general caseimmediately follows from the observation made before the theorem: if K∗ andL∗ are the inversive closures of K and L, respectively, then the replicability ofL∗/K∗ is the same as the replicability of L/K.

To complete the proof of the theorem, assume that L = K〈S〉, where S =a1, . . . , am is a finite set, and that every element of L has a transform in L′,the algebraic closure of K in L. Then there exist τ1, . . . , τm ∈ Tσ such that

5.5. REPLICABILITY 353

τi(ai) ∈ L′ (1 ≤ i ≤ m). Setting τ = τ1 . . . τm we obtain that τ(S) ⊆ L′.Since every element of L has a transform in K〈τ(S)〉, L/K and K〈τ(S)〉/Khave equal replicabilities. Therefore, it remains to prove that the replicabilityof K〈τ(S)〉/K is finite. Let N be any σ-overfield of K. Then distinct σ-K-isomorphisms of K〈τ(S)〉 into N contract to distinct field K-isomorphisms ofK(τ(S)) in to N . Since the number of such K-isomorphisms is finite (it cannotexceed K(S) : K), this completes the proof of the theorem. Corollary 5.5.3 Let K be a difference field with a basic set σ, Ky1, . . . , ys thering of σ-polynomials in σ-indeterminates y1, . . . , ys over K, and P a reflexiveprime σ-ideal in Ky1, . . . , ys. Furthermore, let η = (η1, . . . , ηs) be a genericzero of P and L = K〈η1, . . . , ηs〉. If trdegK∗L∗ > 0, then there exist σ-overfieldsof K containing arbitrarily many generic zeros of P .

PROOF. It is easy to see that if P satisfies the stated condition, thereexists an element of L no transform of which is algebraic over K. ApplyingTheorem 5.5.2 we obtain that L/K has finite replicability. Therefore, there areinfinitely many distinct images of η under σ-K-isomorphisms into σ-overfieldsof K. Since each such an image is a generic zero of P , this completes theproof.

The following statement is a direct consequence of Theorems 5.1.10 and5.4.22.

Proposition 5.5.4 Let L be a difference field extension of an ordinary differ-ence field K with a basic set σ such that LK = K. Then every σ-isomorphism ofK into a σ-overfield M of L extends to a σ-isomorphism of L into a σ-overfieldof M .

It follows from the last proposition that if an ordinary difference field ex-tension L/K is separably algebraic and normal, then its replicability is not lessthan the replicability of LK/K. The following result strengthens this statement.

Proposition 5.5.5 Let K be an ordinary difference field with a basic set σ, letL be σ-field extension of K, and let F be an intermediate σ-field of the extensionL/K. Then the replicability of L/K is greater than or equal to the replicabilityof LF /K.

PROOF. Let M be a σ-overfield of K such that there are k distinct σ-K-isomorphisms φ1, . . . , φk of LF into M . We are going to show that there are kdistinct σ-K-isomorphisms of L into some σ-overfield of K. Actually, we willconstruct inductively a sequence M1 ⊆ · · · ⊆ Mk of σ-overfields of M such thateach φi extends to a σ-K-isomorphism of L into Mi. By Theorem 5.4.22 wemay assume that L ⊆ M . Applying Proposition 5.5.4 we obtain that φ1 has anextension to a σ-K-isomorphism of L into some σ-overfield M1 of M .

Suppose that σ-fields M1, . . . , Mj−1 (2 ≤ j ≤ k) for which each φi (1 ≤ i ≤j − 1) extends to a σ-K-isomorphism of L into Mi has been constructed. ByProposition 5.5.4, there exists an extension of φj to a σ-K-isomorphism of Linto a σ-overfield Mj of Mj−1. This completes the construction of the sequence,and therefore the proof of the proposition.


5.6 Monadicity

Definition 5.6.1 Let K be a difference field with a basic set σ and L a differ-ence (σ-) overfield of K. The difference field extension L/K is called monadicif its replicability is 1. (In this case we also say that L is monadic over K.) Ifthe replicability of L/K is greater than 1, the extension is said to be amonadic.Thus, a difference field extension L/K is monadic if L does not admit two dis-tinct σ-K-isomorphisms into any σ-field extension of K. A monadic differencefield extension L/K is called properly monadic if L = K and not every elementof L has a transform in K.

Example 5.6.2 Let L be the field of rational fractions in one indeterminatex over the field of complex numbers C. Let t = x3 and let K = C(t). Letus treat L as an ordinary difference field with a basic set σ = α such thatα : f(x) → f(x2) for every f(x) ∈ L. Clearly, α is an isomorphism of L intoitself and K is a σ-subfield of L. Let us show that the identical automorphism isthe only σ-K-automorphism of L (despite the fact that there are non-identicalK-automorphisms of L, since the field extension L/K is normal, see Theorem1.6.9). Moreover, we shall show that if M is any σ-overfield of L, then there isat most one σ-K-isomorphism of L into M . Indeed, let φ and ψ be two suchσ-K-isomorphisms. Let y = φ(x) and z = ψ(x). Then y3 = z3 = t, α(y) = y2,and α(z) = z2. Clearly, in order to prove that φ = ψ, it is sufficient to show thatz = y. Since y3 = z3, we obtain that z = ωy where ω3 = 1. Then ω ∈ C ⊆ Kand α(ω) = ω. Furthermore, since α(z) = z2, ωα(y) = ω2y2 hence α(y) = ωy2.On the other hand, α(y) = y2 = 0. It follows that ω = 1 and z = y. Thus, theσ-field extension L/K is monadic.

Definition 5.6.3 A difference field extension L/K with a basic set σ is calledpathological if either L is monadic over K or L/K is incompatible with someother σ-field extension of K.

The first two statements of the next proposition are direct consequences ofProposition 2.1.7(v). The third statement follows from Proposition 5.5.5.

Proposition 5.6.4 Let K be an inversive difference field with a basic set σ.Then

(i) A σ-field extension L of K is monadic if and only if the inversive closureL∗ of L is a monadic extension of K.

(ii) If L1 and L2 are equivalent algebraic finitely generated σ-field extensionsof K, then the extension L1/K is monadic if and only if L2/K is monadic.

(iii) If a σ-field extension L/K is monadic, then the extension LK/K is alsomonadic.

5.6. MONADICITY 355

The following two theorems describe situations when difference field exten-sions are amonadic. Both statements are due to A. Babbitt [2].

Theorem 5.6.5 Let K be an inversive ordinary difference field with a basic setσ = α and let K〈η〉 be a benign σ-field extension of K with minimal normalstandard generator η. Suppose that η /∈ K, and let L be an intermediate σ-fieldof K〈η〉/K. Then L is an amonadic extension of K.

PROOF. Suppose that the extension L/K is monadic. Without loss of gen-erality we can assume that L = K〈ζ〉 for some element ζ ∈ L. let N denotethe inversive closure of K〈η〉 and let M be the inversive closure of L in N .Furthermore, let G = Gal(N/K) be the Galois group of the extension N/K inthe usual algebraic sense and let G(M) be a subgroup of G corresponding to thesubfield M (that is, G(M) = M ′ if one uses the notation of Theorem 1.6.51).

Since K〈η〉/K is a benign extension with minimal normal standard generatorη and K is inversive, for any two distinct integers i and j, the fields K(αi(η))and K(αj(η)) are linearly disjoint and have isomorphic Galois groups over K.Therefore, we can represent G as an infinite direct product of finite groups allisomorphic to the Galois group Gη of K(η) over K. Using this observation wewill denote elements g ∈ G as strings (. . . , g0, g1, g2, . . . ) where gi (i ∈ Z) denoteelements of Gη and their isomorphic images in the components of the mentioneddecomposition of G. Furthermore, for any g = (. . . , g0, g1, g2, . . . ) ∈ G we setg∗ = α−1gα.

Notice that that if gα(x) = αg(x) for every element x ∈ M , then g ∈G(M), since L is a monadic extension of K and by Proposition 5.6.4 the sameis true of M . Also, it is easy to see that if g = (. . . , g0, g1, g2, . . . ), then g∗ =(. . . , g1, g2, g3, . . . ), that is, if g0 acts on η, g1 on α(η), etc., then g1 acts on η, g2

on α(η), etc. in g∗. With this definition of g∗, it is clear that if M is a monadicextension of K and g−1g∗ ∈ G(M), then g ∈ G(M).

Let ζ ∈ K〈αm(η), . . . , αm+i(η)〉 (m ∈ N). Then we can choose elementsgm, . . . , gm+i, where gm+j acts on αm+j(η) (0 ≤ j ≤ i) so that (. . . , gm−1, gm, g2,. . . , gm+i, gm+i+1, . . . ) ∈ G\G(M) for any choice of . . . , gm−1, gm−2, . . . ; gm+i+1,gm+i+2, . . . . It follows that for some integers k (e.g., for k = i) it is possible toselect elements gm, . . . , gm+k acting on αm(η), . . . , αm+k(η), respectively, suchthat g = (. . . , gm−1, gm, . . . , gm+k, gm+k+1, . . . ) ∈ G \ G(M) for any choice of. . . , gm−1, gm−2, . . . ; gm+k+1, gm+k+2, . . . . Let such a k be chosen as small aspossible.

Let g = (. . . , gm−1, gm, . . . , gm+k, gm+k+1, . . . ) be an arbitrary comple-tion of gm, . . . , gm+k to an element (a “string”) of G and let h = g−1g∗.Since g /∈ G(M) and M is a monadic extension of K, h /∈ G(M). Settinghi = g−1

i gi+1 (i ∈ Z) we can easily see that hm, . . . , hm+k−1 are determinedby gm, . . . , gm+k, while the other hi being arbitrary (since any desired sethm−1, hm−2, . . . ;hm+k, hm+k+1, . . . can be obtained by some choice of a com-pletion of gm, . . . , gm+k to a string in G). If k > 0, this contradicts the choice ofk. If k = 0, then all hi are arbitrary and can be chosen so that the correspondingh is the identity e of G. We obtain that e /∈ G(M) which is impossible. Thiscompletes the proof of the theorem.


Theorem 5.6.6 Let K be an inversive ordinary difference field with a basicset σ = α and let L = K〈η〉 be a σ-field extension of K generated by anelement η ∈ L. Furthermore, suppose that the extension L/K is algebraic andld(L/K) > 1. Then L is an amonadic extension of K.

PROOF. Let N be the normal closure of L over K. Then N is a finitely gen-erated σ-field extension of K (it immediately follows from Theorem 1.6.13(iii)).Since ld(K〈η〉/K) > 1, η /∈ NK .

Let elements ξ1, . . . , ξn define a benign decomposition of N over K. For everyi = 1, . . . , n, let Mi denote the inversive closure of NK〈ξ1, . . . , ξi〉 and let M0 =NK . Furthermore, let l denote the smallest positive integer for which η ∈ Ml

(1 ≤ l ≤ n). Since NK〈ξ1, . . . , ξl−1〉〈ξl〉 is a benign extension of NK〈ξ1, . . . , ξl−1〉and the extensions NK〈ξ1, . . . , ξl−1〉/K and Ml−1/K are equivalent, we obtainfrom Theorem 5.4.12 that Ml−1〈αm(ξl)〉 is a benign extension of Ml−1 for somesufficiently large m ∈ N. Since for all sufficiently large r ∈ N we have αr(η) ∈Ml−1〈αm(ξl)〉 \ Ml−1, it follows from Theorem 5.6.5 that Ml−1〈αr(ξl)〉 is anamonadic extension of Ml−1 for such large r. Applying Proposition 5.6.4(ii)we obtain that Ml−1〈η〉 is an amonadic extension of Ml−1. This implies thatL = K〈η〉 is an amonadic extension of K.

The further study of monadic extensions of ordinary difference fields willneed consideration of Galois groups of difference field extensions and the conceptof coordinate extensions (see Definition 5.6.19) below). R. Cohn introduced thisconcept in [41] and used it in his investigation of monadicity. In what followswe present the results of this study.

Let K be an inversive difference field with a basic set σ = α1, . . . , αn and La σ∗-overfield of K. As usual, Gal(L/K) denotes the corresponding Galois group,that is, the group of all automorphisms (not necessarily σ-automorphisms) thatleave the field K fixed. It is easy to see that the mappings αi : θ → α−1

i θαi

(θ ∈ Gal(L/K), 1 ≤ i ≤ n) are automorphisms of the group Gal(L/K); theyare called the induced automorphisms of Gal(L/K).

Definition 5.6.7 With the above notation, a subgroup B of Gal(L/K) is calledσ-stable if αi(B) = B for i = 1, . . . , n. If αi(b) = b for every b ∈ B,αi ∈ σ, thesubgroup B is called σ-invariant. The largest σ-invariant subgroup of Gal(L/K)consists of all σ-automorphisms of L that leave the field K fixed. This group iscalled the difference (or σ-) Galois group of L/K; it is denoted by Galσ(L/K).

Definition 5.6.8 A difference field extension L/K with a basic set σ is calledσ-stable in a σ-overfield M of L if every σ-K-automorphism of M maps L intoitself. The extension L/K is said to be strongly σ-stable in M if every σ-K-isomorphism of L into M maps L onto itself.

A difference (σ-) field extension L/K is called σ-stable (strongly σ-stable) ifit is σ-stable (respectively, strongly σ-stable) in every σ-overfield of L.

Exercise 5.6.9 Show that a difference (σ-) field extension L/K is σ-stable ifand only if for any σ-overfield M of L, every σ-K-automorphism of M maps Lonto itself.

5.6. MONADICITY 357

Exercise 5.6.10 Prove that an ordinary algebraic difference field extensionL/K is σ-stable if and only if it is normal.

The proof of the following statement is an easy exercise that we leave to thereader.

Proposition 5.6.11 Let K be an inversive difference field with a basic set σand let M be a σ∗-overfield of K. Then

(i) If L is an intermediate σ-field of M/K, then the Gal(M/L) is σ-stablesubgroup of Gal(M/K).

(ii) If H is a σ-stable subgroup of Gal(M/K) and F = a ∈ M |φ(a) = a forevery φ ∈ H, then F is a σ∗-overfield of K and H is a subgroup of Gal(M/F ).

(iii) For any intermediate σ∗-field L of the extension M/K, one hasGalσ(M/K)

⋂Gal(M/L) = Galσ(M/L).

Exercise 5.6.12 Let K be a difference (not necessarily inversive) differencefield with a basic set σ, L a σ-overfield of K, and H = g ∈ Gal(L/K) |αg = gαfor every α ∈ σ. Show that

a) If A is a subgroup of H, then L(A) = x ∈ L | g(x) = x for every g ∈ Ais a difference (σ-) field.

b) If the σ-field L is inversive, then L(A) is inversive and H = Gal(L/K∗)where K∗ denotes the inversive closure of K in L.

Example 5.6.13 As in Example 5.1.11, let Q be treated as an inversive or-dinary difference field with the identity translation α, and let a and i de-note the positive fourth root of 2 and the square root of −1, respectively. LetL = Q(i, a) be considered as a σ∗-overfield of Q such that α(i) = −i andα(a) = −a. Then K = Q(i) is an intermediate σ∗-field of L/Q and Gal(L/K)is the cyclic group of order 4 with generator β such that β(a) = ia. In thiscase, α(β)(a) = α−1βα(a) = α−1β(−a) = α−1(−ia) = −ia, hence α(β) = β3.It follows that Galσ(L/Q) consists of the identity automorphism and β2. Thefixed field M(Galσ(L/Q)) of this group is Q〈a2〉, which is the only intermediateσ∗-field of L/Q (except of Q and L).

In what we consider ordinary difference fields. In this case we shall use thenotation introduced in section 5.4: if K is a difference field with a basic setσ, then K will denote the underlying field of K (that is, if a capital letterdenotes some difference field, then the same letter with a “hat” denotes thecorresponding underlying field). The field K will be also denoted by (K, σ) or(K, α) if K is an ordinary difference field with a basic set σ = α. (However, ifno confusion can arise, we still can talk about algebraic, separable, normal, etc.σ-field extensions of K meaning that the underlying fields of such extensionsare algebraic, separable, normal, etc. field extensions of K.)

Proposition 5.6.14 Let K be a difference field with a basic set σ and let L bea σ-overfield of K such that the extension L/K is σ-stable. Then the inversiveclosure of L is algebraic over the inversive closure of K.


PROOF. Without loss of generality we may assume that the σ-field K isinversive and L is a σ∗-overfield of K. Let K denote the algebraic closure ofK in L treated as a σ∗-overfield of K (clearly, α(K) = K for every α ∈ σ).If K = L, it follows from Corollary 5.1.5 (with L2 = L1) that there exists aσ-overfield M of L and a σ-K-isomorphism φ : L → M such that L and φ(L)are quasi-linearly disjoint over K, and M = L〈φ(L)〉. By Theorem 1.6.43, thereis an automorphism ψ of M which contracts to φ on L and to φ−1 on φ(L). Itfollows that ψ is a σ-K-automorphism of M which does not map L into itself.This contradiction with the σ-stability of L/K completes the proof.

Definition 5.6.15 An automorphism φ of a group G with identity e is calledregular if φ(g) = g for every g ∈ G, g = e.

Proposition 5.6.16 Let L be an inversive difference (σ∗-) field extension ofan inversive ordinary difference field K with a basic set σ = α. Then

(i) If the extension L/K is σ-stable and the induced automorphism α of thegroup Gal(L/K) is regular, then L/K is monadic.

(ii) The extension L/K is monadic if and only if it is strongly σ-stable andthe automorphism α is regular.

(iii) If the extension L/K is algebraic and normal, then L/K is monadic ifand only if this extension is σ-stable and α is regular.

PROOF. The first two statements are direct consequences of the definitionsof monadic, σ-stable, and strongly σ-stable extensions. Statement (iii) followsfrom the obvious fact that if L/K is algebraic, normal, and σ-stable, then it isstrongly σ-stable as well.

Proposition 5.6.17 Let K be a difference field with a basic set σ, L a σ-overfield of K, and M a σ-overfield of L. If the σ-field extension L/K is σ-stablein M then Galσ(M/L) is a normal subgroup of Galσ(M/K) and the factor groupGalσ(M/K)/Galσ(M/L) is isomorphic to the group of all σ-K-automorphismsof L which are contractions of σ-K-automorphisms of M .

PROOF. Let a ∈ L, φ ∈ Galσ(M/K), and λ ∈ Galσ(M/L). Since the exten-sion L/K is σ-stable in M , φ(a) ∈ L, hence λφ(a) = φ(a). Then φ−1λφ(a) = a,so that φ−1λφ ∈ Galσ(M/L).

Clearly, the elements of Galσ(M/K) contract to σ-K-automorphisms of L,and two elements of Galσ(M/K) contract to the same element of Galσ(L/K) ifand only if thee lie in the same coset of Galσ(M/L) in Galσ(M/K). We obtaina one-to-one correspondence between elements of Galσ(M/K)/Galσ(M/L) andthe elements of Galσ(L/K) which are contractions of σ-K-automorphisms of Mto L. It is easy to see that this correspondence is a group isomorphism.

Proposition 5.6.18 Let K be a difference field with a basic set σ, M a σ-overfield of K, and H is a normal subgroup of Galσ(M/K). Let L be the fixedfield of H (that is, L = a ∈ M |φ(a) = a for every φ ∈ H). Then

5.6. MONADICITY 359

(i) L/K is a σ-field extension which is σ-stable in M .

(ii) The group consisting of σ-K-automorphisms of L which are contractionsof σ-K-automorphisms of M is a homomorphic image of Galσ(M/K)/H.

PROOF. Obviously, L is a σ-subfield of M containing K. Since the sub-group H is normal, for any λ ∈ Galσ(M/K) and φ ∈ H, one has λ−1φλ ∈ H.Therefore, for every a ∈ L, λ−1φλ(a) = a, hence φλ(a) = λ(a), so that λ(a) ∈ L.Thus, L/K is σ-stable in M .

By Proposition 5.6.17, Galσ(M/L) is a normal subgroup of Galσ(M/K) andGalσ(M/K)/Galσ(M/L) is isomorphic to the group of all σ-K-automorphismsof L which are contractions of σ-K-automorphisms of M . Since H⊆Galσ(M/L),Galσ(M/K)/Galσ(M/L) is a homomorphic image of Galσ(M/K)/H.

Definition 5.6.19 Let K be an inversive ordinary difference field with a basicset σ = α and let L be a field extension of K. Then the set of all σ∗-fieldextensions of K defined on L is called a set of coordinate extensions of L/K (ora set of coordinate extensions of K defined on L). If L is a σ∗-overfield of K,then an inversive difference field extension of K with the underlying field L issaid to be an extension of K coordinate with L/K.

Example 5.6.20 As in Example 5.1.2, let us consider Q as an ordinary dif-ference (σ-) field with the identity translation α, and let L and M be twoσ-overfields of Q with the same underlying field Q(i) (i2 = −1) and transla-tions that send i to i and −i, respectively. Then it is easy to see that the set ofcoordinate extensions of L/Q consists of this extension and the extension M/Q.

The following statement and its corollary are direct consequences of thedefinitions of a coordinate extension and induced automorphism. We leave theproof to the reader as an exercise.

Proposition 5.6.21 Let K be an inversive ordinary difference field with a basicset σ = α, L a σ∗-overfield of K, G = Gal(L/K), and α the automorphism ofG induced by α (α : g → α−1gα). Then the extensions coordinate with L/K areprecisely those defined by the translations from the set αG = Gα. Furthermore,if θ ∈ G, then the automorphism αθ of G induced by αθ is equal to θα.

Corollary 5.6.22 With the notation of Proposition 5.6.21, let Aut(L) be thegroup of all automorphisms of the field L. Then, if there are inversive differencefield extensions of K defined on L, then the translations of these extensionsform a left coset and also a right coset of G in Aut(L) (and, hence, lie in thenormalizer of G in Aut(L)). Furthermore, the automorphisms of the group Ginduced by these translations form the left and a right coset of the group of innerautomorphisms of G in the group of all automorphisms of G.

Exercise 5.6.23 With the notation of Proposition 5.6.21, show that if θ andρ are elements of the group G such that the translations αθ and αρ defineisomorphic σ∗-extensions of K on the field L, then the corresponding difference


isomorphisms φ : (L, αθ) → (L, αρ) should be an element of G satisfying therelationship α(φ)θ = ρφ. Conversely, prove that if the last equality holds forsome element φ ∈ G, then φ is a difference isomorphism of (L, αθ) into (L, αρ).

Note that it is possible that no element of G satisfies the above relationship(for example, this is the case if α and θ are identity mappings while ρ(x) = xfor some x ∈ L).

It is easy to see that every element φ ∈ G = Gal(L/K) defines a differenceisomorphism of the difference field (L, αφ) onto one of the coordinate extensionsof K defined on L. Therefore, there is one-to-one correspondence between differ-ence K-isomorphisms of L/K onto coordinates extensions of L/K and elementsof the group Gal(L/K).

Because of the results of Theorems 5.1.6, 5.4.15, and 5.4.22, the study ofcompatibility of difference field extensions can be reduced to the study of com-patibility of finite separable difference field extensions. In this connection, thefollowing result seems to be quite important.

Proposition 5.6.24 Let K be an inversive ordinary difference field with a basicset σ = α and let L be a σ-overfield of K such that the field extension L/K isfinite and separable. Furthermore, let N be a σ-overfield of L whose underlyingfield is a normal closure of L over K. Then

(i) The σ-field L is inversive.(ii) If L/K is compatible with every extension coordinate with N/K, then L/K

is compatible with every σ-field extension of K.

PROOF. Statement (i) is evident: since α(L) : K = α(L) : α(K) = L : K,one has α(L) = L.

In order to prove (ii), notice, first, that N is a σ∗-field extension of Ksuch that N/K is finite, normal, and separable. Let N = K(a) and let theminimal polynomial f(y) = Irr(a, K) be treated as a σ-polynomial of one σ-indeterminate y over K. Let M be any σ-field extension of K. Then f(y) ∈ Myand we can apply the existence theorem for ordinary difference polynomials (seeTheorem 7.2.1 below), whose proof is independent of the material of this sec-tion and any results used in this section, to obtain that there is a solution bof the difference polynomial p(y) in some σ-field extension of M . Then thereis a difference K-isomorphism of K〈b〉 onto an extension of K coordinate withN/K. Furthermore, by our assumption, there is a σ-K-isomorphism φ of L intoa σ-overfield Ω of K〈b〉. Since the field extension K〈b〉/K is normal and K〈b〉contains a subfield K-isomorphic to L, φ maps L into K〈b〉 and, hence, intoM〈b〉. Thus, the difference field extensions L/K and M/K are compatible.

Let L be a finite normal separable σ-field extension of an inversive ordinarydifference field K with a basic set σ = α. (As we have seen, such extensionsare of primary importance in the analysis of compatibility.) It follows fromProposition 5.6.21 that L/K is one of the n coordinate extensions L1/K =L/K, . . . , Ln/K where n = L : K. We divide these extensions into equivalenceclasses by placing two extensions Li/K and Lj/K in the same class if and onlyif they are σ-isomorphic (that is, there is a difference K-isomorphism of Li

5.6. MONADICITY 361

onto Lj). Let us denote these equivalence classes by K1, . . . ,Kr with L/K ∈K1. Let ki and hi denote, respectively, the number of extensions in the classKi and CardGalσ(L/K) for each extension L/K in Ki, i = 1, . . . , r. (We usethe notation Galσ(L/K) for the set of all difference K-automorphisms of Lno matter what extension of α to the underlying field of L is considered as atranslation of L.) Furthermore, for every j = 1, . . . , n, let Gj = Galσj

(Lj/K),dj = CardGalσj

(Lj/K), αj the translation of Lj (which is an extension of α),and αj the automorphism of the group G = Gal(L/K) induced by αj . (clearly, ifLj/K ∈ Ki, then dj = hi.)

Example 5.6.25 As in Example 5.1.11, let Q be considered as an inversiveordinary difference field with the identity translation α, let a and i denote thepositive fourth root of 2 and the square root of −1, respectively, and let thefield L = Q(i, a) be treated as a σ∗-overfield of Q (where the translation isalso denoted by α) such that α(i) = −i and α(a) = −a. Furthermore, letK = (Q(i), α). Then one can easily see that the set of coordinate extensionsof L/K consists of four elements: L1 = L = (Q(i, a), γ1), L2 = (Q(i, a), γ2),L3 = (Q(i, a), γ3), and L4 = (Q(i, a), γ4) where γ1 = α, and γ2, γ3, and γ3 aredefined by the conditions γ2(a) = a, γ3(a) = ia, and γ4(a) = −ia, respectively.Each of these extensions has two difference automorphisms, the identity andγ23 (see Example 5.6.13). There are two classes of coordinate extensions in this

case: K1 = L1/K, L2/K and K2 = L3/K, L4/K.Theorem 5.6.26 Let K be an inversive ordinary difference field with a basicset σ = α and let L be a σ-overfield of K such that the field extension L/K isfinite, normal, and separable. Let n = L : K. Then, with the notation introducedbefore Example 5.6.25, one has

(i) n = hiki for i = 1, . . . , r,

(ii) r =1n

n∑i=1

di,

(iii)r∑

i=1

1hi

= 1.

PROOF. If an extension L′/K belongs to a class Ki (1 ≤ i ≤ r), then thenumber of difference K-isomorphisms of L′ onto some extension L′′ of the sameclass is equal to CardGalσ(L′/K) = hi. It is easy to see that Gal(L/K) isthe disjoint union of ki sets of such difference K-isomorphisms (where L′′ runstrough the class Ki). Therefore, hiki = CardGal(L′/K) = n.

For every i = 1, . . . , r, let Φi denote the set j |Lj/K ∈ Ki. Then kihi =∑Φi

dj , hence rn =r∑

i=1

∑Φi

dj =n∑

j=1

dj .

In order to prove statement (iii) one just needs to substitute ki =n

hiin the

equalityr∑

i=1

ki = n.


Corollary 5.6.27 With the assumptions of the preceding theorem, either noneof the extension coordinate with L/K is monadic or all are monadic. If theyall are monadic, then they and all their difference subextensions are compatiblewith every difference field extension of K. If there are more than one coordinateextensions of L/K and they are not monadic, then each is incompatible with atleast one of the other coordinate extensions.

PROOF. Since L/K is normal, all coordinate extensions are strongly σ-stable. It follows that the extensions of class Ki are monadic if and only ifhi = 1. Applying statement (iii) of the last theorem we obtain that one of thesecoordinate extensions is monadic if and only if there is but one class. Then theyare all monadic an Proposition 5.6.24 implies that the coordinate extensionsand all their subextensions are compatible with very difference field extensionof K.

In order to prove the last statement, notice that two coordinate extensionsare compatible if and only if they are isomorphic. Indeed, because of the nor-mality of L/K, for every difference overfield M of K and for every difference iso-morphisms φ : Li/K → M/K and ψ : Lj/K → M/K, one has φ(Li) = ψ(Lj).Therefore, φ−1ψ is a difference K-isomorphism of Lj onto Li. If there are morethan one coordinate extensions and they are not monadic, then, as we haveseen, r ≥ 2, and each coordinate extension of some class is incompatible withthe coordinate extensions of another class.

Definition 5.6.28 Let K be an ordinary difference field with a basic set σ =α and L a σ-field extension of K such that L/K is normal and separable. Theextension L/K is called isolated if the induced automorphism α of the groupGal(L/K) is identity, that is, Galσ(L/K) = Gal(L/K).

The following statement is a direct consequence of Corollary 5.6.22.

Proposition 5.6.29 Let L/K be a normal separable ordinary difference fieldextension with a basic set σ = α. Then the set of extensions coordinate withL/K contains an isolated extension if and only if α is an inner automorphismof Gal(L/K), and these extensions are then all isolated if and only if the groupGal(L/K) is commutative.

Proposition 5.6.30 With the assumptions of Proposition 5.6.29 and n =CardGal(L/K), we have the following statements.

(i) If n is prime, then the coordinate extensions of L/K are either allmonadic or all isolated and mutually incompatible.

(ii) If n = pq, where p and q are distinct primes, then there are just thefollowing four possibilities:

(a) All coordinate extensions Li (1 ≤ i ≤ n) of L/K are monadic.(b) All coordinate extensions of L/K are isolated and mutually incompatible.(c) There are p classes of coordinate extensions Li/K such that

CardGal(Li/K)=p or there are q classes of coordinate extensions Li/K withCardGal(Li/K) = q.

5.6. MONADICITY 363

(d) There is one isolated coordinate extension, a classes of coordinate exten-sions Li/K with CardGal(Li/K) = p, and b classes of coordinate extensionsLj/K with CardGal(Lj/K) = q, where a and b are solutions of the Diophantineequation qx + py = n − 1.

PROOF. Statement (i) is a direct consequence of Theorem 5.6.26(i).Suppose that there are no monadic or isolated extensions among the co-

ordinate extensions L1, . . . , Ln. Let a be the number of classes of coordinateextensions Li/K with CardGal(Li/K) = p, and let b be the number of classes

of extensions Lj/K with CardGal(Lj/K) = q. By Theorem 5.6.26,a

p+

b

q= 1,

or aq + bp = pq. Then p|a, q|b and we may write pqm + pqm′ = pq for somepositive nonnegative m and m′. Then m+m′ = 1 hence either m = 0 or m′ = 0,that is, either a = 0, b = q, or a = p, b = 0.

Now suppose that there is an isolated coordinate extension of K. It is easyto see that the number of isolated extensions coordinate with L/K is the orderof the center Z of the group Gal(L/K). It follows from the Sylow theorems thatevery group of order pq is either commutative (assuming that p ≤ q, this is thecase if p = q, or p < q and p q) or has the trivial center. Therefore, either allcoordinate extensions Li/K are isolated or just one is. In the latter case, withthe same a and b as before, Theorem 5.6.26(iii) implies that

a

p+

b

q+

1pq

= 1,

henceaq + bp = n − 1.

It is easy to see that the last equations has just one positive integer solutionwith respect to a and b.

In what follows we are considering finitely generated monadic extensions. Inparticular, it will be shown that if such an extension is separable, then it haslimit degree one. We start with the following result by R. Cohn called in [41] aMapping Theorem.

Theorem 5.6.31 Let K be an inversive ordinary difference field with a basicset σ and let L be a σ∗-overfield of K such that the field extension L/K isnormal and separable. Let G = Gal(L/K) and for any φ ∈ G, let Lφ denotethe difference field with the underlying field L and the basic set σφ = αφ.Furthermore, let F/K be a σ∗-subextension of L/K and let H = Gal(L/F ).Then

(i) The difference K-isomorphisms of F into Lφ (i. e., K-isomorphismsµ : F → Lφ such that µ(α(a)) = αφ(µ(a)) for every a ∈ F ) are the contractionsto F of the K-automorphisms θ ∈ G such that α(θ) ∈ φθH.

(ii) Two mappings θ1, θ2 ∈ G satisfying the above condition yield the samedifference K-isomorphism of F into Lφ if and only if θ1 and θ2 lie in the samecoset of H in G.


(iii) The σ∗-field extension F/K is monadic if and only if for any θ ∈ G, theinclusion α(θ) ∈ HθH implies θ ∈ H.

(iv) If the field extension L/K is finite, then the extension F/K is σ-stableif and only if for any θ ∈ G, the inclusion α(θ) ∈ GθG implies that θ lies in thenormalizer of H.

PROOF. Since the field extension L/K is normal, every K-automorphism ofF into L extends to a K-automorphism of K-automorphism of L (see Theorem1.6.8(v)). Thus, every K-automorphism of F into Lφ (φ ∈ G) is a contraction ofsome automorphism from G. On the other hand, if θ ∈ G, then the contraction ofθ to F is a difference K-isomorphism of F into Lφ if and only if for every a ∈ F ,θα(a) = αφθ(a) or θ−1φ−1α(θ)(a) = a, that is, if and only if θ−1φ−1α(θ) ∈ Hor α(θ) ∈ φθH.

Statement (ii) of the theorem is obvious.Suppose that F/K is monadic and there is an element θ ∈ G \ H such that

α(θ) ∈ HθH, that is, α(θ) = λθµ for some λ, µ ∈ H. By part (i) of our theorem,the contraction of θ to F is a non-identical difference K-isomorphism of F intoLλ. Since the identity mapping is also a difference K-isomorphism of F into Lλ

(α(e) = e ∈ λeH = H), the extension F/K is not monadic, contrary to ourassumption.

Suppose now that F/K is not monadic. Let θ1 and θ2 be elements of Gwhose contractions to F are distinct difference K-isomorphism of F into someLψ, ψ ∈ G. Then θ = θ−1

1 θ2 /∈ H, α(θ)1 = ψθ1λ, and α(θ)2 = ψθ2µ whereλ, µ ∈ H. It follows that α(θ) = (ψθ1λ)−1ψθ2µ = λ−1θ−1

1 θ2µ ∈ HθH. Thiscontradiction completes the proof of statement (iii).

In order to prove the last statement of the theorem, suppose that an elementθ ∈ G does not belong to the normalizer of H and α(θ) = γ1θγ2 for some γ1, γ2 ∈H. Since for any a ∈ F we have θ(a) = αα(θ)(a) = αγ1θγ2(a) = (αγ1)θ(a), thecontraction of θ to F is a σ-K-isomorphism of F into Lγ1 which does not leavethe field F fixed. Since γ1 ∈ H, Lγ1 is a difference overfield of F and we obtainthat the extension F/K is not σ-stable.

Conversely, suppose that there is a difference K-isomorphism ρ of F intosome Lφ, which is a difference overfield of F , such that ρ(F ) = F . Then φ ∈ Hand ρ does not lie in the normalizer of H. By part (i) of our theorem, we haveα(ρ) = φρχ for some χ ∈ H. This completes the proof.

Corollary 5.6.32 Let a difference (σ-) field extension L/K satisfy the assump-tions of the last theorem. If the set of extensions coordinate with L/K containsan isolated extension, then L/K has no properly monadic subextensions.

PROOF. It follows from Proposition 5.6.29 that α is an inner automor-phism of the group G = Gal(L/K). Let φ be an element of G that induces thisautomorphism.

Suppose that F/K is a monadic σ-subextension of L/K. Without loss ofgenerality we may assume that the σ-field F is inversive. Let H = Gal(L/F ).Then for any θ ∈ G, the inclusion φ−1θφ ∈ HθH implies that θ ∈ H. Settingθ = φ we obtain that φ ∈ H, hence φ−1λφ ∈ HλH for every λ ∈ G. It follows

5.6. MONADICITY 365

that H = G. Let us show that the last equality implies that F = K. Indeed,if a ∈ F and b is a conjugate of a in L, then there is a K-isomorphism χof K(a) onto K(b) such that χ(a) = b. By Corollary 1.6.41, χ extends to anisomorphism of L into an overfield of L. Since the field extension L/K is normal,χ ∈ G. Therefore, b = a, so the element a has no conjugates in L, except for aitself. Since L/K is normal and separable, we obtain that a ∈ K. Thus, L/Khas no properly monadic subextensions. Theorem 5.6.33 Let K be an ordinary difference field with a basic set σ andlet L be a σ-overfield of K such that the field extension L/K is algebraic. Fur-thermore, let S denote the separable closure of K in L. Then L/K is monadicif and only if S/K is monadic.

PROOF. Suppose, first, that L/K is monadic. If S/K is not monadic, thenthere is a σ-K-isomorphism φ, distinct from the identity, of S into some σ-overfield M of S. By Theorem 5.4.22, L/S is compatible with every σ-fieldextension of S, so that we may assume that L ⊆ M . Furthermore, since LS = S,φ can be extended to a σ-K-isomorphism φ′ of L into a σ-overfield M ′ of M .Since φ′ is not identity homomorphism, we obtain a contradiction with themonadicity of L/K.

Conversely, if S/K is monadic, then the field extension L/S is purely in-separable, hence it has at most one isomorphism into any overfield of S (seeProposition 1.6.21(ii)). It follows that the extension L/S is monadic, henceL/K is monadic. Theorem 5.6.34 Let K be an inversive ordinary difference field with a basic setσ = α, L a σ-overfield of K, and L∗ the inversive closure of L. Furthermore,suppose that L∗ = K〈A〉 for some set A such that

(a) The field extension K(A)/K is normal and separable.(b) If Aσ denotes the set αi(a) | a ∈ A, i ∈ Z, i = 0, then the fields K(A)

and K(Aσ) are linearly disjoint over K.Then the σ-field extension L∗/K has no properly monadic σ-subextensions.

PROOF. For every i ∈ Z, let Ai = αi(a) | a ∈ A, Aσi = αi(a) | a ∈Aσ, and Gi = Gal(K(Ai)/K). It follows from Theorem 1.6.43 that wheneverg1 ∈ Gi1 , . . . , gk ∈ Gik

(k ≥ 1, i1, . . . , ik ∈ Z), there is a unique automorphismg ∈ Gal(L∗/K) whose restriction on each Gij

(1 ≤ j ≤ k) is gj . Furthermore,condition (a) of our theorem implies that for any g ∈ Gal(L∗/K) and for any i ∈Z, the contraction of g on K(Ai) is a K-automorphism of K(Ai). It follows thatthe group Gal(L∗/K) is isomorphic to the direct sum

⊕i∈Z Gi. Furthermore,

it is easy to see that if g ∈ G0 = Gal(K(A)/K), then αigα−i belongs to Gi andthe mapping φi : G0 → Gi defined by g → αigα−i is a group isomorphism. Letus index the elements of G0 with indices from some set J (so that G0 = gjj∈J )and write gij for φi(gj) (i ∈ Z, j ∈ J). Then Gi = gijj∈J for any i ∈ Z, andelements of the group Gal(L∗/K) have unique representations as strings

(. . . , g−1j−1 , g0j0 , g1j1 , . . . ) (5.6.1)

where . . . , j−1, j0, j1, · · · ∈ J .


Let α denote the inner automorphism of the group Gal(L∗/K) induced byα (that is, α(g) = α−1gα for every g ∈ Gal(L∗/K)). Let us show that if anelement g ∈ Gal(L∗/K) is represented in the form (5.6.1), then

α(g) = (. . . , g−2j−1 , g−1j0 , g0j1 , . . . )

(so that α(g) is obtained from g by moving each coordinate one place to theleft). Indeed, it is clear that one just needs to prove this property for the casewhen g belongs to some Gi. Then for any k ∈ Z and x ∈ Ak we have

α(g)(x) = α−1gα(x) = α−1gk+1,jk+1α(x) = α−1αk+1gjk+1α−k−1α(x)

= αkgjk+1α−k(x) = gkjk+1(x).

It follows that gkjk+1 is the kth component of α(g).Let us show that if F/K is a proper σ-subextension of L/K, then F/K is not

monadic. Without loss of generality we can assume that the σ-field F is inversive.Let H = Gal(L∗/F ). We are going to prove that there exists g ∈ Gal(L∗/K)\Hsuch that α(g) ∈ gH. By Theorem 5.6.31, this implies that there is a differenceK-isomorphism of F into L∗ which is not the identity mapping, so the extensionF/K is not monadic.

First of all, notice that there is p ∈ N and elements θ0, . . . , θp ∈ G such thatthe elements of G of the form (. . . , θ00, . . . , θpp, . . . ) do not belong to H for anychoice of unspecified coordinates. Indeed, since the extension F/K is proper,one can choose an element x ∈ F \ K. Since L = K〈A〉, there exist k, r ∈ Nsuch that αk(x) ∈ K(

⋃ri=0 αi(A)). Furthermore, there is a K-automorphism

λ of the field K(⋃r

i=0 αi(A)) which maps αk(x) to a different element. SinceGal(K(

⋃ri=0 αi(A))/K) is the direct sum of the groups G0, . . . , Gr, there exist

elements λ0, . . . , λr ∈ G such that λ = λ00 . . . λrr. Setting p = r and θi = λi

for i = 0, . . . , p, we obtain the elements θ0, . . . , θp ∈ G which satisfy the desiredcondition.

Considering finite sets of elements of G with the above property, let uschoose such a set θ0, . . . , θp with the smallest possible p. Let Θ denote thecorresponding set of all elements of the form (. . . , θ00, . . . , θpp, . . . ), and let Θ′ =g−1α(g) | g ∈ Θ.

Let µj = θ−1j θj+1 (j = 0, . . . , p − 1). It is easy to check that Θ′ is the

set of elements of the form (. . . , µ00, . . . , µp−1,p−1, . . . ) for all possible choicesof unspecified coordinates. By the minimality of p, there exists and elementν ∈ Θ′ ⋂H. (If p = 0, then Θ′ = G and such an element is identity.) Let g bean element of Θ such that g−1α(g) = ν. Then g /∈ H and g−1α(g) ∈ H.

Theorem 5.6.35 Let K be an inversive ordinary difference field with a basicset σ = α and let L a finitely generated monadic σ-field extension of K.Then the field extension L/LK is purely inseparable. In particular, if the fieldextension L/K is separable, then LK = L.

PROOF. Suppose, first, that L/K is a subextension of a the inversive closureof a benign σ-field extension M/K with minimal standard generator a. Thenthe set A = a satisfies the requirements of Theorem 5.6.34. Indeed, if it is not

5.6. MONADICITY 367

so, then, setting n = K(a) : K, we obtain that there exist positive integers jand k such that

K(α−j(a), . . . , αk(a)) : K(α−j(a), . . . , α−1(a), α(a), . . . , αk(a)) < n.

Since σ-field K is inversive, one has K(αi(a)) : K = n for every i ∈ Z, hence

K(α−j(a), . . . , αk(a)) : K < nj+k+1.

The last inequality, in turn, implies the inequality

K(a, . . . , αj+k(a)) : K < nj+k+1

which contradicts the condition ld(K〈a〉/K) = n. Therefore, in this case wehave L = K.

Let us consider the general case. Let S denote the separable closure of Kin L (treated as a σ-subfield of L). Then Theorem 5.6.33 implies that the σ-field extension S/K is monadic. Let N denote the normal closure of S overK. By Theorem 5.4.13, there exists a chain of σ-field extensions K ⊆ N0 ⊆N1 ⊆ . . . Nm such that N0 = NK = (Nm)K , each σ-field Ni is inversive, eachextension Ni/Ni−1 is equivalent to a benign extension, and Nm/K is equivalentto N/K.

It is easy to see that if j is the smallest integer such that S ⊆ Nj , then j = 0.Indeed, if j > 0, then the extension Nj−1〈S〉/Nj−1 is a monadic subextensionof Nj/Nj−1. As we have shown, in this case Nj−1〈S〉 = Nj−1, so that one hasthe inclusion S ⊆ Nj−1 which contradicts our choice of j. Thus, S = LK hencethe extension L/LK is purely inseparable.

The following two statements are direct consequences of the last theorem.We leave the proof to the reader as an exercise.

Corollary 5.6.36 Let K be an inversive ordinary difference field with a ba-sic set σ and let L a finitely generated monadic σ-field extension of K. Thenrld(L/K) = 1. In particular, if Char K = 0, then ld(L/K) = 1.

Corollary 5.6.37 If an inversive difference field of zero characteristic admitsa finitely generated proper monadic extension, then it admits a proper monadicextension of finite degree.

Theorem 5.6.38 Let C(x) denote the field of rational functions in one variablex with coefficients in the field of complex numbers. Let K denote the ordinarydifference field with the underlying field C(x) and basic set σ = α whereα(f(x)) = f(x + 1) for every f(x) ∈ C(x). Then

(i) K has no incompatible difference field extensions.

(ii) The field K admits no finitely generated proper monadic extensions.

PROOF. Statement (ii) is an immediate consequence of Corollary 5.6.37.To prove part (i), suppose that g is an element of some σ-overfield of K, and let


g be algebraic over C(x). If g /∈ K, then g has finite branch points c1, . . . , ck.Then αi(g) (i = 0, 1, 2, . . . ) has finite branch points c1 + i, . . . , ck + i. There-fore, for any positive integer r, αr(g) has at least one branch point whichis not among the branch points of g, α(g), . . . , αr−1(g). Therefore, αr(g) /∈C(x, g, α(g), . . . , αr−1(g)) hence ld(K〈g〉/K) > 1. It follows from Theorem5.4.22 that K has no incompatible σ-field extensions.

The following result gives a similar characterization of another differencefield with the underlying field C(x). We refer the reader to [41, Chapter 9,Theorem XX] for the proof.

Theorem 5.6.39 Let q be a nonzero complex number such that qn = 1 forn = 1, 2, . . . . Let F be the ordinary difference field with the underlying fieldC(x) and basic set σ = α where the translation α is defined by the conditionα(f(x)) = f(qx) for every f(x) ∈ C(x). Then

(i) A σ-field extension L/F is incompatible with some other σ-field extensionof F if and only if L contains a kth root of x for some positive integer k.

(ii) F admits no finitely generated proper monadic extensions.

Theorem 5.6.40 Let K be an inversive ordinary difference field with a basicset σ and let L = K〈A〉 where the set A is σ-algebraically independent over K.Furthermore, let L∗ be the inversive closure of L. Then

(i) If M is a finitely generated separably algebraic monadic σ-field extensionof L∗, then there exists a finitely generated monadic σ-field extension F/K suchthat M = L∗〈F 〉.

(ii) If F is any monadic σ-field extension of K and M = L∗〈F 〉, then theextension M/L∗ is monadic.

PROOF. If M/L∗ is a finitely generated separably algebraic monadic σ-fieldextension, then Theorem 5.6.35 implies that ML∗ = M . It follows from Theorem4.3.19 that the difference field M , which is a core over an inversive σ-field, isinversive. Applying Theorem 5.4.25 we obtain that the σ-fields M and L∗〈MK〉have the same inversive closure, hence M = L∗〈MK〉.

It follows from Corollary 5.6.36 and Proposition 4.3.12 that M : L∗ < ∞,hence there is a finite set B ⊆ MK such that M = L∗(B). It is easy to see thatMK = K(B). Let us denote this σ-field by F . Then the set A is σ-algebraicallyindependent over F and we obtain σ-field extensions M/L∗, L∗/K, and F/Ksuch that M/L∗ is monadic, M = L∗〈F 〉, and L∗ is the inversive closure ofK〈A〉 where the set A is σ-algebraically independent over F .

On the other hand, if we have a monadic σ-field extension F/K with F,M,K,and L∗ satisfying assumptions of part (ii) of the theorem, then the field extensionF/K is algebraic, hence the set A is σ-algebraically independent over F , L∗ isthe inversive closure of K〈A〉, and M = L∗〈F 〉.

Thus, both parts of the theorem will be proven if we show that if F/K isa σ-field extension, a set A is σ-algebraically independent over F , and L∗ isthe inversive closure of K〈A〉, then L∗〈F 〉/L∗ is monadic if and only if F/K is

5.6. MONADICITY 369

monadic. But this is obvious, since any σ-K-isomorphism of F naturally extendsto a σ-L∗-isomorphism of L∗〈F 〉, so if two σ-L∗-isomorphisms of L∗〈F 〉 coincideon F , they coincide on A and therefore they are equal.

The following result gives a characterization of finite difference field exten-sions of an ordinary difference (σ-) field K which are compatible with everyother σ-field extension of K. In section 8.1 we will return to the study of such“universally compatible” extensions in more general case.

Theorem 5.6.41 Let K be an inversive ordinary difference field with a basicset σ = α and let L be a σ-overfield of K such that the field extension L/Kis finite. Then L/K is compatible with every difference field extension of K ifand only if L/K is monadic.

PROOF. First of all, without loss of generality we may assume that K isinversive. Furthermore, it follows from Theorems 5.1.6 and 5.6.33 that one maysuppose that the extension L/K is separable. Let N be the normal closure ofL over K. Then we may consider N as a finitely generate separably algebraicnormal σ-field extension of K of limit degree 1.

By Proposition 5.6.24(ii), the extension L/K is compatible with every differ-ence field extension of K if and only if L/K is compatible with every extensioncoordinate with N/K.

Let G = Gal(N/K) and let H = Gal(N/L). By Theorem 5.6.31, there is adifference isomorphism of L/K into every extension coordinate with N/K if andonly if for every φ ∈ G there exists γ ∈ G such that α(γ) ∈ φγH. Let us associatewith every coset A = θH of H in G (θ ∈ G) the set A′ = α(θ)Hθ−1. Note, thatH is a σ-stable subgroup of G (see Proposition 5.6.11), so A′ does not depend onthe choice of a representative of the coset A. Furthermore, CardA′ = CardHand if β ∈ A, then α(β) ∈ φβH if and only if φ ∈ A′. Therefore, the conditionthat there is a difference isomorphism of L/K into every extension coordinatewith N/K is equivalent to the condition G =

⋃A′ |A is a coset of H in G.

Since the group G is finite and CardA′ = CardH the last condition isequivalent to the requirement that all sets A′ are disjoint, which, obviously,equivalent to the following condition:

(∗) For any elements θ, β ∈ G, the inclusions α(θ) ∈ φθH and α(β) ∈ φβHimply β ∈ θH.

Let us show that the last condition is equivalent to the condition of Theorem5.6.31(iii): L/K is monadic if and only if for any θ ∈ G, the inclusion α(θ) ∈HθH implies θ ∈ H. Indeed, suppose that the condition of Theorem 5.6.31(iii)holds and let α(θ) = φθγ1, α(β) = φβγ2 for some γ1, γ2 ∈ H. Then α(θ−1β) =γ−11 θ−1βγ2, hence θ−1β ∈ H hence β ∈ θH.

Conversely, suppose that condition (∗) holds. Let α(θ) = γ1θγ2 for someγ1, γ2 ∈ H. Then α(θ) ∈ γ1θH. Since α(e) ∈ γ1eH, where e is the identity ofthe group G, we have θ ∈ eH = H. This completes the proof.

Theorem 5.6.42 Any subextension of a finitely generated monadic ordinarydifference field extension is monadic.


PROOF. Let K be an ordinary difference field with a basic set σ, let Lbe a σ-overfield of K such that the difference field extension L/K is finitelygenerated and monadic, and let F be an intermediate difference (σ-) field ofL/K. Because of the results of Proposition 5.6.4(i) and Theorem 5.6.33, whileproving that the extension F/L is also monadic, we can assume that K is in-versive and the extension L/K is separable. With these assumptions, it followsfrom Theorem 5.6.35 that L is a finite extension of the field K. By Theorem5.6.41, the extension L/K is compatible with every difference field extension ofK. Then F/K is also compatible with every σ-field extension of K. ApplyingTheorem 5.6.41 once again we obtain that F/K is monadic.

Note that the condition that L/K is finitely generated is essential in the lasttheorem. Furthermore, there are finitely generated monadic ordinary differencefield extensions which are not subextensions of normal monadic extensions. Thecorresponding examples are due to R. Cohn (see [41, Chapter 9, Example 5]).

Exercise 5.6.43 Let K be an algebraically closed ordinary difference field.Prove that any two difference field extensions of K are compatible, and K hasno properly monadic extension.

Exercise 5.6.44 Let K be an inversive ordinary difference field with a basicset σ. Prove that the following two conditions are equivalent.

(i) K admits incompatible σ-field extensions or K has a finitely generatedproperly monadic extension.

(ii) There is a σ-field extension L/K, L = K, of finite separable degree.[Hint: Use Theorems 5.4.22 and 5.6.35.]

Exercise 5.6.45 Let K be an ordinary difference field which is not algebraicallyclosed. Prove that if every element of K is periodic, then K admits either in-compatible extensions or a properly monadic extension.

Exercise 5.6.46 Let K be an inversive ordinary difference field with a basicset σ and let L = K〈S〉 where the set S is σ-algebraically independent over K.Furthermore, let L∗ denote the inversive closure of L. Prove that

(i) The σ-field K admits incompatible extensions if and only if L∗ admitssuch extensions.

(ii) There is a finitely generated separable properly monadic extension of Kif and only if there is such an extension of L∗.

Chapter 6

Difference Kernels overPartial Difference Fields.Difference Valuation Rings

6.1 Difference Kernels over Partial DifferenceFields and their Prolongations

In this section we extend the study of difference kernels presented in Section5.2 to the case of partial difference fields. The corresponding theory is due toI. Bentsen [10].

Let K be an inversive difference field with a basic set σ = α1, . . . , αn. Letus fix an index i ∈ 1, . . . , n and let K(i) denote the field K treated as aninversive difference field with a basic set σ \ αi denoted by σi. Then αi canbe viewed as a σi-automorphism of K(i). (Also, in accordance with our basicconventions, K(i)〈S〉 and K(i)〈S〉∗ will denote, respectively, the σi- and σ∗

i - fieldextension of K(i) generated by a set S.)

Definition 6.1.1 With the above notation (and a fixed index i, 1 ≤ i ≤ n), adifference (σ-) kernel R over K is an ordered pair (K(i)〈a0, . . . , ar〉, τ) whereK(i)〈a0, . . . , ar〉 is a σi-field extension of K(i) generated by s-tuples a0, . . . , ar

(s ≥ 1 and each aj is an s-tuple (a(1)j , . . . , a

(s)j ) with coordinates in some

σi-overfield of K) and τ is an extension of αi to a σi-isomorphism of the fieldK(i)〈a0, . . . , ar−1〉 onto K(i)〈a1, . . . , ar〉 such that τ(a(k)

j ) = a(k)j+1 ( 0 ≤ j ≤ r−1,

1 ≤ k ≤ s). The number r is said to be the length of the kernel R. (If r = 0, wemean that τ is the σi-automorphism αi of K(i).)

Note that if the kernel R is of length 0 or 1, then the expressionsK(i)〈a0, . . . , ar−1〉 and K(i)〈a1, . . . , ar−1〉, respectively, are interpreted as K(i).

371

372 CHAPTER 6. DIFFERENCE KERNELS AND VALUATION RINGS

Definition 6.1.2 With the notation of Definition 6.1.1, a prolongation of adifference kernel R over K is a σ-kernel R′ consisting of a σi-overfieldK(i)〈a0, . . . , ar, ar+1〉 of K(i)〈a0, . . . , ar〉 (ar+1 is an s-tuple (a(1)

j , . . . , a(s)j ) over

K(i)) together with an extension τ ′ of τ such that τ ′(a(k)r ) = a

(k)r+1, 1 ≤ k ≤ s. A

prolongation R′ of R is called generic if the inversive closure K(i)〈a0, . . . , ar+1〉∗of the σi-field K(i)〈a0, . . . , ar+1〉 is a free join of the σ∗

i -fields K(i)〈a0, . . . , ar〉∗and K(i)〈a1, . . . , ar+1〉∗ over K(i)〈a1, . . . , ar〉∗.

Notice that the definition of a generic prolongation is not ambiguous (seePropositions 2.1.7 and 2.1.8(ii)). Furthermore, if R′ is a prolongation of R suchthat K(i)〈a0, . . . , ar〉 and K(i)〈a1, . . . , ar+1〉 are algebraically disjoint over thefield K(i)〈a1, . . . , ar〉, then statements (ii) and (iv) of Proposition 2.1.8 implythat R′ is a generic prolongation of R.

Definition 6.1.3 With the above notation, we say that a difference kernelR satisfies property P if there exists a σi-overfield L1 of K(i)〈a0, . . . , ar〉, aσi-subfield L of L1 which contains K(i)〈a0, . . . , ar−1〉, and an extension of τ toa σi-isomorphism τ of L into L1 such that

(i) The field extension L1/L is primary.(ii) L and K(i)〈a0, . . . , ar〉 are algebraically disjoint over K(i)〈a0, . . . , ar−1〉.(iii) If r > 0, then L1 = L〈τ(L)〉; if r = 0, then τ(L) ⊆ L.

Definition 6.1.4 With the above notation, a kernel R is said to satisfy prop-erty P∗ (with respect to (M,M1, τ) ) if there exists an inversive σi-overfieldM1 of K(i)〈a0, . . . , ar〉, an inversive σi-subfield M of M1 which containsK(i)〈a0, . . . , ar−1〉, and an extension of τ to a σi-isomorphism τ of M intoM1 such that

(i) The field extension M1/M is primary.(ii) M and K(i)〈a0, . . . , ar〉∗ are algebraically disjoint over K(i)〈a0, . . . , ar−1〉∗.(iii) If r > 0, then M1 = M〈τ(M)〉; if r = 0, then τ(M) ⊆ M .

Proposition 6.1.5 The properties P and P∗ are equivalent.

PROOF. Let L, L1 and τ be as in Definition 6.1.3 of property P, let M1 = L∗1

(the inversive closure of L1), and let M be the inversive closure of L in M1. ByProposition 2.1.8 (parts (ii) and (iii) ), τ extends uniquely to a σi-isomorphismτ of M into M1 such that if the length r of R is positive, then M1 = M〈τ(M)〉,and if r = 0, then τ(M) ⊆ M by the monotonicity of the operation ∗. ApplyingProposition 2.1.8 (parts (ii) and (iv) ) we obtain that P implies P∗.

Now suppose that M , M1 and τ are as in Definition 6.1.4 of property P∗.Let L denote the separable part of M over K(i)〈a0, . . . , ar−1〉. Then L is a σi-overfield of K(i)〈a0, . . . , ar−1〉 such that L and K(i)〈a0, . . . , ar〉 are algebraicallydisjoint over K(i)〈a0, . . . , ar−1〉.

Let τ be the contraction of τ to a σi-isomorphism of L into M1. If r = 0then for any b ∈ L, the element τ(b) is separably algebraic over τ(K(i)) = K(i)

6.1. DIFFERENCE KERNELS AND THEIR PROLONGATIONS 373

and τ(b) ∈ M . Therefore, τ(L) ⊆ L. Let us set L1 = L〈Ki〈a0〉〉 if r = 0 andL1 = L〈τ(L)〉 if r > 0. In either case L1 is a σi-overfield of K(i)〈a0, . . . , ar−1〉and the extensions M/L and M1/M are primary. By Theorem 1.6.48, M1/Land L1/L are primary as well, so that P∗ implies P.

Definition 6.1.6 With the above notation, we say that a difference kernelR satisfies property L∗ if it satisfies property P∗ strengthen by replacing therequirement that the fields L and K(i)〈a0, . . . , ar〉 are algebraically disjoint overK(i)〈a0, . . . , ar−1〉 by the requirement that these fields are quasi-linearly disjointover K(i)〈a0, . . . , ar−1〉.

Proposition 6.1.7 A difference kernel R = (K(i)〈a0, . . . , ar〉, τ) satisfies prop-erty L∗ if and only if the field extension K(i)〈a0, . . . , ar〉∗/K(i)〈a0, . . . , ar−1〉∗is primary.

PROOF. If the extension K(i)〈a0, . . . , ar〉∗/K(i)〈a0, . . . , ar−1〉∗ is primary,one can set M1 = K(i)〈a0, . . . , ar〉∗, M = K(i)〈a0, . . . , ar−1〉∗ and apply parts(ii) and (iii) of Proposition 2.1.8 to obtain that R satisfies property L∗. Con-versely, if the kernel R satisfies L∗, then Corollary 1.6.50 immediately impliesthat the extension K(i)〈a0, . . . , ar〉∗/K(i)〈a0, . . . , ar−1〉∗ is primary.

Exercise 6.1.8 With the above notation, prove that if the extensionK(i)〈a0, . . . , ar〉∗/K(i)〈a0, . . . , ar−1〉∗ isprimary, theextensionK(i)〈a0, . . . , ar〉∗/K(i)〈a0, . . . , ar−1〉∗ is also primary.

Show that the converse statement also holds if the fields K(i)〈a0, . . . , ar−1〉∗and K(i)〈a0, . . . , ar〉 are quasi-linearly disjoint over K(i)〈a0, . . . , ar−1〉.

The following example presents a kernel which does not satisfy P∗.

Example 6.1.9 As in Example 5.1.17, let us consider Q as a difference fieldwith a basic set σ = α1, α2 where α1 and α2 are the identity automorphisms.Let b and i denote the positive square root of 2 and the square root of −1,respectively (b, i ∈ C), and let F = Q(b, i). We consider F as a σ-overfield of Qwhere the extensions of α1 and α2 to automorphisms of F (denoted by the sameletters) are defined by their actions on b and i as follows: α1(b) = −b, α1(i) =i, α2(b) = b, and α2(i) = −i.

Let F (2) denote the field F treated as an ordinary difference field with thebasic set σ2 = α1 (we keep the notation of Definition 6.1.1). By Corollary5.1.16, there exists an algebraic closure G of the field F that has a structureof a σ2-overfield of F (2). By Proposition 2.1.9(ii), the σ2-field G is inversive.Furthermore, there exists an element a ∈ G \ F such that a2 = b.

Let us consider the difference (σ2-) kernel R = (F (2)〈a〉, τ) where τ = α2

(this mapping is considered as a σ2-automorphism of F (2)). Suppose that thiskernel satisfies P∗. Then there exist inversive σ2-overfields M1 and M of F (2)〈a〉and F (2), respectively, such that the field extension M1/M is primary. It followsthat a ∈ M and there exists a σ2-isomorphism of M into M which coincideswith α2 on F (2). Denoting this isomorphism by α2, we can consider M as a


σ-overfield of F which contains a. On the other hand, Example 5.1.17 showsthat a is not contained in any σ-field extension of F . This contradiction impliesthat the kernel R does not satisfy P∗.

Theorem 6.1.10 (i) If a difference kernel R satisfies P∗, then R has a genericprolongation which satisfies P∗.

(ii) If a generic prolongation R′ of a difference kernel R satisfies P∗, thenthere exists a triple (M,M1, τ) with respect to which R satisfies P∗ and troughwhich a generic prolongation R′′ of R can be obtained such that R′′ is equivalentto R′ in the sense of isomorphism.(iii) If a difference kernel satisfies L∗, then all its generic prolongations are

equivalent and satisfy L∗.

PROOF. Suppose that a difference (σ-) kernel R = (K(i)〈a0, . . . , ar〉, τ) sat-isfies P∗ with respect to the triple (M,M1, τ) (see Definition 6.1.4).By Theorem5.1.8, (with (M,M1, τ) corresponding to (K,L, φ) in Theorem 5.1.8), there ex-ists an inversive σ-overfield M2 of M1 and an extension τ1 of τ such that M2 =M1〈τ1(M1〉 and the field extension M2/M1 is primary. Setting ar+1 = τ1(ar)and denoting by τ ′ the contraction of τ1 to a σi-isomorphism of K(i)〈a0, . . . , ar〉to K(i)〈a1, . . . , ar+1〉, we obtain the prolongation R′ = (K(i)〈a0, . . . , ar+1〉, τ ′)of the kernel R.

Since the fields M and K(i)〈a0, . . . , ar〉∗ are algebraically disjoint overK(i)〈a0, . . . , ar−1〉∗, τ1(M) and K(i)〈a1, . . . , ar+1〉∗ are algebraically disjointover K(i)〈a1, . . . , ar〉∗. Also, the fact that M1 and τ1(M1) are quasi-linearlydisjoint, hence algebraically disjoint, over τ1(M) implies that the fields M1

and τ1(M)〈ar+1〉∗ are algebraically disjoint over τ1(M). Applying Theorem1.6.44(ii) we obtain that K(i)〈a0, . . . , ar〉∗ and K(i)〈a1, . . . , ar+1〉∗ are alge-braically disjoint over K(i)〈a1, . . . , ar〉∗ and also M1 and K(i)〈a0, . . . , ar+1〉∗are algebraically disjoint over K(i)〈a0, . . . , ar〉∗. Therefore, R′ is a genericprolongation of R which satisfies P∗.

Now suppose that R satisfies L∗. Then by Propositions 2.1.8 (parts (ii) and(iii)) and 6.1.7, R satisfies P∗ with M and M1 (see Definition 6.1.4) denotingK(i)〈a0, . . . , ar−1〉∗ and K(i)〈a0, . . . , ar〉∗, respectively. Applying parts (ii) and(iii) of Proposition 2.1.8 once again we obtain (with the notation of the firstpart of the proof) that K(i)〈a0, . . . , ar+1〉∗ = M2. It follows that R′ is a genericprolongation of R which satisfies L∗.

Suppose that R1 = (K(i)〈a0, . . . , ar, b〉, θ) be another generic prolonga-tion of R. then the inversive closure N of K(i)〈a0, . . . , ar, b〉 is the free joinof K(i)〈a0, . . . , ar〉∗1 and K(i)〈a1, . . . , ar, b〉∗1 over K(i)〈a1, . . . , ar〉∗1 where∗1 denotes the inversive closure in N . Furthermore, by Proposition 2.1.7,K(i)〈a0, . . . , ar〉∗1 may be identified with K(i)〈a0, . . . , ar〉∗ and hence N maybe considered as a σ∗

i -overfield of K(i)〈a0, . . . , ar〉∗. By Proposition 2.1.8 (parts(ii) and (iii) ), θ can be extended to an isomorphism θ of K(i)〈a0, . . . , ar〉∗1 ontoK(i)〈a1, . . . , ar, b〉∗1 such that τ1 and θ coincide on K(i)〈a0, . . . , ar−1〉∗. Usingthe arguments of the preceding paragraph and Theorem 5.1.8 we obtain thatthere exists a σ1-M1-isomorphism ψ of M2 onto N such that ψτ1(x) = θψ(x)

6.1. DIFFERENCE KERNELS AND THEIR PROLONGATIONS 375

for any x ∈ M1. Therefore, the contraction of ψ to K(i)〈a0, . . . , ar+1〉 gives theequivalence (in the sense of isomorphism) of the generic prolongations of R.

It remains to prove statement (ii). Let R′ = (K(i)〈a0, . . . , ar+1〉, τ ′) bea generic prolongation of R which satisfies property P∗ with respect to atriple (G1, G2, τ

′1). Let M be the separable part of G1 over K(i)〈a0, . . . , ar−1〉∗.

Then τ ′1(M) ⊆ G1. Indeed, if x ∈ τ ′

1(M), then x is separably algebraic overK(i)〈a1, . . . , ar〉∗ and hence over G1. Since the extension G2/G1 is primary,x ∈ G1. Furthermore, if r = 0, then x is separably algebraic over K henceτ ′1(M) ⊆ M .

Let M1 denote the separable part of G1 over K(i)〈a0, . . . , ar〉∗ if r = 0 andlet M1 = M〈τ ′

1(M)〉 if r > 0. Then K(i)〈a0, . . . , ar〉∗ ⊆ M1 and M ⊆ M1 ⊆ G1.Denoting the restriction of τ ′

1 on M by τ2, one can easily check that R satisfiesproperty P∗ with respect to (M,M1, τ2).

Let M2 = M1〈τ ′1(M1). Then K(i)〈a0, . . . , ar+1〉∗ ⊆ M2 ⊆ G2 and the σi-field

M2 is inversive. Since the field M1 is algebraic over K(i)〈a0, . . . , ar〉∗, the fieldsM1 and K(i)〈a0, . . . , ar+1〉∗ are algebraically disjoint over K(i)〈a0, . . . , ar〉∗.Furthermore, since R′ is a generic prolongation of R, Theorem 1.6.44(ii) im-plies that the fields τ ′

1(M) and K(i)〈a1, . . . , ar+1〉∗ are algebraically disjointover K(i)〈a1, . . . , ar〉∗ and also M1 and K(i)〈a1, . . . , ar+1〉∗〈τ ′

1(M)〉 are alge-braically disjoint over τ ′

1(M). Using this fact, the algebraic disjointness ofK(i)〈a1, . . . , ar+1〉∗〈M1〉 and τ ′

1(M1) over K(i)〈a1, . . . , ar+1〉∗〈τ ′1(M)〉 (which

follows from the fact that τ ′1(M) is algebraic over K(i)〈a0, . . . , ar+1〉∗ and hence

over the field K(i)〈a0, . . . , ar+1〉∗〈τ ′1(M)〉), and Theorem 1.6.44(i), we obtain

that the fields M1 and τ ′1(M) are algebraically disjoint over K(i)〈a0, . . . , ar〉∗.

This completes the proof of the theorem.

Definition 6.1.11 With the notation introduced at the beginning of this sec-tion, a difference kernel R = (K(i)〈a0, . . . , ar〉, τ) is said to satisfy the stepwisecompatibility condition if the σi-field K(i)〈a0, . . . , ar〉 satisfies the stepwise com-patibility condition (see Definition 5.1.13).

Theorem 6.1.12 If a difference kernel R satisfies the stepwise compatibilitycondition, it also satisfies property P∗.

PROOF. Suppose that a kernel R = (K(i)〈a0, . . . , ar〉, τ) satisfies the step-wise compatibility condition. Then the σi-field K(i)〈a0, . . . , ar−1〉∗ also satisfiesthis condition. Furthermore, by Theorem 5.1.15 and Proposition 2.1.9(ii), thereexists an algebraic closure L of K(i)〈a0, . . . , ar−1〉∗ which has a structure of aσ∗

i -overfield of K(i)〈a0, . . . , ar−1〉∗. Since any two σi-field extensions of the lastfield are compatible, we may assume that L and K(i)〈a0, . . . , ar〉∗ have a com-mon σi-overfield M . Furthermore, by Theorem 5.1.9, τ has an extension to anisomorphism τ of L into a σi-overfield N of M . If r = 0, let L1 = K(i)〈a0〉∗〈L〉,and if r ≥ 1, let L1 = L〈τ(L)〉. It follows from the definition of L that the fieldsL and K(i)〈a0, . . . , ar〉∗ are algebraically disjoint over K(i)〈a0, . . . , ar−1〉∗ andthat the field extension L1/L is primary. Thus, R satisfies property P∗.


Note that the converse statement for the last theorem is not true. The fol-lowing example, given in [10], introduces a kernel which satisfies L∗ and doesnot satisfy the stepwise compatibility condition.

Example 6.1.13 Let F be the inversive difference field Q with the basic setσ = α1, α2 described in Example 6.1.9 and let F (2) denote the field F treatedas an ordinary difference field with the basic set σ2 = α1. Let x be any elementtranscendental over F (x belongs to some overfield of F (2)). Then α1 can beextended to an automorphism of F (x) that maps x onto x. This extension (alsodenoted by α1) defines a structure of σ2-overfield of F (2) on F (x); we denotethis σ2-overfield by F (2)〈x〉. Let us consider the kernel R = (F (2)〈x〉, α2). Sincethe kernel in Example 6.1.9 does not satisfy P∗, it follows from Theorem 6.1.12that F (2) does not satisfy the stepwise compatibility condition. On the otherhand the field extension F (x)/F is primary, hence R satisfies L∗ (it follows fromPropositions 6.1.7 and 2.1.9(iii)).

6.2 Realizations of Difference Kernelsover Partial Difference Fields

Let K be an inversive difference field with a basic set σ = α1, . . . , αn, let an in-dex i ∈ 1, . . . , n be fixed, and let K(i) denote the field K treated as a differencefield with the basic set σi = σ \αi. Furthermore, let R = (K(i)〈a0, . . . , ar〉, τ)be a difference kernel over K where a0, . . . , ar are s-tuples (s ∈ N+).

Let η = (η(1), . . . , η(s)) be an s-tuple in a σ-overfield L of K and let θ denotethe extension of αi from K to L (usually we denote such an extension by thesame symbol αi, but in this case it is convenient to use a special symbol). In whatfollows we set η0 = η and ηj = θj(η) (= (θj(η(1)), . . . , θj(η(s))) for j = 1, 2, . . . .

Definition 6.2.1 With the above notation, we say that η is a realization ofthe kernel R in L over K if the set (η0, . . . , ηr) is a specialization over K(i)

of (a0, . . . , ar) with a(k)j → η

(k)j (1 ≤ k ≤ s, 0 ≤ j ≤ r). We also say that θ

is a realization of τ in L over K. If (η0, . . . , ηr) is a generic specialization of(a0, . . . , ar), η is said to be a regular realization of the kernel R.

If there exists a sequence of kernels R0, R1, R2, . . . such that R0 = R,Rj+1 is a generic prolongation of Rj, and η is a regular realization of Rj overK (j = 0, 1, . . . ), then η is called a principal realization of R.

Definition 6.2.2 Two realizations η and ζ of a kernel R over a difference (σ-)field K are called equivalent if there is a σ-K-isomorphism φ : K〈η〉 → K〈ζ〉such that φ(η(k)) = ζ(k) for k = 1, . . . , s.

Remark 6.2.3 Let K be an inversive difference field with a basic set σ =α1, . . . , αn, let ζ be an s-tuple with coordinates in some σ-overfield L of K, andlet θ denote the extension of some αi (1 ≤ i ≤ n) to an injective endomorphism ofL (in most cases this extension is denoted by the same symbol αi). Furthermore,

6.2. REALIZATIONS 377

let K(i) denote the field K treated as a difference field with the basic set σi =σ \ αi and let θ′ denote the σi-isomorphism of K(i)〈ζ, θ(ζ), . . . , θs−1(ζ)〉 ontoK(i)〈θ(ζ), θ2(ζ), . . . , θs(ζ)〉. Then it is easy to see that ζ is a regular realizationof the kernel (K(i)〈ζ, θ(ζ), . . . , θs(ζ)〉, θ′) in L over K.

Theorem 6.2.4 Let K be an inversive difference field with a basic set σ =α1, . . . , αn, let an index i ∈ 1, . . . , n be fixed, and let K(i) denote the fieldK treated as a difference field with the basic set σi = σ \ αi. Furthermore,let R = (K(i)〈a0, . . . , ar〉, τ) be a difference kernel over K where a0, . . . , ar ares-tuples (s ∈ N+). Then the following conditions are equivalent.

(i) R satisfies P∗.(ii) R has a principal realization.(iii) R has a regular realization.

PROOF. Suppose that R satisfies P∗. By Theorem 6.1.10, there exists a se-quence of kernels Rj = (K(i)〈a0, . . . , ar+j〉, τj) (j = 0, 1 . . . ,) such that R0 = Rand for every j ∈ N, Rj+1 is a generic prolongation of Rj which satisfies P∗.

Then the field L =∞⋃

j=0

K(i)〈a0, . . . , ar+j〉, together with the union of the iso-

morphisms τj (j ≥ 0) as an extension of αi, becomes a σ-overfield of K inwhich a0 is a principal realization of R over K. Thus, (i) implies (ii). Since theimplication (ii)⇒(iii) is obvious, it remains to show that (iii) implies (i).

Let η be a regular realization of R and let L be the inversive closure ofK〈η〉. Let θ denote the translation of L which is a realization of τ and letLθ denote the difference field obtained from L by deletion of the translationθ from the basic set of L (the basic set of Lθ will be denoted by σθ). Clearly,the σθ-field Lθ is inversive and θ is a σθ-isomorphism of this field into itself.Let S denote the separable part of Lθ over K〈η0, . . . , ηr−1〉∗ (as before, we setηj = θj(η) for j = 0, 1, . . . ,). If r = 0, let S1 denote the separable part of Lθ

over K〈η0, . . . , ηr〉∗, and if r ≥ 1, let S1 denote the compositum S〈θ(S)〉 in Lθ.Then S and K〈η0, . . . , ηr〉∗ are algebraically disjoint over K〈η0, . . . , ηr−1〉∗, andthe extension Lθ/S, and hence S1/S, is primary. By Proposition 2.1.8 (parts(i) and (v)) and by the uniqueness of the separable part of a field extension,S is inversive, and for r = 0, S1 is inversive. If r ≥ 1, then parts (ii) and (iii)of Proposition 2.1.8 imply that S1 = S〈θ(S)〉 is inversive. If θ0 denotes therestriction of θ to a σθ-isomorphism of S into Lθ, then S1 = S〈θ0(S)〉 if r ≥ 1,and θ0(S) ⊆ S for r = 0.

Since the specialization over K of (a0, . . . , ar) onto (η0, . . . , ηr) is generic,it extends uniquely to a σi-isomorphism of K(i)〈a0, . . . , ar〉 onto K〈η0, . . . , ηr〉which, in turn, extends to a σi-isomorphism of K(i)〈a0, . . . , ar〉∗ onto the fieldK〈η0, . . . , ηr〉∗ and then to a difference isomorphism φ of a difference overfieldN of K(i)〈a0, . . . , ar〉∗ onto S1. If M is a difference subfield of N whose imageunder φ is S, then τ extends uniquely to a difference isomorphism τ of M intoN such that φτ = θ0φ on M . It follows that the kernel R satisfies P∗ withrespect to (M,N, τ ).


Remark 6.2.5 The last part of the proof of the theorem and Theorem 6.1.10(ii)imply that every principal realization of a difference kernel is equivalent to oneconstructed as in the proof of the implication (i) ⇒ (ii) in Theorem 6.2.4.

Corollary 6.2.6 With the notation of the theorem, suppose that a kernelR = (K(i)〈a0, . . . , ar〉, τ) satisfies L∗. Then R has a principal realization overK and all principal realizations of R are equivalent.

PROOF. Since R satisfies L∗, this kernel also satisfies P∗ and hence hasa principal realization. Let η and ζ be two principal realizations of R. LetR0 = R, R1, R2, . . . and R′

0 = R, R′1, R′

2, . . . be the sequences of kernelsassociated with η and ζ, respectively, in the sense of the definition of principalrealization (Definition 6.2.1). Since R satisfies L∗, one can apply Theorem 6.1.10and induction on j and obtain that the kernels Rj and R′

j are equivalent (inthe sense of isomorphism) for every j ∈ N. Since for every j ∈ N, η is aregular realization of Rj and ζ is a regular realization of R′

j , there exists (bya composition of maps) a σi-K(i)-isomorphism φj of K(i)〈η0, . . . , ηr+j〉 ontoK(i)〈ζ0, . . . , ζr+j〉 such that η

(k)l → ζ

(k)l (0 ≤ l ≤ r + j, 1 ≤ k ≤ s). Clearly, the

union of all φj defines a σ-isomorphism of K〈η〉 onto K〈ζ〉 with η → ζ. We need the following two lemmas to prove an important result on principal

realizations (see Theorem 6.2.9 below).

Lemma 6.2.7 Let L and M be fields, let R1 and R2 be two subrings of L, andlet R3 be a subring of R1 ∩ R2. Let R = R1[R2] (the smallest subring of Lcontaining R1 and R2) and let φ be a homomorphism of R into M such that thecontractions of φ to R1 and R2 are isomorphisms. Furthermore, suppose thatthe quotient fields in M of φ(R1) and φ(R2) are algebraically disjoint over thequotient field of φ(R3) in M . Then φ is an isomorphism of R into M .

PROOF. For any subring A of L or M , let A denote the quotient field of A inL or M , respectively. Let R2 = R3[S] and let B be a maximal algebraically inde-pendent over R1 subset of S. Since R3 ⊆ R1, B is also algebraically independentover R3, hence φ(B) is algebraically independent over φ(R3) and therefore overφ(R3). Since φ(R1) and φ(R2) are algebraically disjoint over φ(R3), φ(B) is al-gebraically independent over φ(R1). It follows that no nonzero element of R1[B]is mapped to zero under φ, hence φ can be extended to a homomorphism φ′ ofR1(B)[S] onto φ(R1)(φ(B))[φ(S)]. Since elements of S are algebraic over R1[B],R1(B)[S] is a field that can be identified with R1(S) = R (see Theorem 1.6.4).It follows that φ′ is an isomorphism of R into M hence φ is an isomorphism aswell.

Lemma 6.2.8 With the notation introduced in Theorem 6.2.4, let R1 =(K(i)〈a0, . . . , ar, ar+1〉, τ1) and R′

1 = (K(i)〈b0, . . . , br, br+1〉, τ ′1) (r ≥ 0) be ker-

nels such that R1 is a generic prolongation of a kernel R = (K(i)〈a0, . . . , ar〉, τ)and there exists a specialization φ of (b0, . . . , br, br+1) onto (a0, . . . , ar, ar+1)over K(i). Furthermore, suppose that φ contracts to a generic specialization φ1

of (b0, . . . , br) onto (a0, . . . , ar) over K(i). Then the specialization φ is generic.


PROOF. It is easy to see that φ1 extends to a σi-isomorphism of theσi-field K(i)〈b0, . . . , br〉 onto K(i)〈a0, . . . , ar〉. Because of the isomorphismsτ1 and τ ′

1, the contraction of φ to φ2 : (b1, . . . , br+1) → (a1, . . . , ar+1) is anisomorphism of K(i)b1, . . . , br+1 onto K(i)a1, . . . , ar+1. By Proposition2.1.7(v), φ extends uniquely to a σi-homomorphism of K(i)〈b0, . . . , br+1〉∗ ontoK(i)〈a0, . . . , ar+1〉∗, and φ′ contracts to a σi-isomorphism of K(i)〈b0, . . . , br〉∗onto K(i)〈a0, . . . , ar〉∗ which is an extension of φ1. Also, φ′ contracts to aσi-isomorphism of K(i)〈b1, . . . , br+1〉∗ onto K(i)〈a1, . . . , ar+1〉∗ which is an ex-tension of φ2, and φ′ maps K(i)〈b1, . . . , br〉∗ onto K(i)〈a1, . . . , ar〉∗. Since R1

is a generic prolongation of R, Lemma 6.2.7 and the remark after the proof ofProposition 2.1.11 show that φ′ is an isomorphism. Thus, the specialization φis generic.

Theorem 6.2.9 With the notation of Theorem 6.2.4, if η is a principal realiza-tion of a kernel R = (K(i)〈a0, . . . , ar〉, τ), then η is not a proper specializationover K of any other realization of R.

PROOF. Suppose that η is a principal realization of R and ζ a realization ofR such that there exists a specialization φ over K of ζ onto η with ζ(k) → η(k)

(1 ≤ k ≤ s). Then φ is generic if for every j ∈ N, the contraction of φ toφj : K(i)ζ0, . . . , ζj → K(i)η0, . . . , ηj (ζl → ηl for l = 0, 1, . . . , j) is a σi-isomorphism.

Notice that the composition of the specialization over K(i) of (a0, . . . , ar)onto (ζ0, . . . , ζr) with the mapping φr agrees on K(i)a0, . . . , ar with the as-sumed generic specialization over K(i) of (a0, . . . , ar) → (η0, . . . , ηr) . Therefore,φr is a σi-isomorphism. Suppose that φr+k is a σi-isomorphism for some k ≥ 0.By Lemma 6.2.8, φr+k+1 is also a σi-isomorphism, so one can apply inductionon j and obtain that all φj (j ∈ N) are σi-isomorphisms. This completes theproof.

Exercise 6.2.10 Let K be an inversive difference field with a basic set σ andlet η be an s-tuple with coordinates in some σ-overfield of K (s ≥ 1). Prove thatthere exists a kernel R over K such that η is the unique principal realization ofR over K, and every realization of R over K is a specialization of η over K.[Hint: Generalize the arguments of the proof of Proposition 5.2.11.]

Let K be a difference field with a basic set σ and let a = a(i) | i ∈ I bean indexing of elements in some σ-overfield of K. We say that this indexing isσ-algebraically independent over K if and only if all a(i) (i ∈ I) are distinctand form a σ-algebraically independent set over K (see the definition at thebeginning of section 4.1).

In what follows we keep our previous notation: for any difference kernelR = (K(i)〈a0, . . . , ar〉, τ) (K is an inversive difference field with a basic setσ = α1, . . . , αn and an index i ∈ 1, . . . , n is fixed) and for any subindexingb0 = (a(j1)

0 , . . . , a(jk)0 ) of a0, the corresponding subindexing (a(j1)

l , . . . , a(jk)l ) of

al (1 ≤ l ≤ r) will be denoted by bl.


Lemma 6.2.11 Let R = (K(i)〈a0, . . . , ar〉, τ) be a kernel (we use the aboveconventions about K and denote σ\αi by σi) and let R1 = (K(i)〈a0, . . . , ar+1〉,τ1) be a generic prolongation of R. Furthermore, let b0 be a subindexing of a0.

(i) If br is σi-algebraically independent over K(i)〈a0, . . . , ar−1〉, then br+1

is σi-algebraically independent over K(i)〈a0, . . . , ar〉 and the set⋃r+1

j=0 bj is σi-algebraically independent over K(i).

(ii) If b0 is σi-algebraically independent over K(i)〈a1, . . . , ar〉, then b0 is σi-algebraically independent over K(i)〈a1, . . . , ar+1〉 and the set

⋃r+1j=0 bj is σi-

algebraically independent over K(i).

PROOF. (i) Because of the σi-isomorphism τ1 and our assumption onb0, br+1 is σi-algebraically independent over K(i)〈a1, . . . , ar〉 and hence alsoover K(i)〈a1, . . . , ar〉∗. Since R1 is a generic prolongation of R, br+1 is σi-algebraically independent over K(i)〈a0, . . . , ar〉∗ and hence over K(i)〈a0, . . . , ar〉.

Let us show that for each j, 0 ≤ j ≤ r +1, bj is σi-algebraically independentover K(i)〈a0, . . . , aj−1〉. Indeed, our assumption and the result of the precedingparagraph show that this statement is true for j = r and j = r + 1. Supposethat j < r. Then br is σi-algebraically independent over K(i)〈ar−j , . . . , ar−1〉and the composition of the corresponding restrictions of τ is a σi-isomorphismof K(i)〈a0, . . . , aj〉 onto K(i)〈ar−j , . . . , ar〉. It follows that bj is σi-algebraicallyindependent over K(i)〈a0, . . . , aj−1〉. Also, we obtain that for every j, 0 ≤ j ≤r + 1, bj is σi-algebraically independent over K(i)〈b0, . . . , bj−1〉, hence

⋃r+1j=0 bj

is σi-algebraically independent over K(i).(ii) Since b0 is σi-algebraically independent over K(i)〈a1, . . . , ar〉, it is

σi-algebraically independent over K(i)〈a1, . . . , ar〉∗ and hence over the fieldK(i)〈a1, . . . , ar+1〉∗ (because the prolongation R1 of R is generic) and overK(i)〈a1, . . . , ar+1〉.

Let j be an integer, 0 ≤ j ≤ r such that⋃j

k=0 bk is σi-algebraically inde-pendent over K(i) (obviously, this is true for j = 0). Since the restriction ofτ1 to K(i) is an automorphism of K(i), the set

⋃j+1k=1 bk is also σi-algebraically

independent over K(i). At the same time, the previous paragraph shows thatb0 is σi-algebraically independent over K(i)〈b1, . . . , bj+1〉, hence

⋃r+1j=0 bj is σi-

algebraically independent over K(i).

Theorem 6.2.12 Let R = (K(i)〈a0, . . . , ar〉, τ) be a kernel (we use the no-tation and conventions of the preceding lemma), let η = (η(1), . . . , η(s)) be aprincipal realization of R, and let b0 be a subindexing of a0.

(i) If br is σi-algebraically independent over K(i)〈a0, . . . , ar−1〉, then the cor-responding subindexing ζ of η is σ-algebraically independent over K.

(ii) If b0 is σi-algebraically independent over K(i)〈a1, . . . , ar〉, then the corre-sponding subindexing ζ of η is σ-algebraically independent over K.

PROOF. Since η is a principal realization of R, for every j ∈ N there existsa kernel Rj = (K(i)〈a0, . . . , ar+j〉, τj) obtained from R by a sequence of generic


prolongations and such that η is a regular realization of Rj over K. Using part(i) of Lemma 6.2.11 and induction on j we obtain that the set

⋃r+jk=0 bk is σi-

algebraically independent over K(i) and hence⋃r+j

k=0 τk(ζ), where τ denotes therealization of τ , is also σi-algebraically independent over K(i). Since this is truefor every j ∈ N, ζ is σ-algebraically independent over K. The proof of part (ii)is similar; we leave it to the reader as an exercise.

We conclude this section with an important result on principal realizationsof ordinary difference kernels.

Let K be an inversive ordinary difference field with a basic set σ = α. LetR = (K(a0, . . . , ar), τ) be a difference kernel over K with a principal realizationη = (η1, . . . , ηs) (ai is an indexing (a(1)

i , . . . , a(s)i ) for i = 0, 1, . . . ). The following

example, which is due to R. Cohn [41, Chapter 10, Section 8], shows that if R =(K(a0, . . . , ar), τ) is a kernel which specializes to R, there may be no principalrealization of R which specializes to η. At the same time, Theorem 6.2.14 belowpresents one case where such a specialization of a principal realization does exist.

Example 6.2.13 Let K be an inversive ordinary difference field with a basicset σ = α and let a kernel R = (K(a, b), τ) over K be defined as follows: a andb are 4-tuples whose coordinates a(i) and b(i) (1 ≤ i ≤ 4) are chosen in such away that a(1), b(1), a(2), b(2), and a(3) form an algebraically independent set overK, b(3) = a(3)b(1)(a(2))−1, and a(4) = a(3)b(1)(a(1))−1. It is easy to see that thecoordinates of a, as well as coordinates of b, are algebraically independent overK, so the kernel R is well-defined (with a = a0 and b = a1, and if one uses thestandard notation for kernels, τ maps a(i) to b(i) for i = 1, . . . , 4 and τ |K = α).The realizations of R are solutions of the system

y1(αy4) = y2(αy3) = y3(αy1) = y4(αy2).

(The corresponding σ-polynomials belong to the σ-polynomial ring Ky1, y2, y3,y4.) Since trdegKK(a, b) = 5 and trdegKK(a) = 4, δR = 1. Furthermore,a(3) specializes to 0 over K(a(1), a(2), b(1), b(2)), and this specialization maps(a(4), b(3), b(4)) onto zero. Thus, (a(1), a(2), 0, 0; b(1), b(2), 0, 0) is a specializationof (a, b). It follows that η = (η(1), η(2), 0, 0), with η(1) and η(2) σ-algebraicallyindependent over K, is a realization of R. Since σ-trdegKK〈η〉 = 2 > δR, thisrealization is not principal (see Theorem 5.2.10).

The following result is due to B. Lando [117].

Theorem 6.2.14 Let K be an inversive ordinary difference field with a basic setσ = α. Let R = (K(a0, . . . , ar), τ) and R = (K(a0, . . . , ar), τ) be two kernelsover K (r ∈ N, ai = (a(1)

i , . . . , a(s)i ), and ai = (a(1)

i , . . . , a(s)i ) for i = 0, 1, . . . , r)

such that there is a K-isomorphism of K(a0, . . . , ar−1 onto K(a0, . . . , ar−1)(in this case we write K(a0, . . . , ar−1) ∼=K K(a0, . . . , ar−1)). Furthermore, letη = (η(1)

i , . . . , η(s)i ) be a principal realization of R. Then there exists a principal

realization η = (η(1)i , . . . , η

(s)i ) of R which specializes to η over K.


PROOF. Using Remark 5.2.12 one can assume that r = 0 or r = 1. Weshall give a proof with the assumption that r = 1 (the case r = 0 can betreated similarly). Since K(a0) ∼=K K(a0), we may assume that a0 = a0. Since(a1) specializes to (a1) over K(a0), δR ≥ δR. If δR = δR, then K(a0, a1) ∼=K

K(a0, a1), hence a principal realization of R specializes to η over K. Thus, itremains to consider the non-trivial case a0 = a0, δR = 0 and δR = 1.

Clearly, it is sufficient to show that for any m > 0, a kernel (K(a0, . . . , am), τ)canbe foundwith (K(a0, . . . , ak+1), τ) a genericprolongationof (K(a0, . . . , ak), τ)(k = 1, . . . , m− 1 and the corresponding extensions of τ are denoted by the sameletter τ) such that (a0, . . . , am) specializes to (η, α(η), . . . , αm(η)). Indeed, if thestatement of the theorem is false, then no principal realization of R specializesto η. Since the number of distinct principal realizations is finite (see Theorem5.2.10(iii)) there exists an integer N > 0 such that for any principal realizationη of R, (η, . . . , αN (η)) does not specialize to (η, . . . , αN (η)).

Let L be the algebraic closure of K〈η〉. By taking free joins and iso-morphic kernels if necessary, we may assume that (η, α(η)) = (a0, a1), thattrdegLL(a1) = 1 and that (a0, a1) specializes to (a0, a1) over L (we write this as(a0, a1) →L (a0, a1)). Then for any given m > 1, there exist s-tuples b2, . . . , bm

such that

(i) K(a1, b2, . . . , bm) ∼=K K(a1, α2(η), . . . , αm(η)),

(ii) (a0, a1, b2, . . . , bm) →L (a0, a1, α2(η), . . . , αm(η)),

(iii) trdegLL(a0, a1, b2, . . . , bm) = 1.

(To see this, it is sufficient to take into account the result of Proposition1.6.42 and the K-isomorphism K(a1) ∼=K K(a1).) Now, in order to continuethe proof we need the following fact.

Lemma 6.2.15 Let L be an algebraically closed field and L(b)= L(b(1), . . . , b(n))where the n-tuple b = (b(1), . . . , b(n)) of elements of an overfield of L satisfiesthe condition trdegLL(b) = 1. Furthermore, let b = (b

(1), . . . , b

(n)) be an n-tuple

of elements of L. If b specializes to b over L, then there exists a parametert ∈ L(b), transcendental over L, such that L[b] has a representation in the

power series ring L[[t]] with b(i) = b(i)

+∞∑

j=1

cijtj (i = 1, . . . , n).

PROOF. Clearly, L(b) is a field of algebraic functions of one variable overL in the sense of the definition in the last part of Section 1.6. Using the resultsof this part, we obtain a place p and a valuation ring A of L(b) over L suchthat L[b] ⊆ A and b(i) − b

(i) ∈ p for i = 1, . . . , n (see Proposition 1.6.70). ByProposition 1.6.69(i), A/p is algebraic over L. Moreover, since the field L isalgebraically closed, A/p = L. Applying Proposition 1.6.71 (and taking intoaccount the trivial fact that A/p = L is separable over L) we obtain that every

element c in the p-adic completion of L(b) has a representation c =∞∑

j=r

cjtj with

r ∈ Z and cj ∈ L for all j. Also, if c ∈ A, then r ≥ 0; and if c ∈ p, then r ≥ 1.


Since L[b] ⊆ A, we have L[b] ⊆ L[[t]], and since b(i) − b(i) ∈ p, b(i) − b

(i)=

∞∑j=1

cijtj with b

(i), cij ∈ L. This completes the proof of the lemma.


Applying the last lemma we obtain a parameter t ∈ L(a1, b2, . . . , bm), tran-scendental over L, such that L[a1, b2, . . . , bm] ⊆ L[[t]] with

a(i)1 = a

(i)1 +

∞∑j=1

cijtj (i = 1, . . . , n);

b(i)k = αk(η(i)) +

∞∑j=1

dkijtj (i = 1, . . . , n; k = 2, . . . , m). (6.2.1)

The mapping τ : K(a0) → K(a1) of the kernel R may be extended toan isomorphism of K(a0, a1, α

2(η), . . . , αm−1(η) onto K(a1, b2, . . . , bm) obtainedby the composition of the K-isomorphism from K(a0, a1, α

2(η), . . . , αm−1(η))to the field K(a1, α

2(η), . . . , αm(η)) generated by τ and the K-isomorphismK(a1, α

2(η), . . . , αm(η)) → K(a1, b2, . . . , bm) in (i). This K-isomorphism, aswell as its further extensions, will be denoted by the same letter τ (it cannotcause any confusions). At the next step, one can extend τ to a monomorphism ofL to the algebraic closure L1 of the quotient field of L[[t]]. This monomorphism,in turn, can be extended to a monomorphism of L[[t]] into L1[[t1]] where anelement t1 is transcendental over L1 and τ(t) = t1. Repeating this methodof extension, for each k = 2, . . . , m − 1 we can extend τ to a monomorphismLk−1[[tk−1]] → Lk[[tk]] with Lk−1[[tk−1]] ⊆ Lk and tk transcendental over Lk

(according to our convention, this monomorphism is also denoted by τ).Since a

(i)1 ∈ L[[t]] (i = 1, . . . , s), τk(a0) = τk−1(a1) ∈ Lk−1[[tk−1]] (2 ≤

k ≤ m). Let ak = τk(a0). Then K[a0, a1, . . . , am−1] ⊆ Lm−2[[tm−2]], and τrestricted to K[a0, a1, . . . , am−1] is an isomorphism onto K[a1, . . . , am]. There-fore, (K(a0, a1, . . . , am), τ)) is a kernel which can be obtained from R by m− 1prolongations.

In what follows, let b denote the (m − 1)s-tuple (b2, . . . , bm). we are goingto show that the kernel (K(a0, a1, . . . , am), τ)) is obtained from R by genericprolongations. Indeed, since t ∈ L(a1, b), tk = τk(t) ∈ Lk(ak+1, τ

k(b)), 1 ≤ k ≤m − 1. Also, the fact that tk is transcendental over Lk implies that

trdegLkLk(ak+1, τ

k(b) ) ≥ 1. (6.2.2)

Applying (i), Proposition 1.6.31(iii), and the fact that δR = 0, we obtain that

trdegK(a0,...,ak+1)K(a0, . . . , ak+1, τk(b)) ≤ trdegK(ak+1)K(ak+1, τ

k(b))

= trdegK(a1)K(a1, b) = trdegK(a1)K(a1, α2(η), . . . , αm(η)) = 0. (6.2.3)


Combining (6.2.2), (6.2.3) and the inequality in Proposition 1.6.31(iii), we obtainthat

trdegK(a0,...,ak)K(a0, . . . , ak+1) = trdegK(a0,...,ak)K(a0, . . . , ak+1, τk(b))

≥ trdegLkLk(ak+1, τ

k(b)) ≥ 1.

At the same time, Proposition 1.6.31(iii) implies that

trdegK(a0,...,ak)K(a0, . . . , ak+1) ≤ trdegK(ak)K(ak, ak+1)

= trdegK(a0,...,ak)K(a0, . . . , ak+1, τk(b))= trdegK(a0)K(a0, a1)=δR = 1.

It follows that trdegK(a0,...,ak)K(a0, . . . , ak+1) = 1 for every k ≥ 0, therefore(K(a0, . . . , am), τ) is obtained from R by generic prolongations.

Let φk : Lk[[tk]] → Lk be a Lk-homomorphism defined by φk(tk) = 0,1 ≤ k ≤ m−1. Using the series expansion (6.2.1) we obtain that a

(i)2 = τ(a(i)

1 ) =

τ(a(i)1 )+

∞∑j=1

(τ(cij))tji = b

(i)2 +

∞∑j=1

(τ(cij))tji = α2(η(i))+

∞∑j=1

d2ijtj+

∞∑j=1

(τ(cij))tji .

Therefore,

a(i)k = αk(η(i)) +

∞∑j=1

dkijtj +

∞∑j=1

(τ(dk−1,ij))tj1 + · · · +

∞∑j=1

(τk−1(cij))tjk−1

for k ≥ 2. Furthermore, K[a0, . . . , ak+1] ⊆ Lk[[tk]] and aq ∈ Lk−1[[tk−1]],φk(aq) = aq for q ≤ k. On the other hand,

φk(a(i)k+1) = φk(αk+1(η(i)) +

∞∑j=1

dk+1,ijtj + · · · +

∞∑j=1

τk(cij))tjk ) = αk+1(η(i))

+ · · · +∞∑

j=1

(τk−1(dk+1,ijtj))tjk−1 ∈ Lk−1[[tk−1]].

Thus, φ = φ0 φ1 · · ·φm−1 is a well-defined homomorphism on K[a0, . . . , am]with φ(ak) = αk(η) (k = 0, . . . , m), hence (a0, . . . , am) →K (a0, a1, α

2(η), . . . ,αm(η)). This completes the proof of the theorem.

Corollary 6.2.16 Every regular realization of a kernel is the specialization ofa principal realization.

PROOF. Let R = (K(a0, . . . , ar), τ) be a kernel of length r ≥ 0 over anordinary difference field K with a basic set σ = α, and let η be a regularrealization of R. By Theorem 5.2.10, σ-trdegKK〈η〉 ≤ δR. We proceed byinduction on m = δR− σ-trdegKK〈η〉.

If m = 0, then Theorem 5.2.10(v) implies that η is a principal realization.Suppose that m > 0 and the result is true for any kernel R such that δR− σ-trdegKK〈η〉 < m. For any k ≥ 0, let Rk denote the kernel with the fieldK(η, . . . , αr+k(η)). Then δR ≥ δR1 ≥ · · · ≥ δRk ≥ 0.

6.3. DIFFERENCE VALUATION RINGS 385

Let d be the minimal k ≥ 0 such that δRk = δRk+h for all h ≥ 0. Then η isa principal realization of Rd. Since m > 0, d > 0, and by the minimality of d,δRd−1 > δRd. It follows that Rd is not a generic prolongation of Rd−1.

Let R′ be the generic prolongation with the field K(η, . . . , αr+d−1(η), ζr+d)which specializes to K(η, . . . , αr+d−1(η), αr+d(η)). By Theorem 6.2.14, there is aprincipal realization ζ of R′ which specializes to η. Then ζ is a regular realizationof R. Furthermore, since the specialization ζ →K η is not generic, δR − σ-trdegKK〈ζ〉 < δR−σ-trdegKK〈η〉 = m. By the induction hypothesis, there is aprincipal realization θ of R which specializes to ζ. Therefore, θ →K ζ →K η.

6.3 Difference Valuation Rings and Extensionsof Difference Specializations

Definition 6.3.1 Let K be a difference field with a basic set σ = α1, . . . , αn.A difference (or σ-) specialization of K is a σ-homomorphism φ of a σ-subringR of K onto a difference (σ-) domain Λ. If φ cannot be extended to a σ-homomorphism of a larger σ-subring of K onto a domain which is a σ-overringof Λ, then φ is said to be a maximal difference (or σ-) specialization of K.

It is easy to see that if φ is a maximal σ-specialization of a difference (σ-)field K, then the image Λ of the corresponding σ-subring R of K is a σ-field.Indeed, if φ(x) = 0 (x ∈ R), then one can define φ(x−1) as the element (φ(x))−1

in the quotient field of Λ. Since φ is maximal, one should have (φ(x))−1 ∈ Λ.Also, if φ is a maximal σ-specialization of a difference field K with a domain

R ⊆ K, then M = Ker φ is a prime reflexive difference ideal of R which consistsof the nonunits of R. Thus, R is a difference local ring with maximal ideal M .

Definition 6.3.2 Let K be a difference field with a basic set σ and let φ be amaximal σ-specialization of K. Then the domain R of φ is called a maximaldifference (or σ-) ring of K. If K is the quotient field of R, we say that R is adifference (or σ-) valuation ring of K, and φ is called a difference (or σ-) placeof the σ-field K.

If a difference field K is inversive, then every its maximal σ-ring is alsoinversive. It follows from the fact that every difference homomorphism of adifference subring R of K can be extended to a homomorphism of the inversiveclosure of R in K.

Definition 6.3.3 A difference ring R with a basic set σ is called a local differ-ence (or σ-) ring if the nonunits of R form a σ-ideal. This ideal is denoted byM(R).

As we have mentioned after Proposition 2.1.10, for any difference ring Awith a prime difference ideal P , the ring AP is a local difference ring if and onlyif P is reflexive. Therefore, the maximal ideal M(R) of a local difference ring Ris reflexive (in this case R = RM(R)).


Definition 6.3.4 Let R be a local difference ring with a basic set σ and maximal(σ∗-) ideal M(R). If a σ-subring R0 of R is a local σ-ring with a maximal idealM(R0)such that M(R)

⋂R0 = M(R0), then R is said to dominate R0.

In what follows, we present some results on extensions of difference special-izations of difference fields which are natural generalizations of the correspond-ing statements proved in [118] for ordinary difference rings and fields.

Proposition 6.3.5 Let K be a difference field with a basic set σ and R a localσ-subring of K. Then the following statements are equivalent.

(i) R is a maximal σ-ring of K.(ii) R is maximal among local σ-subrings of K ordered by domination.(iii) If x ∈ K \ R, then the perfect σ-ideal RxM(R) generated by M(R)

in Rx coincides with Rx.PROOF. (i)⇒ (ii). If R′ is a local σ-subring of K dominating R, then the

natural σ-epimorphism R′ → R′/M(R′) (M(R′) is the maximal ideal of thering R′) would be an extension of the maximal σ-specialization of K with the do-main R.

Since the implications (ii)⇒ (i) and (iii)⇒ (ii) are obvious, it is sufficient toprove that (i) implies (iii). Let R be as in (ii) and x ∈ K\R. If the perfect σ-idealRxM(R) of Rx does not contain 1, it is contained in a reflexive primeideal P of Rx. Then one can extend the σ-specialization R → R/M(R) of theσ-field K to a natural σ-specialization Rx → Rx/P of K. This contradictsthe fact that R is a maximal σ-ring of K.

Let K be a difference field with a basic set σ and R a difference valuationring of K. Then the set U of units of R forms a subgroup of K′ = K\0 and onemay define the natural homomorphism v : K ′ → K ′/U . Let K ′/U be denotedby Υ, with the operation written as addition. Then v is called a difference (orσ-) valuation of K. Let Υ+ = v(M(R)\0); then for a ∈ Υ+ we have −a /∈ Υ+.For any a, b ∈ Υ we define a < b if b − a ∈ Υ+. Then Υ becomes a partiallyordered group which is not necessarily linearly ordered. Clearly, x ∈ R \ 0 ifand only if v(x) ≥ 0, and x ∈ M(R) \ 0 if and only if v(x) > 0.

The following example, due to R. Cohn, shows that there are differencevaluation rings which are not valuation rings (and, thus, difference valuationswhich are not valuations).

Example 6.3.6 Let us consider the field of complex numbers C as an ordinarydifference field whose basic set σ consists of an identity automorphism α. Let abe transcendental over C and let C(a) be considered as a σ-overfield of C suchthat αa = −a (thus, C〈a〉 = C(a)). Let C〈a〉y be the ring of σ-polynomials inone σ-indeterminate y over C〈a〉 and let A = y2 − (αy)2 +ay2(αy)2 ∈ C〈a〉y.Then A + αA = (y + α2y)(y − α2y)(1 + a(αy)2) and the variety M(A) of thedifference polynomial A has two components, one satisfying y + α2y, and theother satisfying y − α2y. Let η be a generic zero of an irreducible componentwith α2η + η = 0. Clearly, η is a solution of a σ-polynomial F ∈ C〈a〉y if and


only if η is a solution of a first-order σ-polynomial F ′ obtained by substituting−y for α2y. Since F ′ is a multiple of A, we obtain that η specializes to 0 overC〈a〉. This specialization can be extended to a maximal one and, hence, thereis a difference valuation ring R of C〈a, η〉 with η ∈ R. On the other hand, R isnot a valuation ring, because the element ζ = α(η)η−1 is integral over R (it isa solution of the polynomial X2 − (1 + a(αη)2) ∈ R[X]), but ζ /∈ R. If φ is themaximal specialization and φ(ζ) is defined, then φ(ζ2) = φ(1+a(αη)2) = 1 andφ(ζα(ζ)) = 1. But ζα(ζ) = ((αη)η−1)(α2η)(αη)−1 = −1. Thus, ζ is not in thedomain of φ.

Theorem 6.3.7 Let R be a local difference subring of a difference field K witha basic set σ, M(R) the maximal ideal of R, and x ∈ K. Then the natural σ-homomorphism φ : R → R/M(R) extends to one sending x to 0 if and only if1 /∈ [x] in Rx.

PROOF. If φ extends to a σ-homomorphism φ′ of Rx such that φ′(x) = 0,then [x] ⊆ Ker φ′ whence 1 /∈ [x]. Conversely, let N denote the difference idealof Rx generated by M(R) and x, that is, N = RxM(R)+[x] = M(R)+[x].Note that if φ cannot be extended to a σ-homomorphism of Rx sending xto 0, then 1 ∈ N. (If 1 /∈ N, then N is contained in a prime σ-idealP of Rx. Since P ∩ R = M(R), φ can be extended to a σ-homomorphismRx → Rx/P ). Recall (see Section 2.3) that N =

⋃∞k=0 Nk where N0 = N

and Nk = [Nk−1]′ for k = 1, 2, . . . . (As in Section 2.3, if S is a subset of adifference ring A with a basic set σ and Tσ is the free commutative semigroupgenerated by σ, then S′ denote the set of all elements a ∈ A such that someproduct of the form τ1(a)i1 . . . τm(a)im with τ1, . . . , τm ∈ Tσ and i1, . . . , im ∈ Nbelongs to S.)

Let us show that if there is an element c ∈ N such that c = u+ z for someu ∈ R \M(R) and z ∈ [x], then 1 ∈ [x]. Then the proof will be completed, since1 ∈ N and 1 can be expressed in this form (with z = 0). Considering therepresentation of N as a union of Nk, k = 0, 1, . . . we obtain that c ∈ [Nk] forsome k. Now we proceed by induction on k. If c ∈ [N0] = N = M(R)+ [x], thenc = u + z = m + w where m ∈ M(R), w ∈ [x]. Since u /∈ M(R), u−m /∈ M(R)hence (u − m)−1 ∈ R. Then 1 = (u − m)−1(u − m) = (u − m)−1(w − z) ∈ [x].

Let k ∈ N and c ∈ [Nk+1]. Then c =p∑

i=1

digi where di ∈ Rx and gi ∈Nk+1. For each i, di = ri + zi and gi = si + ti where ri, si ∈ R and zi, ti ∈ [x].

It follows thatp∑

i=1

digi =p∑

i=1

(ri + zi)(si + ti) = v + w where v =p∑

i=1

risi

and w ∈ [x]. Thus, c = u + z = v + w. If v ∈ M(R), then, as above, (u −v)−1 ∈ R and 1 ∈ [x]. Otherwise, si /∈ M(R) for some i′, 1 ≤ i′ ≤ p. Sincegi′ = si′ + ti′ ∈ Nk+1, there is a product of powers of transforms of gi′ whichlies in [Nk]: π(gi′) = (si′ + ti′)l0(τ1(si′) + τ1(ti′))l1 . . . (τq(si′) + τq(ti′))lq ∈ [Nk]for some τ1, . . . , τq ∈ Tσ. Setting u′ = sl0

i′ (τ1(si′))l1 . . . (τq(si′))lq we obtain thatπ(gi′) = u′ + z′ where z′ ∈ [x]. Since si′ /∈ M(R) and the ideal M(R) is primeand reflexive, u′ ∈ R \M(R). Applying the induction hypothesis we obtain that1 ∈ [x].


The following statement is a direct consequence of Theorem 6.3.7.

Corollary 6.3.8 Let R be a maximal difference ring of a difference field K. LetM(R) be the maximal ideal of R and let x ∈ K. Then x ∈ M(R) if and only if1 /∈ [x].

Let K be a difference field with a basic set σ and Ky a ring of σ-polynomials in one σ-indeterminate y over K. Let R be a σ-subring of K andlet g be a σ-polynomial in Ry with a constant term b ∈ R (by a constantterm of a σ-polynomial we mean a term that does not contain any transformof a σ-indeterminate). Furthermore, let gR and gK denote perfect σ-idealsgenerated by g in the σ-rings Ry and Ky, respectively.

Proposition 6.3.9 With the above notation, let the σ-ideal gK be prime,x a general zero of gK , and φ : R → Λ a difference specialization of Kwith φ(b) = 0. If gK

⋂Ry = gR, then φ can be extended to Rx with

φ(x) = 0.

PROOF. Let us show, first, that if f ∈ gR and c is the constant term

of the σ-polynomial f , then φ(c) = 0. Indeed, since gR =∞⋃

k=0

[g]k, f ∈ [g]k

for some k ≥ 0. We proceed by induction on k. If k = 0, then f =p∑

i=1

hiτi(g)

where hi ∈ Ry, τi ∈ Tσ (1 ≤ i ≤ p). Comparison of the constant terms in the

last equality yields c =p∑

i=1

ciτi(b) (ci ∈ R is a constant term of hi, 1 ≤ i ≤ p)

whence φ(c) = 0.In order to make the induction step, it is sufficient to prove that if k ≥ 1

and f =p∑

i=1

higi, where hi, gi ∈ Ry and some product π(gi) of powers of

transforms of gi lies in [g]k−1 (1 ≤ i ≤ p), then φ(c) = 0. Let ai denote theconstant term of gi (1 ≤ i ≤ p). Then π(gi) has constant term π(ai) andby the induction hypothesis φ(π(ai)) = 0 whence φ(ai) = 0. It follows that

φ(c) = φ

(p∑

i=1

ciai

)= 0.

In particular, we have shown that if f ∈ gR, then its constant term is notequal to 1.

By extending φ in K, one may assume that R is a localσ-subring of K andhence of Kx, with gK

⋂Ry = gR. (If P = Ker φ and R is replaced

by S = RP , then gK

⋂Sy = gS .) By Theorem 6.3.7, if φ does not

extend to a σ-homomorphism that sends x to 0, then 1 ∈ [x] in Rx, that is,

1 =q∑

i=1

bi(x)τi(x) where τi ∈ Tσ and bi(x) ∈ Rx is the result of substitution

of x for y in some σ-polynomial bi(y) ∈ Ry (1 ≤ i ≤ q).


It follows that f = 1−q∑

i=1

bi(y)τiy ∈ gK

⋂Ry. Since the constant term

of f is 1, f /∈ gR. Therefore, if the specialization φ cannot be extended toRx with φ(x) = 0, then gK

⋂Ry = gR.

Proposition 6.3.10 Let K be a difference field with a basic set σ and let R bea maximal difference ring of K. Then

(i) If S is another maximal difference ring of K such that R ⊆ S, thenM(S) ⊆ M(R). (Therefore, in this case every ideal of S is an ideal of R.)

(ii) Let P be a prime σ∗-ideal of R. Then there is a maximal difference ringR1 of K such that R ⊆ R1 and M(R1) = P .(iii) The prime reflexive σ-ideals of R are linearly ordered by inclusion.(iv) Let A be a maximal difference ring of K with a specialization φ : A → Λ,

and let R ⊆ A. Then φ(R) is a maximal difference ring of Λ.

PROOF. (i). By Corollary 6.3.8, if x ∈ M(S), then 1 /∈ [x]S (where [x]Sdenotes the σ-ideal generated by x in Sx). Since R ⊆ S, one also has 1 /∈ [x]Rwhence x ∈ M(R).

(ii). Let R1 be a maximal local difference ring of K dominating RP . By (i),M(R1) ⊆ M(R). Therefore, M(R1) = M(R1)

⋂R = (M(R1)

⋂RP )

⋂R =

PRP

⋂R = P .

(iii). Let P be a prime σ-ideal of R. If x ∈ R\P and a ∈ P , then ax−1 ∈ P . (Itfollows from the fact that x is a unit in any maximal σ-ring A of K dominatingRP and so ax−1 ∈ M(A); but by part (ii), M(A) = P .) Now, let P and Q betwo reflexive prime σ-ideals of R. Suppose that Q P , and let b ∈ P, c ∈ Q.As we have seen, bc−1 ∈ P hence b ∈ cP ⊆ Q. Since this is true for every b ∈ P ,P ⊆ Q.

(iv). Let ψ : R → Ω be a maximal specialization of K with domain R. SinceKer φ = M(A) ⊆ M(R) = Ker ψ, one can define the σ-homomorphism θ :φ(R) → Ω by θ(φ(r)) = ψ(r), r ∈ R. Clearly, Ker θ = φ(M(R)). Let x ∈ Λ.If x /∈ φ(R), then there exists z ∈ A \ R such that φ(z) = x. Since ψ ismaximal, 1 ∈ RzM(R) and thus 1 ∈ φ(R)xKer θ. Therefore, θ cannotbe extended to x.

Theorem 6.3.11 Let K be a difference field with a basic set σ and R0 a σ-subring of K with prime reflexive σ-ideals P and Q such that P ⊆ Q. Let S bea proper maximal σ-ring of K with R0 ⊆ S and M(S)

⋂R0 = P . Then there

exists a proper maximal σ-ring R of K such that R0 ⊆ R and M(R)⋂

R0 = Q.Furthermore, if S is a σ-valuation ring of K then R is also.

PROOF. Let L = S/M(S) and let φ : S → L be the canonical σ-epimorphism. Then φ(R0) ⊆ L and the inclusion P ⊆ Q implies that theσ-homomorphism θ0 : φ(R0) → R0/Q (φ(x) → x + Q for every x ∈ R0) iswell-defined. Let us extend θ0 to a maximal σ-specialization θ of L. Let Rθ

be the domain of θ, let φ = θ φ and R = φ−1(M(Rθ)). Then R is a local


σ-subring of K with the maximal σ-ideal φ−1(M(Rθ)). Clearly, R0 ⊆ R ⊆ Sand M(R)

⋂R0 = Q. It is easy to check that R is a maximal σ-ring of K. The

fact that the quotient field of R is K follows from the fact that M(S) ⊆ R.

Corollary 6.3.12 Let K be a difference field with a basic set σ and R0 a localσ-subring of K. Let L be a σ-overfield of K and S a proper maximal σ-ring(valuation ring) of L containing R0. Then there exists a proper maximal σ-ring(valuation ring) R of L dominating R0.

PROOF. Clearly, M(S)⋂

R0 is a prime reflexive σ-ideal of R0 which iscontained in M(R0). It remains to apply Theorem 6.3.11.

The existence of S in the corollary is equivalent to the condition that Lhave a subring S0, S0 ⊆ R0, which contains a proper nonzero prime σ∗-ideal.This condition does not always hold. For example, if Q is treated as an ordinarydifference (σ-) field with the identity translation α and Qb is a σ-overring of Qsuch that b is transcendental over Q and α(b) = b, then the σ-specialization b →1 does not extend to a σ-place Q〈a〉 where a2 = b and α(a) = −a (see Example5.3.4). However, the condition does hold in the situations of the Theorem 6.3.14below. To prove this theorem we need the following statement that can be provedby using the arguments of the proof of Theorem 5.3.2 and applying Proposition1.6.64. We leave the details to the reader as an exercise.

Lemma 6.3.13 Let R be an ordinary difference integral domain with quotientfield K. If L = K〈a1, . . . , am〉 is a difference field extension of K such thatthe field extension L/K is primary, then there exists a nonzero element u ∈ Rsuch that any specialization φ of R with φ(u) = 0 can be extended to the ringRa1, . . . , am.

Theorem 6.3.14 Let R0 be a local ordinary difference ring with a basic setσ = α and K the difference quotient field of R0.

(i) Suppose that R0 does not contain minimal nonzero prime reflexive σ-ideals. If L is a primary finitely generated σ-field extension of K, then thereexists a difference valuation ring R of L dominating R0.

(ii) Let b be an element from some σ-overfield of K which is σ-algebraicallyindependent over K. If N = K〈b, a1, . . . , as〉 is a primary σ-field extension ofK〈ζ〉, then there exists a difference valuation ring of N dominating R0.

PROOF. (i). Let x1, . . . , xm be elements of L such that L = K〈x1, . . . , xm〉.Furthermore, let u be an element of R0 with the following property: for everyprime σ-ideal P of R0 not containing u, there is a prime σ-ideal P ′ in thering R0x1, . . . , xm such that P ′ ⋂R0 = P . (The existence of such an elementfollows from the preceding lemma.) By the hypothesis of the theorem, the in-tersection of all nonzero prime reflexive σ-ideals of R0 is (0). Therefore, one canchoose a prime reflexive σ-ideal Q of R0 not containing u and hence obtain aprime reflexive σ-ideal Q′ of R0x1, . . . , xm such that Q′ ∩ R0 = Q.


Let S be a σ-valuation ring of L such that M(S)⋂

R0x1, . . . , xm = Q′.Applying Corollary 6.3.12 we obtain that there exists a σ-valuation ring R of Lcontained in S and dominating R0.

(ii). It is easy to see that the ideals Pk = b − αk(b) (k = 1, 2, . . . )form a descending chain of prime reflexive σ-ideals in R0b. Therefore, ifu ∈ R0b, u = 0, then u /∈ Pk for some k. Now the proof can be completed bythe arguments of the proof of part (i) (with the use of Lemma 6.3.13).

Chapter 7

Systems of AlgebraicDifference Equations

7.1 Solutions of Ordinary DifferencePolynomials

In this section we start the discussion of varieties of ordinary difference polyno-mials. In what follows we use the terminology introduced for the ordinary casein Section 4.5. The whole section, as well as Sections 7.2 and 7.4 - 7.6, is basedon the results of R. Cohn. Some of them were obtained in [28], but the mainpart of the theory first appeared in [41].

Proposition 7.1.1 Let K be an ordinary difference field with a basic set σ =α and let φ be a specialization of an s-tuple a = (a(1), . . . , a(s)) over K.Let i1, . . . , iq be a subset of 1, . . . , s, a denote the set a(i1), . . . , a(iq), andφa = φa(i1), . . . , φa(iq). Then

(i) If φa is σ-algebraically independent over K, so is a. Thus, σ-trdegK

K〈φa(1), . . . , φa(s)〉 ≤ σ-trdegKK〈a(1), . . . , a(s)〉.(ii) If φa is a transcendence basis of φa(1), . . . , φa(s) over K, then ord

K〈φa(1), . . . , φa(s)〉/K〈φa〉 ≤ ordK〈a(1), . . . , a(s)〉/K〈a〉, EordK〈φa(1), . . . ,φa(s)〉/K〈φa〉 ≤ EordK〈a(1), . . . , a(s)〉/K〈a〉, and the equality occurs if andonly if the specialization φ is generic.

PROOF. Part (i) follows from Proposition 5.3.1. If B ⊆ K〈a(1), . . . , a(s)〉 andevery element of K〈a(1), . . . , a(s)〉 is algebraic over K〈a〉(B), then every elementof K〈φa(1), . . . , φa(s)〉 is algebraic over K〈φa〉(φ(B)). Now the first statementof (ii) can be derived from Theorem 1.6.30. The last part of (ii) can be obtainedsimilarly; we leave the details to the reader as an exercise.

Proposition 7.1.2 Let K be an ordinary difference field with a basic setσ = α, Ky1, . . . , ys the ring of σ-polynomials in s σ-indeterminates

393

394 CHAPTER 7. SYSTEMS OF DIFFERENCE EQUATIONS

y1, . . . , ys over K, and η = (η1, . . . , ηs) a solution of a nonzero σ-polynomialA ∈ Ky1, . . . , ys (coordinates of η lie in some σ-overfield of K). Then either

(I) σ-trdegKK〈η〉 < s − 1;or

(II) σ-trdegKK〈η〉 = s−1 and for every j ∈ 1, . . . , s such that the elementsη1, . . . , ηj−1, ηj+1, . . . , ηs, are σ-algebraically independent over K, the followingconditions hold:

(a) the σ-polynomial A contains yj,(b) ordK〈η〉/K〈η1, . . . , ηj−1, ηj+1, . . . , ηs〉 ≤ ordyj

A, and(c) EordK〈η〉/K〈η1, . . . , ηj−1, ηj+1, . . . , ηs〉 ≤ Eordyj

A.

PROOF. Since A(η) = 0, the elements η1, . . . , ηs are σ-algebraically depen-dent over K. Therefore, σ-trdegKK〈η1, . . . , ηs〉 ≤ s − 1.

Suppose that σ-trdegKK〈η1, . . . , ηs〉 = s − 1, and for some j, 1 ≤ j ≤ s,the set ηi | 1 ≤ i ≤ s, i = j is σ-algebraically independent over K. Thenthe (s − 1)-tuple η′ = (η1, . . . , ηj−1, ηj+1, . . . , ηs) annuls no σ-polynomial inKy1, . . . , yj−1, yj+1, . . . , ys, hence A should contain some transform of yj . LetA′ be the σ-polynomial in K〈η1, . . . , ηj−1, ηj+1, . . . , ηs〉yj obtained from Aby replacing every transform τyi with i = j (τ ∈ Tσ) by τηi. Then ordyj

A =ordyj

A′ and EordyjA = Eordyj

A′. Since ηj is a solution of A′, Corollaries 4.1.18and 4.1.20 imply conditions (b) and (c).

Let K be an ordinary difference field with a basic set σ, Ky1, . . . , ys thering of σ-polynomials in σ-indeterminates y1, . . . , ys over K, and M a nonemptyirreducible variety over Ky1, . . . , ys. If (η1, . . . , ηs) is a generic zero of M,then σ-trdegKK〈η1, . . . , ηs〉, ordK〈η1, . . . , ηs〉/K, EordK〈η1, . . . , ηs〉/K, andld(K〈η1, . . . , ηs〉/K) are called the dimension, order, effective order, and limitdegree of the variety M, respectively. They are denoted by dimM, ordM,EordM, and ldM, respectively. If M = ∅, we set dimM = ordM =EordM = −1 and ldM = 0. Obviously, if dimM > 0, then ordM = ∞.

If P is a prime inversive difference ideal of Ky1, . . . , ys then the dimension,order, effective order, and limit degree of P (they are denoted by dimP , ordP ,EordP , and ld P , respectively) are defined as the corresponding values of thevariety M(P ). (Thus, these concepts are determined as above through the σ-field extension K〈η1, . . . , ηs〉/K where (η1, . . . , ηs) is a generic zero of P .)

Let M be a nonempty irreducible variety over Ky1, . . . , ys, (η1, . . . , ηs) ageneric zero of M, and yi1 , . . . , yiq

is a subset of the set of σ-indeterminatesy1, . . . , ys. Then the dimension, order, effective order, and limit degreeof M relative to yi1 , . . . , yiq

are defined as σ-trdegK〈ηi1 ,...,ηiq 〉K〈η1, . . . , ηs〉,ordK〈η1, . . . , ηs〉/K〈ηi1 , . . . , ηiq

〉, EordK〈η1, . . . , ηs〉/K〈ηi1 , . . . , ηiq〉, and

ld(K〈η1, . . . , ηs〉/K〈ηi1 , . . . , ηiq〉), respectively. These characteristics of M are

denoted by dim (yi1 , . . . , yiq)M, ord (yi1 , . . . , yiq

)M, Eord (yi1 , . . . , yiq)M, and

ld (yi1 , . . . , yiq)M, respectively.

If P is a prime inversive difference ideal of Ky1, . . . , ys then the di-mension, order, effective order, and limit degree of P relative to yi1 , . . . , yiq

7.1. SOLUTIONS OF ORDINARY DIFFERENCE POLYNOMIALS 395

are defined as the corresponding characteristics of M(P ) (the notation isthe same: dim (yi1 , . . . , yiq

)P, ord (yi1 , . . . , yiq)P, Eord (yi1 , . . . , yiq

)(P ), andld (yi1 , . . . , yiq

)(P ), respectively).

Definition 7.1.3 With the above notation, a subset yi1 , . . . , yiq of y1, . . . , ys

is called a set of parameters of P (or M(P )) if P contains no nonzero σ-polynomial in Fyi1 , . . . , yiq

. A set of parameters of P which is not a propersubset of any set of parameters of P is called complete.

Clearly, any reflexive prime σ-ideal P of Ky1, . . . , ys has at least onecomplete set of parameters, and every set of parameters can be extended toa complete one.

Remark 7.1.4 If yi1 , . . . , yiq is a complete set of parameters of P , r =

ord (yi1 , . . . , yiq)P , and η = (η1, . . . , ηs) is a generic zero of P , then there is

an r-element transcendence basis B of K〈η1, . . . , ηs〉 over K〈ηi1 , . . . , ηiq〉 whose

elements lie in the set τηj | τ ∈ Tσ, 1 ≤ j ≤ s, j = iν for ν = 1, . . . , q(the existence of such a transcendence basic is stated by Theorem 1.6.30(ii)).Let B = τ1ηj1 , . . . , τrηjr

(τ1, . . . , τr ∈ Tσ, 1 ≤ j1 ≤ · · · ≤ jr ≤ s) and letzk = τkyjk

(1 ≤ k ≤ r). Then

P ∩ Kyi1 , . . . , yiq[z1. . . . , zr] = (0) . (7.1.1)

Moreover, it is easy to see that ord (yi1 , . . . , yiq)P is the maximum integer r

such that there exists an r-element subset z1, . . . , zr of the set

τyj | τ ∈ Tσ, 1 ≤ j ≤ s, j = iν for ν = 1, . . . , q (7.1.2)

with condition (7.1.1).

Remark 7.1.5 With the above notation, let R = Ky1, . . . , ys and letyi1 , . . . , yiq

be a complete set of parameters of a prime σ∗-ideal P of R.Let u1 = yj1 , . . . , us−q = yjs−q

(1 ≤ j1 < · · · < js−q ≤ s) denote the σ-indeterminates of the set y1, . . . , ys\yi1 , . . . , yiq

and let R′ = K〈yi1 , . . . , yiq〉

u1, . . . , us−q. Furthermore, let P ′ = PR′. Then it is easy to check that P ′

is a prime σ-ideal of R′ and if (η1, . . . , ηs) is a generic zero of P over K, then(ηq+1, . . . , ηs) is a generic zero of P ′ over K〈yi1 , . . . , yiq

〉 and ord (yi1 , . . . , yiq)P =

ord (u1, . . . , us−q)P ′ = ordP ′.

Proposition 7.1.6 With the above notation, a set of parameters of a reflexiveprime difference ideal P of Ky1, . . . , ys is complete if and only if it containsdimP elements.

PROOF. Let η = (η1, . . . , ηs) be a generic zero of the σ∗-ideal P . Thenyi1 , . . . , yiq

(1 ≤ i1 < · · · < iq ≤ s) is a set of parameters of P if and only ifthe set ηi1 , . . . , ηiq

is σ-algebraically independent over K. Furthermore, it iseasy to see that such a set of parameters is complete if and only if one has theequalities q = σ-trdegKK〈η1, . . . , ηs〉 = dimP .


Proposition 7.1.7 Let M1 and M2 be two irreducible varieties over the ringof σ-polynomials Ky1, . . . , ys (we use the above notation and conventions)such that M1 ⊆ M2. Then

(i) dimM1 ≤ dimM2.(ii) If yi1 , . . . , yiq

is a complete set of parameters of Φ(M1) (we use the no-tation of Section 2.6), then ord (yi1 , . . . , yiq

)M1 ≤ ord (yi1 , . . . , yiq)M2, and the

equality occurs if and only if M1 = M2. Furthermore, Eord (yi1 , . . . , yiq)M1 ≤

Eord (yi1 , . . . , yiq)M2, and the equality occurs if and only if M1 = M2.

PROOF. Let Pi = Φ(Mi) (i = 1, 2). Then P1 and P2 are reflexive primedifference ideals of Ky1, . . . , ys and P1 ⊇ P2. Therefore, every complete setof parameters of P2 is a set of parameters of P1. Applying Proposition 7.1.6 weimmediately obtain the inequality in part (i).

Let us prove the first part of (ii), the inequality and equality of orders. Letyi1 , . . . , yiq

be a complete set of parameters of P1 (and therefore, a set ofparameters of P2). Without loss of generality we may assume that yi1 , . . . , yiq

is also a complete set of parameters of P2 (otherwise, ord (yi1 , . . . , yiq

)M2 = ∞and the statement is obvious). Let d = ord (yi1 , . . . , yiq

)P1 and let z1, . . . , zd beelements of the set (7.1.2) such that P1∩Kyi1 , . . . , yiq

[z1. . . . , zd] = (0). ThenP2 ∩ Kyi1 , . . . , yiq

[z1. . . . , zd] = (0), and the description of ord (yi1 , . . . , yiq)P

given in Remark 7.1.4 shows that ord (yi1 , . . . , yiq)P1 ≤ ord (yi1 , . . . , yiq

)P2.Now, in order to complete the proof of the first part of (ii), we should show

that if r = ord (yi1 , . . . , yiq)P2 and P1 P2, then ord (yi1 , . . . , yiq

)P1 < r.But this inequality is an immediate consequence of Remark 7.1.5 and Propo-sition 4.2.22 (with the notation of Remark 7.1.5, ord (u1, . . . , us−q)P ′ is theσ-dimension polynomial of the ideal P ′, see Definition 4.2.21).

We leave the proof of the statement about effective orders to the readeras an exercise (one can use an analog of Remark 7.1.5 and the second part ofProposition 4.2.22).

Let K be an ordinary difference field with a basic set σ = α and R =Ky1, . . . , ys the ring of σ-polynomials in σ-indeterminates y1, . . . , ys over K.Let A ∈ R be an irreducible σ-polynomial, that is, A /∈ K and A is irreducibleas a polynomial in indeterminates αjyi (1 ≤ i ≤ s; j = 0, 1, . . . ). As before,M(A) will denote the variety of the σ-polynomial A, that is, a variety of theperfect σ-ideal A.

Definition 7.1.8 With the above notation, an irreducible component M of thevariety M(A) is called a principal component of this variety if whenever Acontains a transform of yi (1 ≤ i ≤ s), the family y1, . . . , yi−1, yi+1, . . . , ysis a complete set of parameters of M and Eord (y1, . . . , yi−1, yi+1, . . . , ys)M isthe effective order of A in yi. (This implies that dimM = s − 1.) Irreduciblecomponents of M which are not principal are called singular.

Proposition 7.1.9 With the above notation, let A be a nonzero irreducible σ-polynomial in Ky1, . . . , ys and η = (η1, . . . , ηs) a solution of A. Furthermore,


suppose that σ-trdegKK〈η1, . . . , ηs〉 = s − 1, and for each j (1 ≤ j ≤ s) suchthat A contains a transform of yj,

EordK〈η1, . . . , ηs〉/K〈η1, . . . , ηj−1, ηj+1, . . . , ηs〉 = EordyjA. (7.1.3)

Then η is a generic zero of a principal component of M(A).

PROOF. Clearly, η is contained in some irreducible component M of M(A).Let ζ be a generic zero of M. Then ζ specializes to η. By Propositions 7.1.1 and7.1.2, σ-trdegKK〈ζ〉 = s − 1.

Suppose that some transform of yj (1 ≤ j ≤ s) appears in A. Then equal-ity (7.1.3) shows that ηj is σ-algebraic over K〈η1, . . . , ηj−1, ηj+1, . . . , ηs〉. Sinceσ-trdegKK〈η1, . . . , ηs〉 = s − 1, the elements η1, . . . , ηj−1, ηj+1, . . . , ηs form aσ-transcendence basis of K〈η1, . . . , ηs〉 over K. Applying Propositions 7.1.1and 7.1.2 once again we obtain that EordK〈ζ〉/K〈ζ1, . . . , ζj−1, ζj+1, . . . , ζs〉 =Eordyj

A. Since y1, . . . , yj−1, yj+1, . . . , ys is a complete set of parameters ofM, this variety is a principal component of M(A) (see Definition 7.1.8). Also,statement (ii) of Proposition 7.1.1 implies that the specialization of ζ to η isgeneric. This completes the proof.

Remark 7.1.10 Let K be an ordinary difference field with a basic set σ =α and let Ky1, . . . , ys be an algebra of σ-polynomials in σ-indeterminatesy1, . . . , ys over K. Then the set of prime reflexive σ-ideals of Ky1, . . . , ys ofdimension 0 is big enough in the following sense: if J is a perfect σ-ideal inKy1, . . . , ys and A ∈ Ky1, . . . , ys \ J , then there exists a reflexive σ-idealP of Ky1, . . . , ys such that J ⊆ P , dimP = 0, and A /∈ P . Indeed, it followsfrom Theorem 5.4.28 (see also Lemma 5.4.31) that the ideal J has a solutionη = (η1, . . . , ηs) which does not annul A. Then the prime σ∗-ideal P with genericzero η satisfies the required conditions.

As a consequence we obtain that every proper perfect σ-ideal J of the alge-bra Ky1, . . . , ys is the intersection of all prime σ∗-ideals of Ky1, . . . , ys ofdimension 0 which contain J .

We conclude this section with a result which a strengthened version ofTheorem 2.6.5 (the “difference Nullstellensatz”), see Theorem 1.7.13 below.

Let K be an ordinary difference field with a basic set σ = α, Ky1, . . . , ysthe ring of σ-polynomials in σ-indeterminates y1, . . . , ys over K, M a non-emptyirreducible variety over Ky1, . . . , ys, and N a variety over Ky1, . . . , ys whichdoes not contain M.

Proposition 7.1.11 With the above notation, the variety M contains a solu-tion which is σ-algebraic over K and does not belong to N .

PROOF. By Theorem 2.6.5, there is a σ-polynomial A ∈ Φ(N ) \ Φ(M)(we use the notation of Section 2.6). Let η = (η1, . . . , ηs) be a generic zero ofM and let K ′ be the algebraic closure of K in K〈η〉. Furthermore, let L be acompletely aperiodic σ-field extension of K ′ such that the extension L/K ′ is


σ-algebraic. (To construct L, one can consider the ring of σ-polynomials K ′zin one σ-indeterminate z over K ′ and adjoin to K ′ a generic zero of the varietyof the σ-polynomial αz − z − 1.) Applying Theorem 5.1.6 we obtain that the σ-field extensions L/K and K〈η〉/K are compatible. Let ζ = ηi1 , . . . , ηik

be aσ-transcendence basis of η over K, let λ = η1, . . . , ηs \ ζ, and let µ be theelement of Kη obtained by substituting ηj for yj in A (1 ≤ j ≤ s).

Since A /∈ Φ(M), µ = 0. By Theorem 5.4.28 there exists a nonzero σ-polynomial B ∈ Kyi1 , . . . , yik

such that if η is an indexing of k elements ofa σ-overfield of K, B(η) = 0, and K〈η〉/K is compatible with K〈ζ, λ〉/K, thenthere exists a σ-specialization ζ, λ of ζ, λ such that λ is σ-algebraic over K〈ζ〉,and the σ-specialization of µ is not 0. Thus, ζ, λ lies in M, but not in N , sincethis s-tuple is not a solution of B. Furthermore, the fact that the extensionsL/K and K〈η〉/K are compatible and Proposition 4.5.3 imply that ζ may bechosen as a set of elements of L. Then ζ is σ-algebraic over K, hence ζ, λ isσ-algebraic over K as well.

Using the previous notation, let us denote by Ma the set of all solutions ofa difference variety M over Ky1, . . . , ys which are σ-algebraic over K. ThenProposition 7.1.11 immediately implies the following statement.

Corollary 7.1.12 If M and N are two difference varieties over Ky1, . . . , yssuch that Ma = Na, then M = N .

Theorem 7.1.13 Let K be an ordinary difference field with a basic set σ = α,Ky1, . . . , ys the ring of σ-polynomials in σ-indeterminates y1, . . . , ys over K,and Φ ⊆ Ky1, . . . , ys. Then a σ-polynomial A ∈ Ky1, . . . , ys belongs to Φif and only if A is annulled by every solution of Φ which is σ-algebraic over K.

PROOF. Since any σ-polynomial in Φ is annulled by any solution of Φ, onejust needs to show that if A(η) = 0 for every η ∈ M(Φ)a, then A ∈ Φ.Suppose that A /∈ Φ. Then Theorem 2.6.5 implies that M(Φ) = M(Φ

⋃A).Applying Proposition 7.1.11 we obtain that M(Φ)a M(Φ

⋃A)a. Therefore,there is a solution of Φ which is σ-algebraic over K and does not annul A. Thiscontradiction completes proof.

Let K be an ordinary difference field with a basic set α = α, L a σ-overfield of K, and Ky1, . . . , ys and Ly1, . . . , ys the rings of σ-polynomialsin σ-indeterminates y1, . . . , ys over K and L, respectively. In what follows weshall denote these rings by Ky and Ly, respectively. Furthermore, for anyset S ⊆ Ky, [S] and S will denote, respectively, the difference (σ-)ideal andthe perfect σ-ideal of Ky generated by S, while [S]L and SL will denote,respectively, the σ-ideal and the perfect σ-ideal of Ly generated by the set S.The following results, describing some relationships between characteristics ofa prime σ∗-ideal P of Ky and the essential prime divisors of PL, are dueto R. Cohn.

Theorem 7.1.14 With the above notation, let P be a prime σ∗-ideal of Ky,let P1, . . . , Pk be the essential prime divisors of PL, and let η = (η1, . . . , ηs)be a generic zero of P . Then


(i) dimPi ≤ dimP for i = 1, . . . , n.(ii) If Σ = yj1 , . . . , yjq

is a complete set of parameters of some Pi

(1 ≤ i ≤ k), then Σ is a (not necessarily complete) set of parameters of Pand Eord (Σ)Pi ≤ Eord (Σ)P . Furthermore, if the σ-field extensions L/K andK〈η〉/K are incompatible, then the last inequality is strict.(iii) If L/K and K〈η〉/K are compatible, then there exists i, 1 ≤ i ≤ k, with

the following properties:(a) dimPi = dimP .(b) If Σ is a complete set of parameters of P , then Σ is a complete set of

parameters of Pi and Eord (Σ)Pi = Eord (Σ)P .(iv) If L/K and K〈η〉/K are compatible and some divisor Pj (1 ≤ j ≤ k) has

properties (a) and (b) of (iii), then Pj

⋂Ky = P .

PROOF. Let Pi be a prime divisor of P and let η′ be a generic zero of Pi.Then η′ annuls every σ-polynomial in P , hence η′ is a specialization of η over K.Let ζ and ζ ′ be subindexings of η and η′, respectively, consisting of coordinateswith the same indices. Applying Propositions 4.1.10(ii) and 7.1.1(i), we obtainthat σ-trdegLL〈η′〉 ≤ σ-trdegKK〈η′〉 ≤ σ-trdegKK〈η〉, hence dimPi ≤ dimP .If the equality holds and ζ ′ is a σ-transcendence basis of η′ over K, then Propo-sition 7.1.1(ii) implies that

EordL〈η′〉/L〈ζ ′〉(Pi) ≤ EordK〈η′〉/K〈ζ ′〉 ≤ EordK〈η〉/K〈ζ〉. (7.1.4)

Suppose that the σ-field extensions L/K and K〈η〉/K are incompatible.Then η′ is not a generic zero of P and, therefore, not a generic σ-specialization ofη over K. Let Σ be a complete set of parameters of Pi. Then ζ ′ is σ-algebraicallyindependent over L and, hence, over K. We obtain inequalities (7.1.4), the lastof which must be strict, as it follows from Proposition 7.1.1(ii). This completesthe proof of parts (i) and (ii) of the theorem.

Let L/K and K〈η〉/K be compatible. By Corollary 5.1.7, there exists aσ-overfield L〈η′〉 of L and a σ-K-isomorphism ρ of K〈η〉 into L〈η′〉 with thefollowing properties:

1) ρ(η) = η′;2) Let ζ be a σ-transcendence basis of η over K. Then ζ ′ = ρ(ζ) is a σ-

transcendence basis of η′ over L and EordL〈η′〉/L〈ζ ′〉 = EordK〈η〉/K〈ζ〉.Clearly, ζ ′ is a solution of some Pi (1 ≤ i ≤ k). Then dimPi ≥ σ-

trdegKK〈η〉 = dimP .Let Σ be a complete set of parameters of P and let ζ be the corresponding

subindexing of η. Since ζ ′ = ρ(ζ) is σ-algebraically independent over L, Σ iscontained in a set of parameters of Pi, hence Σ is a complete set of parameters.

It is easy to see that there is a generic zero θ of Pi whose coordinates withthe same indices as elements of Σ are the coordinates of ζ ′. Then η′ is a σ-specialization of θ over L〈ζ ′〉. Applying Proposition 7.1.1 we obtain that

Eord (Σ)Pi = EordL〈θ〉/L〈ζ ′〉 ≥ EordL〈η′〉/L〈ζ ′〉= EordK〈η〉/K〈ζ〉Eord (Σ)P,

so statement (iii) is proved.


In order to prove the last statement of the theorem, let Pj be an essentialprime divisor of PL in Ly satisfying properties (a) and (b) of statement(iii), and let Q = Pj

⋂Ky. Then Q is a prime σ∗-ideal of Ky containing P

and the perfect σ-ideal Q′ generated by Q in Ly. It follows that Q containssome essential prime divisor P ′ of Q′. Applying Proposition 7.1.7 and the al-ready proven part of our theorem, we obtain that Eord (Σ)Pj ≤ Eord (Σ)P ′ ≤Eord (Σ)Q ≤ Eord (Σ)P . It follows from Proposition 7.1.7(ii) that Q = P . Thiscompletes the proof of the theorem.

Exercise 7.1.15 With the notation of the last theorem, give an example of aσ-field extension L/K and proper prime σ∗-ideal P of Ky such that 1 ∈ PL.

Proposition 7.1.16 Let K be an ordinary difference field with a basic setσ = α, let L be a σ-overfield of K, and let Ky and Ly be rings ofσ-polynomials in σ-indeterminates y1, . . . , ys over K and L, respectively. Let Jbe a perfect σ-ideal of the ring Ky and P1, . . . , Pk the essential prime divisorsof J in Ky. Then JL is the intersection of the essential prime divisors ofPiL (1 ≤ i ≤ k).

PROOF. Let Qi1, . . . , Qimibe the esential prime divisors of the per-

fect σ-ideal PiL in Ly (1 ≤ i ≤ k). Then JL = P1

⋂ · · ·⋂PkL =

P1 . . . PkL = P1L

⋂ · · ·⋂PkL =k⋂

i=1

mi⋂j=1

Qij (see Theorem 2.3.3). Since

the opposite inclusion is obvious, the proposition is proved.

Theorem 7.1.17 With the notation of Proposition 7.1.16, let P be a primeσ∗-ideal of Ky and let η be a generic zero of P . Furthermore, suppose thatL is a σ-overfield of K such that L and K〈η〉 are quasi-linearly disjoint overK. (Actually, the last condition refers to the underlying fields of L, K〈η〉 andK, respectively, but we have agreed not to use special notation for underlyingfields if no confusion can occur.) Then PL is a prime σ∗-ideal of the samedimension as P .

If Σ is a complete set of parameters of P , then Σ is a complete set of para-meters of PL and Eord (Σ)PL = Eord (Σ)P .

PROOF. For any k ∈ N, let Rk and Sk denote the algebras of polynomialsK[αiyj | 0 ≤ i ≤ k, 1 ≤ j ≤ s] and L[αiyj | 0 ≤ i ≤ k, 1 ≤ j ≤ s],respectively, and let Pk = P

⋂Rk. Then Pk is a prime ideal of Rk with generic

zero (η, αη, . . . , αkη) = (η1, . . . , ηs, α(η1), . . . , α(ηs), . . . , αk(ηs)). Furthermore,by Theorem 1.2.42(v), the radical r(PkSk) is a prime ideal of the ring Sk. Wedenote this radical by P ′

k.

It is easy to see that if A ∈ P ′k, then α(A) ∈ P ′

k+1. It follows that Q =∞⋃

k=0

P ′k

is a prime σ-ideal of Ly. Let Q′ be the reflexive closure of Q. Then P ⊆ Q′ ⊆PL, hence Q′ = PL. Thus, PL is a prime σ∗-ideal of Ly.


By Theorem 1.2.42(iii), one has Q⋂

Ky = P . Furthermore, Q′ ⋂Kyis the reflexive closure of Q

⋂Ky. Since the σ-ideal P is reflexive, we obtain

that Q′ ⋂Ky = P . It follows that if ζ is a generic zero of Q′, then there isa σ-K-isomorphism of K〈ζ〉 onto K〈η〉, hence the σ-field extensions K〈η〉/Kand L/K are compatible. The statements about dimension and relative effectiveorders are direct consequences of Theorem 7.1.14.

Exercise 7.1.18 Use the last theorem to prove the ordinary version of Theorem5.1.4 without the assumption that the the difference field K is inversive.

Corollary 7.1.19 Let K, L, Ky, and Ly be as in Proposition 7.1.16 andlet the field extension L/K be primary. Furthermore, let P be a prime σ∗-idealof Ky. Then

(i) PL is a prime σ∗-ideal in Ly of the same dimension as P .(ii) If Σ is a complete set of parameters of P , then Σ is a complete set of

parameters of PL, P = PL

⋂Ky, and Eord (Σ)PL = Eord (Σ)P .

PROOF. By Theorem 1.6.40(ii) and Proposition 1.6.49(i), there exists ageneric zero η of P such that K〈η〉 and L are quasi-linearly disjoint over K.Applying Theorem 7.1.17 we obtain the conclusion of the corollary.

Theorem 7.1.20 Let K be an ordinary difference field with a basic set σ = α,L a σ-overfield of K, and K ′ the algebraic closure of K in L (clearly, K ′ is aσ-field). Let Ky and Ly be the rings of σ-polynomials in σ-indeterminatesy1, . . . , ys over K and L, respectively, and let P be a prime σ∗-ideal in Ky.Furthermore, let P1, . . . , Pk be essential prime divisors of PL in Ly and letP ′

1, . . . , P′m be essential prime divisors of PK′ in K ′y. Then k = m and,

after a suitable reordering of P1, . . . , Pk one has(i) dimPi = dimP ′

i for i = 1, . . . , k.(ii) Every complete set Σ of parameters of P ′

i is a complete set Σ of parametersof Pi, and Eord (Σ)Pi = Eord (Σ)P ′

i (1 ≤ i ≤ k).(iii) P ′

i = Pi

⋂K ′y (1 ≤ i ≤ k).

PROOF. By Corollary 7.1.19, each P ′iL is prime σ∗-ideal of the ring

Ly. Since PL = PK′L, one can apply Proposition 7.1.16 to obtain

that PL =k⋂

i=1

P ′iL.

If 1 ≤ i, j ≤ k and i = j, then P ′iL

⋂K ′y = P ′

i and P ′jL

⋂K ′y = P ′

j ,hence P ′

iL P ′jL. It follows that P ′

iL, 1 ≤ i ≤ k, are the essential primedivisors of PL in Ly and coincide (up to the order) with P1, . . . , Pk. Theother conclusions of the theorem follow from Corollary 7.1.19.

Theorem 7.1.21 Let K be an ordinary difference field with a basic set σ = α,K∗ the inversive closure of K, Ky and K∗y the rings of σ-polynomials inσ-indeterminates y1, . . . , ys over K and K∗, respectively, and P a prime σ∗-idealof Ky. Then


(i) PK∗ is a prime σ∗-ideal of K∗y.(ii) P and PK∗ have the same complete sets of parameters and equal effec-

tive orders with respect to those sets.(iii) PK∗

⋂Ky = P .

PROOF. Let A,B ∈ K∗y and AB ∈ PK∗ . Then there exists m ∈N such that αm(A), αm(B) ∈ Ky and the σ-polynomial αm(A)αm(B) isobtained from elements of P with the use of shuffling and taking linear combi-nations over K (see the description of the construction of a perfect σ-ideal con-sidered after Definition 2.3.1). Since the ideal P is perfect, αm(A)αm(B) ∈ P .Then αm(A) ∈ P or αm(B) ∈ P , hence A ∈ PK∗ or B ∈ PK∗ . It followsthat the σ∗-ideal PK∗ is prime. Since the compatibility condition required inthe assumptions of Theorem 7.1.14(iii) is obviously satisfied, the other state-ments of our theorem follow from Theorem 7.1.14(iii).

7.2 Existence Theorem for Ordinary AlgebraicDifference Equations

The following fundamental result is an abstract form of existence theorem forordinary algebraic difference equations. This theorem was first formulated andproved by R. Cohn in [28] where he used the technique of characteristic setsof ordinary difference polynomials. We present here another proof obtained byR. Cohn, the proof he used in his book [41]. It is based on the technique of differ-ence kernels developed in Section 5.2. (Historically, the first attempt to obtainan existence theorem for an algebraic difference equation was made by J. Ritt[162] who considered the case of difference polynomial of first order. Despite thefact that J. Ritt did not obtain a satisfactory general result, his idea of usingthe technique of characteristic sets developed in [165] (where they are called“basic sets”) appeared to be very fruitful for the study of abstract algebraicdifference equations of any order. In particular, it has led to the developmentof the contemporary technique of characteristic sets of partial difference poly-nomials, which is an efficient tool in the study of abstract algebraic differenceequations.)

Theorem 7.2.1 Let K be an ordinary difference field with a basic set σ,Ky1, . . . , ys the ring of σ-polynomials in σ-indeterminates y1, . . . , ys over K,and A an irreducible σ-polynomial in Ky1, . . . , ys. Then

(i) The variety M(A) has principal components.(ii) Suppose that A contains a transform of some yi (1 ≤ i ≤ s). Then

(a) If A contains a transform of order 0 of some σ-indeterminate yj andM is a principal component of M(A), then ord (y1, . . . , yi−1, yi+1, . . . , ys)M =ordyi

A.

7.2. EXISTENCE THEOREM 403

(b) Let M1, . . . ,Mk be the principal components of M(A) and let d ande denote, respectively, the degree and reduced degree of A with respect to thehighest transform of yi which appears in A. Then

k∑j=1

ld (y1, . . . , yi−1, yi+1, . . . , ys)Mj ≤ d and

k∑j=1

rld (y1, . . . , yi−1, yi+1, . . . , ys)Mj = e.

(c) If d1 and e1 denote, respectively, the degree and reduced degree of A withrespect to the lowest transform of yi contained in A, then

k∑j=1

ild (y1, . . . , yi−1, yi+1, . . . , ys)Mj ≤ d1 and

k∑j=1

irld (y1, . . . , yi−1, yi+1, . . . , ys)Mj = e1.

(d) Let q = EordykA (1 ≤ k ≤ s) and let M be a component of M(A)

such that y1, . . . , yk−1, yk+1, . . . , ys is a complete set of parameters of Mand Eord (y1, . . . , yk−1, yk+1, . . . , ys)M = q. Then M a principal componentof M(A).

PROOF. We divide the proof in two parts. The first part is devoted to theproof of statement (i) and statements (a) - (c) of (ii) in the case of inversivedifference field K. The second part completes the proof of the theorem in thegeneral case.

PART I. Suppose, first, that the σ-field K is inversive and for every i =1, . . . , s, some transform of yi appears in A. Without loss of generality we canassume that A contains transforms of order zero of some σ-indeterminates yj .(Otherwise, A can be replaced by an appropriate σ-polynomial of the formα−k(A) where k is a positive integer.)

Thus, without loss of generality, we assume that A is of order zero withrespect to y1, . . . , yp (1 ≤ p ≤ s) and of positive order with respect to every yj

with p < j ≤ s.We are going to reduce the problem of solving the equation A = 0 to finding

a realization of certain kernel R = (K(a0, a1), τ) of length one. The following isa description of R.

Suppose, first, that p < s, that is, A is not of order zero with respectto every yj (1 ≤ j ≤ s). Let mi = ordyi

A for i = p + 1, . . . , s and letm′ = p+mp+1 + · · ·+ms. We introduce a0 and a1 as m′-tuples (a(1)

0 , . . . , a(m′)0 )

and (a(1)1 , . . . , a

(m′)1 ), respectively and say that the first p coordinates of a0 are

assigned, respectively, to y1, . . . , yp, next mp+1 coordinates of a0 are assigned,respectively, to yp+1, αyp+1, . . . , α

mp+1−1yp+1, . . . , the last ms coordinates of


a0 are assigned, respectively, to ys, αys, . . . , αms−1ys. Similarly, we say that the

first p coordinates of a1 are assigned, respectively, to αy1, . . . , αyp, next mp+1 co-ordinates of a1 are assigned, respectively, to αyp+1, α

2yp+1, . . . , αmp+1yp+1, . . . ,

the last ms coordinates of a1 are assigned, respectively, to αys, α2ys, . . . , α

msys.Now we define the isomorphism τ sending a

(i)0 to a

(i)1 (1 ≤ i ≤ m′) as follows.

First, if a coordinate a(j)1 of a1 is assigned to some αkyi and also some coordinate

a(j′)0 of a0 is assigned to αkyi, then we set a

(j)1 = a

(j′)0 . Let

Y = yi | 1 ≤ i ≤ p ∪ αkyj | p + 1 ≤ j ≤ s, 0 ≤ k ≤ mj − 1and for every αkyj ∈ Y (including the case k = 0 in which we obtain elementsin yi | 1 ≤ i ≤ p), let bij denote the coordinate of a0 assigned to αjyi, or thecoordinate of a1 assigned to αjyi if no coordinate of a0 corresponds to this term.

To complete the description of τ (that is, a description of a0 and a1) itremains to define bij . Since A is irreducible, (A) is a prime ideal of the ringK[Y ] consisting of multiples of A. Let us define bij as coordinates of an m-tuple which is a generic zero of (A). Then A vanishes when every term αkyj

in A is replaced by its assigned coordinate of a0 or a1. Let us show that thecoordinates of a0 are algebraically independent over K. Indeed, if it is not so,then the coordinates of a0 annul some polynomial B ∈ K[Y ] which does notcontain any of the terms αmiyi, p + 1 ≤ i ≤ s. Then B is annulled by bij

whence B is a multiple of A. However, this is impossible, since A contains termsαmiyi. Similarly, using the fact that A contains y1, . . . , yp, we obtain that thecoordinates of a1 are algebraically independent over K. Now we can define τ asan isomorphism of K(a0) onto K(a1) which maps each component of a0 to thecorresponding component of a1 and whose restriction on K coincides with α.The kernel (K(a0, a1), τ) will be denoted by R.

Now, let us consider the case p = s, that is, assume that ordyiA = 0 for

i = 1, . . . , s. Let a0 = (a(1)0 , . . . , a

(s)0 ) be a generic zero of the ideal (A) in K[Y ]

(a coordinate a(i)0 , 1 ≤ i ≤ s, is assigned to yi). Then we define R to be the

kernel of length zero formed by the field K(a).It is easy to see that the solutions of the equation A = 0 are determined by

realizations of the kernel R. (If ζ is a realization of R, then a solution of theequation A = 0 can be obtained by equating each yi, 1 ≤ i ≤ s, to the coordinateof a0 assigned to yi; and all solutions of the equation can be obtained in thisway.)

Let M1, . . . ,Md be the varieties over K whose generic zeros are the distinctprincipal realizations of the kernel R we have constructed. Obviously, every Mi,1 ≤ i ≤ d, is contained in M(P ). We are going to show that M1, . . . ,Md areprincipal components of M(P ), while all other irreducible components of M(P )are singular.

Suppose that ζ is a principal realization of R which gives a generic zeroof a principal component M of M(P ). Then ζ is a regular realization of R.Indeed, if it is not, then M annuls a σ-polynomial B ∈ Ky1, . . . , ys such thatordyi

B ≤ ordyiA for i = 1, . . . , s and B /∈ (A). Taking i such that yi appears in

A, let us consider the resultant R of A and B with respect to yi (in this case A


and B are treated as polynomials in several variables of the form αkyj with yi

among them). Clearly, R = 0 and EordyiR < Eordyi

A. Applying Proposition7.1.2 we obtain that either y1, . . . , yi−1, yi+1, . . . , ys is not a set of parametersof M or Eord(y1, . . . , yi−1, yi+1, . . . , ys)M < Eordyi

A that contradicts the factthat M is a principal component of M(P ).

Since ζ is a regular realization of R, it is a principal realization. (If it isnot, then Theorem 5.2.10(v) implies that s − 1 = σ-trdegKK〈ζ〉 < δR. Thisis, however, impossible, since a principal realization of R gives a solution ofA = 0 and hence, by Proposition 7.1.2, generates a σ-field extension of Kof σ-transcendence degree less than s. By Theorem 5.2.10(i), we should haveδR < s.)

Now we are going to prove that M1, . . . ,Md are principal components ofM(P ) that satisfy conditions (a) - (c) in part (ii) of the theorem.

Let M denote one of the Mi (1 ≤ i ≤ d), let ζ be the realization of R fromwhich M is obtained, and let η be the generic zero of M consisting of the scoordinates of ζ corresponding to the coordinates of a0 assigned to the yi. Letus construct a special set b0 for R. If p < s, we choose k ∈ N, p + 1 ≤ k ≤ sand define b0 to be a subindexing of a0 consisting of components assigned toy1, . . . , yp and αmi−1yi with i = p + 1, . . . , s, i = k. The coordinates of a0

and b1 are the elements bij other than bkmkused in the construction of a0

and a1 at the beginning of the proof. Since αmkyk appears in A, an argumentused in the construction of the kernel R shows that these coordinates form anindexing over K whose coordinates are algebraically independent over K. Itfollows that coordinates of b1 are algebraically independent over K(a0). On theother hand, our construction of the elements bij shows that bkmk

is algebraicallydependent on the set bij | (i, j) = (k,mk) over K. Therefore, all bij , and henceall coordinates of a1, are algebraic over K(a0, b1). Thus, b0 is a special set forR. (If p = s, any s−1 coordinates of a0 form a special set.) Since our special sethas s−1 coordinates, δR = s−1, hence dimM = s−1 (see Theorem 5.2.10(i)).

Let us show that any s − 1 σ-indeterminates in the set y1, . . . , ys forma set of parameters of M. (The coordinates of a0 assigned to such a set ofparameters may not form a special set, as it happens when 1 ≤ p ≤ s andthe omitted coordinate is some yk with k ≤ p.) To prove this, let us take anyk ∈ 1, . . . , s and consider A as a σ-polynomial in one σ-indeterminate yk withcoefficients in the σ-ring Ky1, . . . , yk−1, yk+1, . . . , ys. Since these coefficientsare not multiples of A, they do not vanish when all yj are replaced by ηj . Itfollows that ηk is σ-algebraically independent over K〈η1, . . . , ηk−1, ηk+1, . . . , ηs〉,hence the elements η1, . . . , ηk−1, ηk+1, . . . , ηs form a σ-transcendence basis ofK〈η1, . . . , ηs〉 over K. Applying Proposition 7.1.2 we obtain that that ordyk

Aand Eordyk

A are upper bounds for ord(y1, . . . , yk−1, yk+1, . . . , ys)M andEord(y1, . . . , yk−1, yk+1, . . . , ys)M, respectively. This determines ordyk

A andEordyk

A for 1 ≤ k ≤ p.Now suppose that p < s (that is, A is not of order zero in every yi, 1 ≤ i ≤ s)

and let k ∈ p + 1, . . . , s. Let c0 be a subindexing of a0 consisting of thecoordinates assigned to terms αjyi with k ∈ N, 1 ≤ i ≤ s, i = k. Then c0


contains a special set. The s− 1 coordinates of c1 assigned to αy1, . . . , αyp andto αmiyi with p + 1 ≤ i ≤ s, i = k, form an indexing e0 whose coordinates arealgebraically independent over K(c0) and also over K(a0). By the construction,the remaining coordinates of c1 are equal to the corresponding coordinates ofc0, whence trdegK(c0)K(c0, c1) = s − 1. The definition of e0 and Proposition1.6.31(iii) imply that

trdegK(c0,c1)K(a0, c1) = trdegK(c0,e0)K(a0, e0) = trdegK(c0)K(a0) .

Since coordinates of a0 are algebraically independent over K and mk of thesecoordinates do not belong to c0, trdegK(c0)K(a0) = mk. It follows that ordc0Ris defined and

ordc0R = trdegK(c0,c1)K(a0, a1) = trdegK(c0,c1)K(a0, c1) = mk.

By Theorem 5.2.10(ii),

ord(y1, . . . , yk−1, yk+1, . . . , ys)M = trdegK〈η1,...,ηk−1,ηk+1,...,ηs〉K〈η〉

= trdegK〈c0〉K〈ζ〉 = ordc0R = mk.

Let us show that, without loss of generality, we can assume that EordykA =

ordykA. Indeed, let q = Eordyk

A and t = mk − q. Replacing each αiyk in Awith αi−tyk we obtain a σ-polynomial A′ such that ordyk

A′ = EordykA′ = q.

Now we can construct a kernel R′ for A′ as the kernel R was constructed forA. Dropping the coordinates of ζ assigning to yk, αyk, . . . , αt−1yk one obtainsa regular realization ζ ′ of R′. Since σ-trdegKK〈ζ ′〉 = σ-trdegKK〈ζ〉 = s − 1and δR′ = s − 1, ζ ′ is a principal realization of R′ (see Theorem 5.2.10(v)).This realization gives a solution η′ = (η1, . . . , ηk−1, ηk+1, . . . , ηs) of A′. Fur-thermore, because of the equality EordK〈η〉/K〈η1, . . . , ηk−1, ηk+1, . . . , ηs〉 =EordK〈η′〉/K〈η1, . . . , ηk−1, ηk+1, . . . , ηs〉, the consideration of the effective or-der of A can be replaced by the study of the effective order of A′. Also, applyingCorollaries 4.3.6 and 4.3.8, we obtain that

ld(K〈η〉/K〈η1, . . . , ηk−1, ηk+1, . . . , ηs〉) = ld(K〈η′〉/K〈η1, . . . , ηk−1, ηk+1,

. . . , ηs〉) and rld(K〈η〉/K〈η1, . . . , ηk−1, ηk+1, . . . , ηs〉)= rld(K〈η′〉/K〈η1, . . . , ηk−1, ηk+1, . . . , ηs〉).

Thus, A′ can be used instead of A in the proof of part (b), so in the followingdiscussion of (b) we assume that Eordyk

A = ordykA, that is, mk = q.

Let u0 be the subindexing of a0 consisting of coordinates assigned to y1, . . . ,yk−1, yk+1, . . . , ys and let v0 denote the subindexing of a0 consisting of coordi-nates assigned to the terms of the sety1, . . . , yp

⋃αmi−1yi | p+1 ≤ i ≤ s, i =k. As before, let c0 be the subindexing of a0 consisting of coordinates assignedto the set of all transforms of yi with i = k. Then u0 ⊆ c0, v0 ⊆ c0, coordinatesof u0 are algebraically independent over K(a1), and v0 is a special set.


As we have seen, ordc0R is defined and trdegK(c0,c1)K(a0, a1) = q. ByTheorem 5.2.8(i), for any positive integers r, n and for any s-tuples a0, . . . , ar+n

obtained by successive prolongations of the kernel R, one has

trdegK(cr,...,cr+n)K(ar, . . . , ar+n) = q.

Repeatedly applying Theorem 5.2.7(ii), we obtain that the set Ur =r−1⋃i=0

ui is

algebraically independent over K(ar, . . . , ar+n), whence

trdegK(cr,...,cr+n,Ur)K(ar, . . . , ar+n, Ur) = q.

By Theorem 5.2.4, the set Vr+n =∞⋃

i=r+n+1

vi is algebraically independent over

K(ar, . . . , ar+n) and, therefore, over K(ar, . . . , ar+n, Ur). Since∞⋃

i=0

ci coincides

with the union of Ur, Vr+n, cr, . . . , cr+n, we have

q = trdegK(cr,...,cr+n,Ur,Vr+n)K(ar, . . . , ar+n, Vr+n)

= trdegK(⋃∞

i=0 ci)K(ar, . . . , ar+n,

∞⋃i=0

ci).

The fact that n is any positive integer implies that

trdegK(K(⋃∞

i=0 ci)K(∞⋃

i=0

ci

⋃ ∞⋃j=r

aj) = q,

and since r is also an arbitrary positive integer, we obtain that

EordK〈η〉/K〈η1, . . . , ηk−1, ηk+1, . . . , ηs〉 = q.

By Proposition 7.1.9, η is a generic zero of a principal component of M(A).Thus, the varieties M1, . . . ,Md are principal components of M(A) and satisfycondition (a).

Considering the subindexing v0 of a0, one can easily see that dv0R andrdv0R are defined and are the degree and the reduced degree, respectively, of Ain αmkyk. Let w0 be the family of coordinates of ζ that correspond to coordi-nates of v0. Since ld(K〈η〉/K〈η1, . . . , ηk−1, ηk+1, . . . , ηs〉 = ld(K〈η〉/K〈w0〉 andrld(K〈η〉/K〈η1, . . . , ηk−1, ηk+1, . . . , ηs〉 = rld(K〈η〉/K〈w0〉 (as it follows fromCorollaries 4.3.6 and 4.3.8), the inequality and equality in part (b) follow fromTheorem 5.2.10(iii) for yi with p + 1 ≤ i ≤ s. Similarly one can obtain property(b) when ordyi

A = 0 for all i = 1, . . . , s.Let m = maxmp+1, . . . , ms and let K ′ denote the inverse difference field of

K, that is, the field K treated as a difference field with the basic set σ′ = α−1.(Recall that this concept was introduced in Section 4.3 (before Exercises 4.3.11)


and used in the definition of inverse limit degree and inverse reduced limitdegree of a difference field extension). Let K ′z1, . . . , zs be the ring of differencepolynomials in difference indeterminates z1, . . . , zs over K ′, and let A′ be thedifference polynomial in this ring obtained from A by the substitution of αm−izk

for the αiyk (1 ≤ k ≤ s, 0 ≤ i ≤ mi for i = p + 1, . . . , s). Let L be a σ-overfield of K and let L′ denote the inverse field of the inversive closure ofL. (Clearly, L′ can be considered as a difference overfield of K ′.) If an s-tupleη = (η1, . . . , ηs) with coordinates in L is a generic zero of a principal componentM of M(A), then αmη = (αm(η1), . . . , αm(ηs)) is a generic zero of a principalcomponent M′ of M(A′). Indeed, it is clear that αmη annuls A′, and, wheneverλ is a subindexing of the coordinates of η and λ′ the corresponding subindexingof the coordinates of αmη, we have σ-trdegKK〈λ〉 = σ′-trdegK′K ′〈λ′〉 andEordK〈η〉/K〈λ〉 = EordK ′〈αmη〉/K ′〈λ′〉.

It is easy to see that we have obtained an one-to-one correspondence betweenprincipal components of M(A) and principal components of M(A′). Further-more, this correspondence preserves the relative inverse limit degrees and inversereduced limit degrees of the corresponding principal components. Applying thepart of (b) which has already be proven for A′, we obtain (c) except for those yi

for which ordyiA > 0 but Eordyi

A = 0. In these cases, however, the statements(b) and (c) are equivalent, as it follows from Proposition 4.3.12, and it is easyto check that either (b) or (c) holds in each of such a case. This completes PartI of the proof.

Part II. Now we do not assume that the σ-field K is inversive.Part II(a). Let us still suppose that for every i = 1, . . . , s, some transform of

yi appears in A.Let K∗ denote the inversive closure of K. Since the σ-polynomial A is irre-

ducible in the ring Ky1, . . . , ys, each irreducible factor of A in K∗y1, . . . , yscontains precisely those αjyi which appear in A. Using equality (1.6.2) we obtainthat the sum of reduced degrees in any αjyi of the distinct irreducible factors isthe reduced degree of A in αjyi; also the sum of degrees in αjyi of the distinctirreducible factors is less than or equal to degαjyi

A.Let η be an s-tuple with coordinates in a σ-overfield of K. Then σ-

trdegK∗K∗〈η〉 = σ-trdegKK〈η〉, and for any subindexing λ of coordinates of η,EordK∗〈η〉/K∗〈λ〉 = EordK〈η〉/K〈λ〉 (since the effective orders are defined interms of the inversive closures and (K〈η〉)∗ = (K∗〈η〉)∗, (K〈λ〉)∗ = (K∗〈λ〉)∗ ).It follows that η is a generic zero of a principal component of the variety M(A)of A over K if and only if η is a generic zero of a principal component of thevariety of an irreducible factor of A in K∗y1, . . . , ys. Clearly s-tuples η′, η′′

are generic zeros of distinct varieties over K if and only if they are generic zerosof distinct varieties over K∗. (Indeed, if B is a σ-polynomial in K∗y1, . . . , ysand just one of η′, η′′ annuls B, then for sufficiently large r ∈ N, αr(B) lies inKy1, . . . , ys and is annulled by just one of η′, η′′.) Thus, the generic zeros ofthe principal components of the variety of A as a σ-polynomial in Ky1, . . . , ysare the same as the generic zeros of the principal components of the irreduciblefactors of A in K∗y1, . . . , ys. Also, no two such factors share a principalcomponent. (Such a component would annul the resultant R of the factors with


respect to the highest transform of ys that appears in A. Since R = 0 andEordys

R < EordysA, this is impossible.) Statements (b) and (c) for A now

follow from the already proven corresponding statements for the irreduciblefactors of A in K∗y1, . . . , ys. To prove (a) suppose that A contains a trans-form of order 0 of some yj (1 ≤ j ≤ s). Then, with η and λ as before, we haveordK〈η〉/K〈λ〉 ≥ ordK∗〈η〉/K∗〈λ〉 (this is just the inequality in Proposition1.6.31(iii)). Therefore, if M is a principal component of M(A) and 1 ≤ i ≤ s,then ord (y1, . . . , yi−1, yi+1, . . . , ys)M ≥ ordyi

A. Applying Proposition 7.1.2 weobtain that the last inequality is actually an equality. This completes the proofof (a) and the whole theorem with the restriction stated at the beginning ofPart II of the proof.

Part II(b). Now we are going to remove the last restriction and assume thatthere are σ-indeterminates yi such that no transform of yi appears in A. Letyj1 , . . . , yjt

be all σ-indeterminates yj such that some transform of yj appearsin A and let A denote the σ-polynomial A considered as an element of thering of σ-polynomials Kyj1 , . . . , yjt

. Furthermore, let ξ = (ξ1, . . . , ξs−t) be ageneric zero of a principal component of M(A) and, let θ be an indexing withs − t coordinates which are σ-algebraically independent over K〈ξ〉. Then thecoordinates of ξ and θ arranged in proper order form an s-tuple η in M(A).

Since σ-trdegKK〈ξ〉 = t − 1, we have σ-trdegKK〈η〉 = s − 1. Furthermore,if λ is a subindexing of ξ, then any set of elements of the form αjξi (1 ≤ i ≤s − t, j ∈ N) algebraically independent over K〈λ〉 is algebraically independentover K〈λ, θ〉. Therefore, for any k ∈ N, any transcendence basis of the setαjξi | j ≥ k over K〈λ〉 is also a transcendence basis of this set over K〈λ, θ〉.Since K〈η〉 = K〈λ, θ〉〈ξ〉 and K〈ξ〉 = K〈λ〉〈ξ〉, we obtain that

ordK〈η〉/K〈λ, θ〉 = ordK〈ξ〉/K〈λ〉(if one takes k = 0) and

EordK〈η〉/K〈λ, θ〉 = EordK〈ξ〉/K〈λ〉(if we take k to be sufficiently large).

Furthermore, it follows from Theorem 1.6.28(v) that

ld(K〈η〉/K〈λ, θ〉) = ld(K〈ξ〉/K〈λ〉)and the corresponding equalities hold for the reduced, inverse, and inversereduced limit degrees. Therefore, η is a generic zero of a principal componentof M(A), and to verify (a), (b), and (c) it is sufficient to show that any genericzero of a principal component of M(A) can be obtained in the same way as weobtained η in the above considerations. In other words, given such a generic zeroη, one should show that the subindexing ξ of η corresponding to yj1 , . . . , yjt

isa generic zero of a principal component of A, and the subindexing θ consistingof the remaining components η is σ-algebraically independent over K〈ξ〉.

Since ξ is a solution of A and θ has s − t coordinates, σ-trdegKK〈ξ〉 < tand σ-trdegK〈ξ〉K〈ξ, θ〉 ≤ s − t. Also, σ-trdegK〈ξ〉K〈ξ, θ〉 + σ-trdegKK〈ξ〉 = σ-trdegKK〈η〉 = s − 1 whence σ-trdegK〈ξ〉K〈ξ, θ〉 = s − t and σ-trdegKK〈ξ〉 =


t − 1. It follows that the coordinates of θ are σ-algebraically independentover K〈ξ〉, and ξ satisfies the condition on the σ-transcendence degree for ageneric zero of a principal component of M(A) stated in Proposition 7.1.9.If λ is any subindexing of ξ, then one can proceed as before and obtain thatEordK〈η〉/K〈λ, θ〉 = EordK〈ξ〉/K〈λ〉, so that ξ satisfies condition (7.1.3).Applying Proposition 7.1.9 we complete the proof of parts (a) - (c) of ourtheorem in the general case.

To prove part (d) we first notice that the justification presented at the be-ginning of the proof allows one to assume that K is inversive, Eordyk

= ordyk

and p < s. (One can easily modify the proof for p = s; we leave this modificationto the reader as an exercise.)

Let η be a generic zero of M (we use the notation of statement (d) ofthe theorem). If M is not a principal component of M(A), then η is not aregular realization of the kernel R (see Theorem 5.2.10(v)). In this case η annulssome polynomial B in the set of indeterminates αjyi |, 1 ≤ i ≤ p, j = 0, 1 orp + 1 ≤ i ≤ s, j = 0, . . . , mi such that A does not divide B. Let R be theresultant of A and B with respect to αqyk. Then R = 0 and Eordyk

R < EordykA

if q > 0. If q = 0, then the resultant R1 of R and α(A) with respect to αyk

is a nonzero polynomial in y1, . . . , yk−1, yk+1, . . . , ys. Since η annuls R and R1,we arrive at a contradiction with the result of Proposition 7.1.2. This completesthe proof of the theorem.

Definition 7.2.2 With the notation of Theorem 7.2.1, let q = EordyiA and let

M be a component of M(A) such that y1, . . . , yi−1, yi+1, . . . , ys is a completeset of parameters of M and Eord (y1, . . . , yi−1, yi+1, . . . , ys)M = q. Then M iscalled a principal component of M(A) with respect to yk. (By Theorem 7.2.1(d),M is a principal component of M(A). )

R. Cohn [28] suggested the following procedure for determining the prin-cipal components of M(A) with respect to some yk, 1 ≤ k ≤ s. (We keepthe above notation; in particular A is still an irreducible σ-polynomial inKy1, . . . , ys.) First, one should replace each yi in A with an element ηi

such that the set ηi | 1 ≤ i ≤ s, i = k is σ-algebraically independentover K. Let A′ denote the resulting σ-polynomial in the σ-polynomial ringK〈η1, . . . , ηk−1, ηk+1, . . . , ηs〉yk and let L denote the inversive closure ofK〈η1, . . . , ηk−1, ηk+1, . . . , ηs〉. Applying the procedure described in Example5.2.6 we can find principal components of the irreducible factors of A′. If ηk isa generic zero of such a principal component, then (η1, . . . , ηs) is a generic zeroof a principal component of M(A) with respect to some yk.

Unfortunately, this procedure does not determine whether the principal com-ponents with respect to different yi are identical.

The following statement is an application of Theorem 7.2.1 to the descriptionof proper irreducible varieties of maximal dimension.

Theorem 7.2.3 Let K be an ordinary difference field with a basic set σ =α, Ky1, . . . , ys the algebra of σ-polynomials in σ-indeterminates y1, . . . , ys


over K, and M an irreducible variety over Ky1, . . . , ys. The the followingconditions are equivalent.

(i) dimM = s − 1.(ii) M is a principal component of the variety of an irreducible σ-polynomial

in Ky1, . . . , ys.PROOF. Since the implication (ii) ⇒ (i) is obvious, we just need to show

that if M is an irreducible variety of dimension s − 1, then there is an irre-ducible σ-polynomial A ∈ Ky1, . . . , ys such that M is a principal componentof M(A). Without loss of generality we can assume that y1, . . . , ys−1 is theset of parameters of M. Then every nonzero σ-polynomial in the σ-ideal Φ(M)contains some transform of ys.

Let η = (η1, . . . , ηs) be a generic zero of M. By Corollary 4.1.20, if q =Eord(y1, . . . , ys−1)M, then ηs annuls a nonzero σ-polynomial of effective orderq in K〈η1, . . . , ηs−1〉ys and therefore some σ-polynomial C of effective orderq in Kη1, . . . , ηs−1ys. Replacing every ηi in C with yi (1 ≤ i ≤ s − 1) weobtain a σ-polynomial B ∈ Φ(M) whose effective order in ys is q. Let A be anirreducible factor of B which lies in Φ(M). Then A contains some transformof ys, Eordys

A ≤ q, and M is contained in an irreducible component M′ ofM(A), since a generic zero of M lies in one of such components. By the firstpart of Proposition 7.1.7, dimM′ = s − 1, while the second part of the sameProposition leads to the inequality Eord(y1, . . . , ys−1)M ≥ q. Now Proposition7.1.2 implies that the last inequality is actually an equality, so one can apply thelast statement of Theorem 7.2.1 and obtain that M′ is a principal componentof M(A). By Proposition 7.1.7, M′ = M.

Generally speaking, it is not true that limit degrees of the different principalcomponents of the variety of an irreducible difference polynomial are equal (seeExercise 4.3.27). At the same time we have the following statement.

Proposition 7.2.4 Let K be an ordinary difference field with a basic set σ =α, let Ky be an algebra of σ-polynomials in one σ-indeterminate y overK, and let A be an irreducible σ-polynomial in Ky of order 0. Then, if anycomponent of the variety M(A) has limit degree 1, then all components of M(A)are of limit degree 1.

PROOF. Let M1, . . . ,Mr be the components of M(A) and let η(1), . . . , η(r)

be their generic zeros, respectively. (η(i) = (η(i)1 , . . . , η

(i)s ) for i = 1, . . . , r; as

usual, by a transform αj(η(i)) we mean the s-tuple (αj(η(i)1 ), . . . , αj(η(i)

s ). ) SinceEordA = 0, all components Mi are principal ones.

For every i = 1, . . . , r, let Ni denote the normal closure of the field Ki =K(η(i), α(η(i)), α2(η(i)), . . . ) over K. As the splitting field over K of the sys-tem A,α(A), . . . regarded as polynomials in indetermiantes y, α(y), . . . , respec-tively (see Theorem 1.6.13(iii) ), the fields Ni are K-isomorphic. By Proposition4.3.12(ii), ld(K〈η(i)〉/K) = 1 if and only if Ki : K < ∞. It remains to noticethat Ki/K is finite if and only if Ni/K is finite (see Theorem 1.6.13(iv)) andthe last condition is satisfied for every Ni or for none.


Exercise 7.2.5 Show that statement similar to Proposition 7.2.4 hold for thereduced limit degree, inverse limit degree, and inverse reduced limit degree.

7.3 Existence of Solutions of DifferencePolynomials in the Case of Two Translations

Unfortunately, the result of Theorem 7.2.1 cannot be extended to the case ofpartial difference polynomials. The following example is due to I. Bentsen [10].

Example 7.3.1 As in Example 5.1.17, let us consider Q as a difference fieldwith a basic set σ = α1, α2 where α1 and α2 are the identity automorphisms.Let b and i denote the positive square root of 2 and the square root of −1,respectively (we assume b, i ∈ C), and let us treat F = Q(b, i) as a σ-overfieldof Q where the extensions of α1 and α2 to automorphisms of F are defined asfollows: α1(b) = −b, α1(i) = i, α2(b) = b, α2(i) = −i. It is easy to show that theσ-polynomial A = y2 − b in one σ-indeterminate y over F is irreducible in Fyand A has no solutions. Indeed, if η is a solution of A, then η2 = b which, asdemonstrated in Example 5.1.17, is impossible.

The following theorem is a version of the existence theorem for the case ofdifference polynomials over an inversive difference field with two translations.

Theorem 7.3.2 (Bensten). Let K be an inversive difference field with a basicset σ consisting of two translations. Let R = Ky1, . . . , ys be the ring of σ-polynomials in σ-indeterminates y1, . . . , ys over K. Furthermore, suppose thatthere exists a translation α ∈ σ such that if K is treated as an ordinary differencefield with the basic set σ′ = σ \ α (we denote this σ′-field by K ′), then everytwo σ′-field extensions of K ′ are compatible.

Then every irreducible σ-polynomial A ∈ R\K has a solution ζ = (ζ1, . . . , ζs)with the following properties.

(i) ζ is not a proper specialization over K of any solution of A.(ii) If A contains a transform of some yk (1 ≤ k ≤ s), then elements

ζ1, . . . , ζk−1, ζk+1, . . . , ηs are σ-algebraically independent over K. Furthermore,if a σ-polynomial B ∈ R\K is annulled by η and contains only those transformsof yk which are contained in A, then B is a multiple of A.

PROOF. Let α and β denote the translations of K (so that σ = α, β).Without loss of generality we can assume that K ′ in the condition of the theoremis the field K treated as a difference field with the basic set σ′ = α and theσ-polynomial A has the following property: there exist p, q ∈ 1, . . . , s suchthat some transforms of the form αiyp and βjyq (i, j ∈ N) appear in A. (SinceK is inversive, one can apply an appropriate transform αdβe (d, e ∈ Z) to A andobtain a σ-polynomial with the described condition.) We say that a differencepolynomial with this property is in standard position.

7.3. EXISTENCE THEOREM. THE CASE OF TWO TRANSLATIONS 413

Let us start with the case with the following condition: for every k = 1, . . . , s,some transform of yk appears in A. Let W denote the set of all transforms αiβjyk

(i, j ∈ N, 1 ≤ k ≤ s) which appear in A. Furthermore, for every k ∈ 1, . . . , s,let mk denote the maximum value of j such that αiβjyk ∈ W for some i. Finally,we denote by h an integer between 0 and s such that mk = 0 for k ≤ h andmk > 0 for k > h.

Assuming, first, that h < s, consider the set

Y = βjyk | k ≤ h; j = 0, 1 ∪ βjyk | k > h, 1 ≤ j ≤ q

with t =s∑

k=2h+1

(mk + 1) elements. Let us order the elements of Y according

to the lexicographic order with respect to (k, j) (we denote the elements of Yarranging according to this order by zν , 1 ≤ ν ≤ t): z1 = y1, z2 = βy1, z3 =y2, . . . , z2h = βyh (so that βjyk = z2k−1+j for k = 1, . . . , h; j = 0, 1), z2h+1 =yh+1, z2h+2 = βyh+1, . . . , z2h+(mh+1+1) = βmh+1yh+1, . . . , zt = βmsys (so that

βjyk = zν with ν = k + h + j +k−1∑

i=h+1

mi if k > h, 0 ≤ j ≤ mk). Denoting

the indexing (zν)1≤ν≤t by z we obtain an ordinary difference polynomial ringK ′z with the basic set σ′ = α (z1, . . . , zt are σ′-indeterminates in this ring).Clearly, W ⊆ K ′z and A can be considered as a σ′-polynomial in K ′z. Letξ = (ξ1, . . . , ξt) be a generic zero of a principal component of A over K ′z (theexistence of such a generic zero follows from Theorem 7.2.1).

Let Z0 be the subindexing of z consisting of all coordinates zν except thecoordinates that are equal to βyk with k ≤ h or βmj yj with j > h. Further-more, let Z1 be the subindexing of z consisting of all coordinates zν except thecoordinates that are equal to yk (1 ≤ k ≤ s). Notice that both Z0 and Z1 havel = t − s coordinates and the union of their sets of coordinates is the set ofcoordinates of z. Furthermore, with respect to the ordering of coordinates of Z0

and Z1 induced by the ordering of coordinates of z, if the νth coordinate of Z0

is βjyk, then νth coordinate of Z1 is βj+1yk (1 ≤ ν ≤ s).Let ai = (ai1, . . . , ail) (i = 0, 1) denote the indexing obtained from z by

replacing all coordinates belonging to Zi by the corresponding coordinates ofξ. Then each of the indexings a0 and a1 is σ′-algebraically independent overK ′. Indeed, if r ∈ h + 1, . . . , s, then there exists p ∈ N (p depends on r)such that αpβmryr appears in A. Setting m = r + h + mr +

∑r−1i=h mi, we

obtain that the term αpzm appears in A. Therefore, by the definition of ξ, itfollows that ξ1, . . . , ξm−1, ξm+1, . . . , ξt are distinct and form a σ′-algebraicallyindependent set over K ′. Since m is not in the domain of Z0, we obtain that a0

is σ′-algebraically independent over K ′.Since A is in standard position in the ring Ky1, . . . , ys, there exists k′ ∈

1, . . . , s such that for some p′ ∈ N (p′ depends on k′), the term αp′yr′ appears

in A; that is, for m′ = 2r′ − 1 if r′ ≤ h or m′ = p′ + h +r′−1∑i=h

mi if r′ > h,


the elements ξ1, . . . , ξm′−1, ξm′+1, . . . , ξt are distinct and form a σ′-algebraicallyindependent set over K ′. By the definition of Z1, it follows that a1 is also σ′-algebraically independent over K ′.

Since a0 and a1 are σ′-algebraically independent over K ′, the automorphismβ of K can be extended to an isomorphism τ of K ′〈a0〉 onto K ′〈a1〉 such thatτ(αia0ν) = αia1ν for i = 0, 1, 2, . . . , 1 ≤ ν ≤ l. (As usual, the extension of thetranslation α of K ′ to any σ′-overfield of K ′ is denoted by the same letter α.) Itis easy to see that τ is actually a σ′-isomorphism of K ′〈a0〉 onto K ′〈a1〉 whichcontracts to β on K ′. Furthermore, τ(a0ν) = a1ν , 1 ≤ ν ≤ l. Thus, for the caseh < s we have a difference kernel R = (K ′〈a0〉, τ) for A which is similar to thedifference kernel for the irreducible ordinary difference polynomial in the proofof Theorem 7.2.1.

If h = s, then A can be considered as an ordinary σ′-polynomial in thering K ′y1, . . . , ys. Letting a0 = (a01, . . . , a0s) be a generic zero of a principalcomponent of the variety of A over K ′y1, . . . , ys and setting τ = β (in this casewe view this mapping as a σ′-automorphism of K ′) we again have the desiredkernel R = (K ′〈a0〉, τ).

The fact that every two σ′-field extensions of K ′ are compatible implies thatthe kernel R of length 1 constructed above satisfies the stepwise compatibilitycondition. Furthermore, the compatibility condition on K ′ trivially implies thestepwise compatibility condition on the kernel of length 0. Therefore, in eithercase, Theorems 6.1.12 and 6.2.4 imply that there exists a principal realizationη of the kernel R.

Since for each k ∈ 1, . . . , s, there exists a coordinate a0νkof a0 which is

assigned to yk, a solution for A over Ky1, . . . , ys can be obtained by assigningto yk (1 ≤ k ≤ s) the coordinate of η which corresponds to a0νk

. Let us denotethis solution by ζ = (ζ1, . . . , ζs) (with ζk = ηνk

).Let B be a σ-polynomial in Ky1, . . . , ys\K which contains only terms from

the set W and B is annulled by η. Since η is a regular realization of R, one cantreat B as a σ′-polynomial in K ′z which is annulled by ξ. Since A is in stan-dard position, this σ-polynomial contains a σ-indeterminate of the form βjyr =zµ. If the resultant R over K of A and B with respect to βjyr is not zero, thenR is a nontrivial σ-polynomial containing terms from the set W excluding βjyr,and R is annulled by ξ. Then either ξ1, . . . , ξµ−1, ξµ+1, . . . , ξt are σ′-algebraicallydependent over K ′ or these elements are σ′-algebraically independent over K ′ (inparticular, they are distinct) and EordK ′〈ξ〉/K ′〈ξ1, . . . , ξµ−1, ξµ+1, . . . , ξt〉 <Eordzµ

A. In either case we obtain a contradiction with the definition of ξ. Itfollows that R = 0, and, since A is irreducible, B is a multiple of A.

Recall that for every k ∈ 1, . . . , s, the σ-polynomial A contains some trans-form of yk and is annulled by ζ. Furthermore, if A is considered as a σ-polynomialin yk with coefficients in Ky1, . . . , yk−1, yk+1, . . . , ys, then, as we have seen,these coefficients are not annulled by ζ1, . . . , ζk−1, ζk+1, . . . , ζs. It follows thatthe element ζk is σ-algebraic over K〈ζ1, . . . , ζk−1, ζk+1, . . . , ζs〉 for 1 ≤ k ≤ s.

Let us show that the elements ζ1, . . . , ζk−1, ζk+1, . . . , ζs are σ-algebraicallyindependent over K. To prove this, we consider the kernel R = (K ′〈a0〉, τ)constructed above, and define a subindexing b0 of a0 as follows. If h < k ≤


s, then b0 = (a01, . . . , a0h, a0,h+mh+1 , a0,h+mh+1+mh+2 , . . . , a0,h+mh+1+···+mk−1 ,a0,h+mh+1+···+mk+1 , . . . , a0,h+mh+1+···+ms

); if k ≤ h ≤ s, then b0 = (a01, . . . ,a0,k−1, a0,k+1, . . . , a0h, a0,h+mh+1 , a0,h+mh+1+mh+2 , . . . , a0,h+mh+1+···+ms

).In the first case, τ(b0) = b1 is a subindexing of a1 which is σ′-algebraically

independent over K ′〈a0〉 (it follows from the definition of ξ and the fact that Acontains αiβmkyk for some i, so one can apply Theorem 6.2.12(i)).

In the second case, since some term of the form βiyk appears in A, thecoordinates of b0 are σ′-algebraically independent over K ′〈a1〉 if h < s, and ifh = s, the coordinates of b0 are σ′-algebraically independent over K ′. ApplyingTheorem 6.2.12(ii) we obtain that the subindexing θ of η corresponding to b0 isσ-algebraically independent over K.

Now let us show that ζ is not a proper specialization over K of any solutionof A. Suppose that λ = (λ1, . . . , λs) is a solution of A such that λ (σ-) special-izes over K to the s-tuple ζ. Furthermore, suppose that h < s and consider thel-tuple ω = (ω1, . . . , ωl) of those transforms of coordinates of λ which are sub-stituted into the coordinates of Z0 with ων substituted into the νth coordinateof Z0 (1 ≤ ν ≤ l). Let ω1 be defined in the same way as ω, with the Z0 replacedby Z1. We construct a kernel R′ by contracting the translation β of K〈λ〉 toa σ′-isomorphism of K ′〈ω〉 onto K ′〈ω1〉. With the obvious correspondence ofcoordinates, η is a specialization of ω over K. Therefore, with the appropriateidentification of coordinates indicated in the definitions of z, Z0 and Z1, (η, η1)is a specialization of (ω, ω1) over K ′. The last specialization is generic, since(η, η1) is equivalent to a generic zero of a principal component of the varietyof A over K ′. It follows that ω is a regular realization of R in K〈λ〉. ApplyingTheorem 6.2.9 we obtain that η is a generic specialization of ω over K, henceζ is a generic specialization of λ over K. The case h = s can be treated in thesame way except for the notation changes.

Now we can complete the proof by removing the restriction that each yk

(1 ≤ k ≤ s) has a transform which appears in A. In this case one can proceedas in Part II(b) of the proof of Theorem 7.2.1, with λ interpreted as a solutionof the kind whose existence has been established above for the restricted case.Using Lemma 6.2.7 one easily obtains that ζ is not a proper specialization overK of any other solution of A. Then the verifications of properties (i) and (ii)of the theorem are straightforward (we leave the details to the reader as anexercise).

Corollary 7.3.3 Let a difference field K with two translations and an irre-ducible difference polynomial A in Ky1, . . . , ys be as in Theorem 7.3.2. ThenA has at most finitely many nonequivalent solutions of the type described in thetheorem.

PROOF. By Theorems 2.5.11 and 2.6.3, Ky1, . . . , ys is a Ritt differencering and the variety M(A) of the perfect σ-ideal of Ky1, . . . , ys generatedby A has a unique irredundant representation as the union of finitely manyirreducible varieties. Any solution of A which is not a proper specialization ofany other solution of A is a generic zero of one of these irreducible components


of M(A). It remains to notice that all generic zeros for each such componentof M(A) are equivalent.

Corollary 7.3.4 Let K be an inversive difference field with two translationsand let K be algebraically closed or separably algebraic closed. Then every ir-reducible difference polynomial A of positive degree over K has a solution thathas properties (i) and (ii) of Theorem 7.3.2.

PROOF. Applying Theorem 5.1.6 we obtain that the field K satisfies thehypothesis of Theorem 7.3.2.

A weakening of the hypothesis of Theorem 7.3.2 leads to the following result.

Theorem 7.3.5 Let K be a difference field whose basic set σ consists of twotranslations α and β, and let Ky1, . . . , ys be the ring of σ-polynomials in σ-indeterminates y1, . . . , ys over K. Then the following conditions are equivalent.

(i) K has an algebraic closure which is a σ-field extension of K.(ii) Every algebraically irreducible σ-polynomial A ∈ Ky1, . . . , ys \ K has a

solution η with the properties stated in Theorem 7.3.2.(iii) If a is any element separably algebraic and normal over K, then K may

be extended to a σ-field K〈a〉.PROOF. (i)⇒ (ii). Let L be a σ-overfield of K which is also an algebraic

closure of K. By Proposition 2.1.9(i), the inversive closure L∗ of L is alge-braically closed. Let A ∈ Ky1, . . . , ys \K be an irreducible σ-polynomial andlet B be a nontrivial irreducible factor of A in L∗y1, . . . , ys. Furthermore, letλ = (λ1, . . . , λs) be a solution of B of the type whose existence is established inCorollary 7.3.4. Then λ is a solution of A.

Let M denote an irreducible component of the variety M(A) of the σ-polynomial A over Ky1, . . . , ys which contains λ, and let η = (η1, . . . , ηs) be ageneric zero of M. Then η is a solution of A which specializes to λ over K, andη is not a proper specialization over K of any other solution of A. It is easy tosee that if a term αiβjyk appears in A, it appears in B as well. Therefore, forevery k ∈ 1, . . . , s, such that some transform of yk appears in A, the elementsλ1, . . . , λk−1, λk+1, . . . , λs are σ-algebraically independent over L∗ (see Corollary7.3.4) an hence over K. It follows that the elements η1, . . . , ηk−1, ηk+1, . . . , ηs

are also σ-algebraically independent over K.Let W be the set of all terms αiβjyk that appear in the σ-polynomial A and

let C ∈ Ky1, . . . , ys \K be a σ-polynomial annulled by η and containing onlyterms from W . Let us show that the resultant of A and C over K with respectto any term αpβqyr ∈ W is identical zero. Indeed, let R be such a resultant.Then R is annulled by η and hence by λ. Applying Corollary 7.3.4 we obtainthat R, considering as a nontrivial σ-polynomial over L∗, is a multiple of B.This is, however, impossible, since B contains αpβqyr and R does not. Thus,R = 0.

It follows that A and C have a common divisor in Ky1, . . . , ys \ K. SinceA is irreducible in Ky1, . . . , ys, C is a multiple of A.


(ii) ⇒ (iii) is obvious.(iii) ⇒ (i). Since every finitely generated separably algebraic field exten-

sion is simple (see Theorem 1.6.17), condition (iii) implies the following fact: ifa1, . . . , am are any elements of an overfield of K such that each ai, 1 ≤ i ≤ m,is separably algebraic and normal over K, then one can extend K to a σ-fieldK〈a1, . . . , am〉.

Let us show that condition (iii) implies that K can be extended to a σ-fielddefined on the separable closure of K. (If the field K is perfect (in particular,if Char K = 0), this will complete the proof.) Let S be the separable part ofan algebraic closure K of K, and let N denote the family of finite normal fieldextensions of K in S. Let us choose one primitive (over K) element for eachextension in N, and let Z be this collection of elements of S. Let X be a setof elements σ-algebraically independent over K such that CardX = CardZ,and let the one-to-one correspondence zi xi between the elements of Z andX be fixed. Furthermore, for every xi ∈ X, let Ai(xi) denote a zero order σ-polynomial in KX (that is, Ai do not contain any nontrivial transforms ofelements of X) such that Ai, when regarded as an algebraic polynomial overK, is a minimal polynomial of zi over K. The system of all such σ-polynomialsAi(xi) ∈ KX will be denoted by F .

Let us show that if the system F has a solution over K, then there is astructure of a σ-overfield of K on the field S. To prove this, suppose F has asolution ξ = ξi. Then the field K〈ξ〉 is separably algebraic over K. Indeed,if η ∈ K〈ξ〉, then there exists a field K ′ between K and K〈ξ〉 generated overK by finitely many transforms of elements of ξ and such that η ∈ K′. Sinceevery ξi is separably algebraic over K, any transform αpβqξi (p, q ∈ N) isseparably algebraic over αpβq(K) and hence also over K. It follows that K ′/Kis a separably algebraic field extension, hence η is separably algebraic over K.

Thus we may without loss of generality assume that K ⊆ K〈ξ〉 ⊆ S. Inorder to complete the proof of our statement about the difference structure onS, we are going to show that S = K〈ξ〉. Indeed, if ζ ∈ S, then K(ζ)/K hasa finite separable normal closure N/K in S/K. Then there exists an elementzi ∈ Z which generates N over K, that is, N is a splitting field over K forAi(xi). By the uniqueness of a splitting field (see Theorem 1.6.5), we haveK(ζ) ⊆ K(ξi) ⊆ K〈ξ〉. Thus, S is contained in K〈ξ〉 whence S = K〈ξ〉.

In order to complete the proof of the implication (iii)⇒(i) (and hence theproof of the theorem) we need the following generalization of Theorem 2.6.5 tothe case of difference polynomial rings with arbitrary (not necessarily finite) setof difference indeterminates.

Lemma 7.3.6 Let K be a difference field with a basic set σ and let U be a familyof elements in some σ-overfield of K such that U is σ-algebraically independentover K. Let Φ be any system of σ-polynomials in the σ-polynomial ring KU.Then the following statements are equivalent.

(a) Φ has a solution in some σ-overfield of K.(b) 1 /∈ Φ. (As usual, Φ denotes the perfect ideal of KU generated

by Φ.)


PROOF. Since the implication (a) ⇒ (b) is obvious, it is sufficient to provethat (a) implies (b). Suppose that 1 /∈ Φ. By Zorn’s lemma, there exists amaximal proper perfect ideal P of KU containing Φ. It is easy to see thatP is a prime reflexive difference ideal. (If a, b ∈ P and a /∈ P, b /∈ P , then we canapply Theorem 2.3.3 and arrive at a contradiction: 1 ∈ P, a, 1 ∈ P, b, but1 /∈ P, aP, b, since P, aP, b ⊆ P, ab ⊆ P .) Then K can be consideredas a σ-subfield of the quotient σ-field L of KU/P , and the set U of residueclasses of elements of U is a generic zero of Φ in L (and hence a solution of Φ).


Let us show that condition (iii) of the theorem implies that the system Fhas a solution. Indeed, if it is not so, then the perfect ideal F generated byF coincides with the whole ring KX. Therefore, there exists a finite subsetX ′ ⊆ X such that the subset Φ′ of Φ corresponding to X ′ generates the wholering KX ′. By the above lemma, we obtain that Φ′ is a finite set of irreducible,normal, separable difference polynomials over K which has no solutions in anyσ-overfield of K. This is, however, impossible, as it is explained in the firstparagraph of the proof (i) ⇒ (ii). Thus, condition (iii) implies that Φ has asolution over K and there is a structure of a σ-overfield of K on S.

It remains to prove that if Char K = p > 0, then one can extend S toa σ-overfield of K which is an algebraic closure of K. This can be done byintroducing a translation τ : a → ap on S. Clearly, τ commutes with α and β onS, and τ maps K into K. Thus, one can consider K and S as difference fieldswith the basic set σ1 = α, β, τ such that S is a σ1-overfield of K. It is easy tosee that the σ1-inversive closure S∗ of S is perfect and hence contains a perfectclosure S1 of S. Clearly, S1 is an algebraic closure of K and all three translationsof S∗ map S1 into S1; in particular S1 can be considered as a σ-overfield of K.This completes the proof of the theorem.

Exercise 7.3.7 Prove that the equivalence of conditions (i) and (iii) of the lasttheorem holds for difference fields with any (finite) basic sets of translation.

The following example shows that there exists a difference polynomial Aover an inversive partial difference field K such that the order of A with respectto every translation is positive and A has a solution of the type described inTheorem 7.3.2, though the the field K does not satisfy condition (i) of Theorem7.3.5 (see Example 5.1.17).

Example 7.3.8 (Bentsen). Let K be an inversive difference field with a basicset σ = α, β and let A = αy + βy + αβ2y be a difference polynomial in thering of σ-polynomials in one σ-indeterminate y over K. Let K ′ denote the fieldK treated as an ordinary difference field with the basic set σ′ = α, and leta0 = (a00, a01, a02) be a generic zero of a principal component of the varietyof A considered as an ordinary σ′-polynomial in K ′αy, βy, αβ2y. Then thereexists a σ′-isomorphism τ of K ′〈a00, a01〉 onto K ′〈a00, a02〉 such that a00 → a01

and a01 → a02 and whose contraction to K is β. Then R = (K ′〈a00, a01, a02〉, τ)is a difference (σ′-) kernel.


By Theorem 7.2.1, ordK ′〈a00, a01, a02〉/K ′〈a00, a01〉 = 1, so that a02 is tran-scendental over K ′〈a00, a01〉. It follows that the field extension K ′〈a00, a01, a02〉/K ′〈a00, a01〉 is primary, hence the kernel R satisfies property L∗ (see Proposition6.1.7). By Corollary 6.2.4, R has a principal realization and all principal real-izations are equivalent of this kernel are equivalent. If η is a principal realizationof R, then η is a solution of A satisfying the properties stated in Theorem 7.3.2.

We complete this section with a theorem on the number of generators ofa perfect ideal in a ring of difference polynomials over completely aperiodicordinary difference field.

Theorem 7.3.9 (R. Cohn). Let K be a completely aperiodic ordinary differ-ence field with a basic set σ = α and let Ky1, . . . , yn be the ring of σ-polynomials in σ-indeterminates y1, . . . , yn over K. Then every perfect σ-idealof Ky1, . . . , yn has a basis consisting of n + 1 elements.

PROOF. Let Q be a nonzero perfect ideal of Ky1, . . . , yn and letA1, . . . , Ar be nonzero σ-polynomials such that Q = A1, . . . , Ar. (We usethe notation of Section 2.3, so for any Φ ⊆ Ky1, . . . , yn, Φ denotes theperfect ideal generated by the set Φ.) Let us consider (n +1)r σ-indetermianteszij (1 ≤ i ≤ n + 1, 1 ≤ j ≤ r) over Ky1, . . . , yn and let Kz denote the cor-responding algebra of σ-polynomials over K. Furthermore, let Ky, z denotethe algebra of σ-polynomials in n + nr + r σ-indeterminates zij , yk (1 ≤ k ≤ n)over K.

For every i = 1, . . . , n + 1, let us set Bi =r∑

i=1

zijAj and show that there is

a nonzero σ-polynomial C ∈ Kz such that every (n + nr + r)-tuple η, whichannuls every Bi (1 ≤ i ≤ n + 1), is either a solution of C or a solution of everyAj , 1 ≤ j ≤ r.

Let us adjoin to K elements aij , 1 ≤ i ≤ n + 1, 2 ≤ j ≤ r and bi, 1 ≤ i ≤ n,that form a σ-algebraically independent set over K. Then one can find an (n+1)-tuple (a11, . . . , an+1,1) as a solution of the σ-polynomial obtained from Bi bysetting yj = bj (1 ≤ j ≤ n) and zij = aij (1 ≤ j ≤ r). (The fact that such an(n + 1)-tuple (a11, . . . , an+1,1) exists is the consequence of Theorem 7.2.1.)

It is easy to see that the elements a11, . . . , an+1,1 belong to the σ-field ob-tained by adjoining the set aij | 1 ≤ i ≤ n + 1, 2 ≤ j ≤ r⋃b1, . . . , bn toK. In what follows we write a, b, y, and z for the indexings (aij)1≤i≤n+1, 1≤j≤r,(b1, . . . , bn), (y1, . . . , yn), and (zij)1≤i≤n+1, 1≤j≤r, respectively.

Let J be the reflexive prime σ-ideal of Ky, z with generic zero y=b, z= a.Then Bi ∈ J for i = 1, . . . , n + 1. Since dimJ = n + (n + 1)(r − 1) < (n + 1)r,J⋂

Kz contains a nonzero difference polynomial C1. Successively applyingthe process of reduction, described in the proof of Theorem 2.4.1, we obtain thatthere exist a linear combination B of B1, . . . , Bn+1 with coefficients in Ky, z,a difference polynomial D, which does not contain transforms of z11, . . . , zr1, ,and some product I of transforms of A1 such that IC1 − B = D. Then D ∈ Jand, moreover, D = 0, since otherwise D could not be annulled by the generic


zero (y = b, z = a) of the ideal J . Thus, every common solution of B1, . . . , Bn+1

annuls either C1 or A1. Similar arguments show the existence of nonzero dif-ference polynomials C2, . . . , Cr ∈ Kz such that every common solution ofB1, . . . , Bn+1 annuls either Cj or Aj (2 ≤ j ≤ r). Then C = C1 . . . Cr has thedesired property: every common solution of B1, . . . , Bn+1 annuls either C orevery Aj , 1 ≤ j ≤ r.

Since the difference field K is completely aperiodic, there exist elementsζij ∈ K (1 ≤ i ≤ n = 1, 1 ≤ j ≤ r) such that B does not become zero ifone replaces each zij by ζij . Let Bi become Bζi after this replacement. ThenBζi ∈ Q (1 ≤ i ≤ n+1). Furthermore, if (y1, . . . , yn) = (η1, . . . , ηn) is a solutionof every Bζi, then yi = ηi, zij = ζij (1 ≤ i ≤ n + 1, 1 ≤ j ≤ r) is a commonsolution of B1, . . . , Bn+1 not annulling C. Therefore, (η1, . . . , ηn) annuls everyAj , 1 ≤ j ≤ r.

The last observation shows that the difference polynomials Bζi ∈ Q (1 ≤i ≤ n + 1) generate Q as a perfect difference ideal of Ky1, . . . , yn.

7.4 Singular and Multiple Realizations

In what follows we apply the results of Section 7.2 to the study of non-principalrealizations of difference kernels over ordinary difference fields.

Definition 7.4.1 A realization η of a difference kernel R over an ordinarydifference field K is called singular if η is not a specialization (over K) of aprincipal realization of R. A realization of R is called multiple if it is a special-ization of two non-isomorphic principal realizations of R.

The following two examples are due to R. Cohn [41, Chapter 6, Section 21].

Example 7.4.2 Let K be an ordinary difference field with a basic set σ = α.Let us consider an algebraically irreducible σ-polynomial A = (α2y)y + αy inthe ring of σ-polynomials Ky in one difference indeterminate y over K. Thenyα(A)−A = αy( (α3y)y− 1). If η = 0 is a generic zero of an irreducible compo-nent M of M(A), then η must annul (α3y)y − 1, whence (α3y)y − 1 ∈ Φ(M),0 /∈ M. Thus, the solution 0 of A itself constitutes an irreducible component ofM(A). Clearly, it is a singular component and furnishes a singular realizationof the kernel produced for A by the procedure described in Example 5.2.6.

Example 7.4.3 As in the preceding example, let Ky be the ring of differencepolynomials in one difference indeterminate y over an ordinary difference fieldK with a basic set σ = α. Let A = (αy)2 + y2 ∈ y. Since α(A) − A =(α2y − y)(α2y + y), the variety M(A) has two principal components M1 andM2 which annul α2y − y and α2y + y, respectively.

Let η be a generic zero of M1 and let x be an element in some overfield ofK〈η〉 which is transcendental over K〈η〉. Then the field L = K〈η〉(x) can betreated as a σ-overfield of K〈η〉 if we set α(x) = x. Clearly, the element xη ∈ Lis a solution of A which is not algebraic over K. Therefore, xη is contained

7.4. SINGULAR AND MULTIPLE REALIZATIONS 421

in a principal component of M(A). Since xη is not a solution of α2y + y, thiselement belong to M1. Furthermore, since 0 is a solution of any σ-polynomialwith solution xη, 0 ∈ M1. Similar arguments show that 0 ∈ M2. We obtainthat 0 is a multiple realization of the kernel R formed for A as in the proof ofTheorem 7.2.1.

The following theorem of R. Cohn is a fundamental result on singular andmultiple realizations.

Theorem 7.4.4 Let R = (K(a0, . . . , ar), τ) be a difference kernel over an ordi-nary inversive difference field K of zero characteristic with a basic set σ = α.Let S denote the polynomial ring in s(r + 1) indeterminates K[αjyi|1 ≤ i ≤s, 0 ≤ j ≤ r] ⊆ Ky1, . . . , ys, and let P be the prime ideal of S with thegeneral zero (a0, . . . , ar). Then there exists a σ-polynomial A ∈ S \ P such that

(a) Every singular realization of R annuls A.(b) If r = 0, then R has no singular realization (even if Char K = 0).(c) Every multiple realization of R annuls A.(d) Every regular realization of R is a specialization of one and only one

principal realization.

PROOF. Using Remark 5.2.12 we can replace R by a kernel of length 0 or1. We start with the case of a kernel R of length 1: R = (K(a0, a1), τ).

Let b0 be a special set for R (see the corresponding definition before The-orem 5.2.8) and let c0 be a maximal subindexing of the coordinates of a0 =(a(1)

0 , . . . , a(s)0 ) algebraically independent over K(b0). Then W = b0 ∪ b1 ∪ c0 is

algebraically independent over K, and every coordinate a(j)i (i = 0, 1; 1 ≤ j ≤ s)

is algebraic over K(W ).In what follows, for any subindexing dk = (a(i1)

k , . . . , a(ip)k ) of the indexing

ak = (a(1)k , . . . , a

(s)k ) (k = 0, 1) and for any j ∈ N such that k ≤ 1 − j, dkj

will denote the subindexing (a(i1)k+j , . . . , a

(ip)k+j) of ak+j . Furthermore, d∗k will

denote the subindexing (αkyi1 , . . . , αkyip

) of the indexing (αky1, . . . , αkys). If

Σ ⊆ Ky1, . . . , ys, then Σj (j ∈ N) will denote the set αj(f) | f ∈ Σ.Finally, we set S = K[y1, . . . , ys, αy1, . . . , αys] (this agrees with the no-tation in the condition of the theorem) and Sk = K[αjyi | 1 ≤ i ≤ s,0 ≤ j ≤ k + 1] for every k ∈ N (S0 = S).

If a(i)0 /∈ b0 ∪ c0 for some i ∈ 1, . . . , s, then the ideal P contains an irre-

ducible polynomial Bi = 0 which lies in K[b∗0, c∗0, yi]. Also, if a

(i)0 ∈ c0 \ b0, then

P contains an irreducible polynomial Ci = 0 which lies in K[b∗0, b∗1, c

∗0, αyi]. By

Theorem 1.2.44, there is a polynomial D ∈ S \ P such that if a pair of s-tuplesη0, η1 is a solution of P not annulling D, then P has solution of the form

αkyi = f(i)k + η

(i)k (k = 0, 1; 1 ≤ i ≤ s) (7.4.1)

where the elements f(i)k which correspond to a

(i)k ∈ W form an algebraically

independent set B over the field K ′ = K(η0, η1), while the remaining elementsf

(i)k are series in positive integral powers of the f

(ν)µ ∈ B with coefficients in K ′.


Let A be the product of D and the formal partial derivatives ∂Bi/∂yi and∂Cj/∂(αyj) for all i, j for which Bi and Cj are defined. These partial derivativesare not in P , since the resultant of Bi and ∂Bi/∂yi with respect to yi and theresultant of Cj and ∂Cj/∂(αyj) with respect to αyj are nonzero polynomials inK[b∗0, b

∗1, c

∗0]. Therefore, A /∈ P . We are going to show that A satisfies condition

(a) of the theorem.Let η be a realization of the kernel R not annulling A. Then there is a

solution of P of the form (7.4.1). Indeed, if f(i)0 /∈ B, then its series expansion

does not contain elements f(j)1 which lie in B. Applying Theorem 1.2.44 one

obtains a solution of Bi of the form yi = f(i)0 + η

(i)0 where f

(i)0 is a series in

powers of some f(j)0 . Since the solution is unique by Theorem 1.2.44, f

(i)0 = f

(i)0 .

In what follows, we define a transform α(f (i)0 ) (written also as αf

(i)0 ) as

a formal power series obtained by replacing each f(j)0 occurring in the power

series for f(i)0 by a symbol g

(j)1 and each coefficient c in the series for f

(i)0 by

α(c). (If f(i)0 ∈ B, then α(f (i)

0 ) = g(i)1 .) Furthermore, let (f (i)

0 )′ denote the seriesobtained by replacing each g

(j)1 in α(f (i)

0 ) by the series for f(i)1 . (If f

(i)1 ∈ B, this

replacement is just f(i)1 itself.)

Let us show that (f (i)0 )′ = f

(i)1 . Indeed, if f

(i)0 ∈ B, the equality is obvious.

Let f(i)0 /∈ B. Then Bi is defined and α(Bi) ∈ P ∩ K[b∗1, c

∗1, αyi]. If we replace

each αyj , j = i, with the series for f(j)1 + η

(j)1 , we shall obtain a polynomial

Bi in αyi with power series coefficients. Since ∂(αBi)/∂αyi is not annulled bythe η

(j)1 , it follows from Theorem 1.2.45 that Bi has at most one solution for

αyi as a sum of η(i)1 and a series in positive integral powers of some elements

of B. Since α(Bi) is annulled by (7.4.1), this solution is f(i)1 . Furthermore, by

Theorem 1.2.44, α(Bi) has a unique formal solution for αyi as a sum of η(i)1

and a series in positive integral powers of the remaining αyj − η(j)1 occurring in

α(Bi). This series is α(f (i)0 ), except that αyj − η

(j)1 is written for g

(j)1 for each

j. Replacing every αyj − η(j)1 in this series by the series for f

(j)1 one obtains

the unique series solution f(i)1 of Bi. On the other hand, the result of such a

replacement is (f (i)0 )′ whence (f (i)

0 )′ = f(i)1 .

For every i such that a(i)0 ∈ b0 we define f

(i)k (k = 2, 3, . . . ) so that these f

(i)k

form an algebraically independent set L over K ′(f (1)0 , . . . , f

(s)0 , f

(1)1 , . . . , f

(s)1 ).

Furthermore, for every i such that a(i)0 /∈ b0 we define inductively the set f

(i)j

(j = 2, 3, . . . ) as series in powers of elements of finite subsets of B∪L consideredas parameters. Suppose that for some integer k ≥ 2, the f

(i)j have been defined

for all j < k. As before, we define α(f (i)j ) by replacing formally each f

(µ)ν in the

series for f(i)j with g

(µ)ν+1, and each coefficient c with α(c). Then (f (i)

j )′ is formed

by replacing every g(µ)ν+1 with the series for f

(µ)ν+1. In any case either f

(µ)ν+1 ∈ B∪L

or ν = 0, so that these series have already been defined. We set f(i)j+1 = (f (i)

j )′.

(Thus, we obtain that f(i)j+1 = (f (i)

j )′ for j = 0, 1, . . . .)

7.4. SINGULAR AND MULTIPLE REALIZATIONS 423

Let E be a σ-polynomial in Ky1, . . . , ys which is annulled by the f(i)j +η

(i)j ,

then α(E) is annulled by the substitution of the α(f (i)j )+η

(i)j+1 for the αj+1yi, and

therefore by the substitution of the (f (i)j )′ + η

(i)j+1 for the αj+1yi. It follows that

the substitution of the f(i)j + η

(i)j annuls α(E), so that the set of σ-polynomials

of Ky1, . . . , ys annulled by the substitution of the f(i)j +η

(i)j is a prime σ-ideal

Q containing P .Let us show that Q

⋂K[c∗i ; b∗i , b∗i+1, . . . ] = (0) for i = 0, 1, . . . . We proceed

by induction on i. If i = 0, the result follows from the construction of f(µ)ν .

Suppose that the statement is true for all i < j where j > 0. Clearly, theintersection Q(j) = Q

⋂K[αj−1y1, . . . , α

j−1ys, αjy1, . . . , α

jys] is a prime idealof the ring K[αj−1y1, . . . , α

j−1ys, αjy1, . . . , α

jys] containing αj−1(P ). By theinduction hypothesis, Q(j)

⋂K[c∗j−1, b∗j−1, b∗j , . . . ] = (0).

Since the cardinality of the set c∗j−1

⋃b∗j−1

⋃b∗j is the dimension of αj−1(P ),

we have the inequality dimQ(j) ≥ dimαj−1(P ), whence Q(j) = αj−1(P ). Fur-thermore, since P

⋂K[c∗0, b

∗0] = (0), one has P

⋂K[c∗1, b

∗1] = (0). Therefore,

αj−1(P )⋂

K[c∗j , b∗j ] = (0) and, hence, Q

⋂K[c∗j , b

∗j ] = (0). It follows from the

construction of the f(i)k (k > j) that Q

⋂K[c∗i ; b∗j , b∗j+1, . . . ] = (0).

Let us prove that the difference ideal Q is reflexive. Let a finite sequenceof s-tuples a0, . . . , ak+1 be a generic zero of the ideal Qk = Q ∩ Sk in thering Sk. In accordance with the previous conventions, for every subindexing λof a0 and for every i = 0, 1, . . . , λi will denote the corresponding subindex-ing of ai. Letting e0 = c0 ∪ b0 ∪ · · · ∪ bk+1 and using the result of the previ-ous paragraph we obtain that the set e0 is algebraically independent over K.Since a0, a1 is a generic zero of P , every coordinate of a0, a1 is algebraic overK(c0, b0, b1). Since a1, a2 is a generic zero of P1 = α(P ) considered as an idealin K[αy1, . . . , αys, α

2y1, . . . , α2ys], every coordinate of a1, a2 is algebraic over

K(c1, b1, b2) and also over K(c0, b0, b1, b2) (since c1 is a subset of the coordi-nates of a1). Continuing in this way we obtain that e0 is a transcendence basisof the coordinates of a0, . . . , ak+1 over K.

Now, in order to show that the σ-ideal Q is reflexive, we assume that α(F ) ∈Q for some σ-polynomial F ∈ Ky1, . . . , ys\Q and obtain a contradiction. Let kbe a positive integer such that F ∈ Sk. Since F /∈ Qk, every essential prime divi-sor of the ideal (F,Qk) in Sk contains a nonzero polynomial of K[c∗0; b

∗0, . . . , b

∗k+1]

(it follows from Remark 1.2.39). Therefore, there is a nonzero polynomial N ∈(F,Qk)

⋂K[c∗0; b

∗0, . . . , b

∗k+1]. Then α(N) ∈ (α(F ), Qk+1)

⋂K[c∗1; b

∗1, . . . , b

∗k+2].

On the other hand, we have shown that Q⋂

K[c∗1; b∗1, . . . , b

∗k+2] = (0). This

contradiction shows that the σ-ideal Q is reflexive.Since Q

⋂K[b∗0, b

∗1, . . . ] = (0), we have dimQ > δR. Furthermore, since

Q ∩ K[c∗0, b∗0, b

∗1] = (0), a generic zero of Q is a regular realization of the kernel

R. By Theorem 5.2.10(v), a generic zero of Q is a principal realization of R. IfF ∈ Q, then F is annulled by the substitution of the f

(i)j + η

(i)j for the αjyi.


Applying Theorem 1.2.43 we obtain that F is annulled by the substitution ofthe η

(i)j for the αjyi. Thus, η is a solution of Q which is not a singular realization

of R.This completes the proof of statement (a). (It has been proved for kernels of

length 1, and the consideration of kernels of length 0 is a trivial particular caseof statement (b) proved below.)

PROOF OF STATEMENT (b). With the above notation, let a kernel R begiven by K(a0) and let η be a realization of R. Replacing K by its σ-overfield,if necessary, we assume that η ∈ K and the field K is algebraically closed.Successive generic prolongations of R lead to kernels Rr = (K(a0, . . . , ar), τr),r = 0, 1, . . . , and it is sufficient to prove that one can specialize a0, a1, . . . over Kto η0 = η, η1 = α(η), . . . . We know that a0 specializes to η0. Suppose that it hasbeen shown that there is a specialization φ of a0, . . . , ar to η0, . . . , ηr = αr(η)(r ≥ 1). Clearly, there is also a specialization ψ of ar+1 to ηr+1 = αr+1(η).Considering the dimensions one obtains that K(a0, . . . , ar+1) is the free joinover K of K(a0, . . . , ar) and K(ar+1). It follows (see Proposition 1.6.49) thatthe field extensions K(a0, . . . , ar)/K and K(ar+1)/K are quasi-linearly disjoint.Applying Proposition 1.6.65 we obtain that φ and ψ yield a specialization ofa0, . . . , ar+1 to η0, . . . , ηr+1, so one has a specialization over K of a0, a1, . . . toη0, η1 = α(η), . . . .

PROOF OF STATEMENT (c). As before, we may assume that the kernel Ris of length 0 or 1. We shall give the proof for length 1. (The case of length 0 canbe considered in a similar way; we leave the corresponding trivial modificationsto the reader as an exercise.)

Let R = (K(a0, a1), τ) and let b0, c0, and a σ-polynomial A ∈ Ky1, . . . , ysbe defined as in the proof of part (a). Let η be a realization of R which do notannul A. As before, there is a series f

(i)j + η

(i)j (i = 1, . . . , s; j = 0, 1, . . . ) and a

reflexive σ-ideal Q of Ky1, . . . , ys annulled by the series whose generic zero isa principal realization of R. Let Q′ be a reflexive prime σ-ideal of Ky1, . . . , ysannulled by η whose generic zero is a principal realization of R. We should provethat Q = Q′.

Suppose that Q = Q′. By Theorem 5.2.10, Q and Q′ have a common com-plete set of parameters and the same relative order with respect to this set.Furthermore, it follows from Proposition 7.1.7(ii) that Q Q′, so that thereexists k ∈ N and a σ-polynomial H such that H ∈ (Q

⋂Sk) \ Q′. By Theorem

1.2.43(ii), Q′ ⋂Sk has a solution not annulling H of the form αjyi = g(i)j + η

(i)j

(1 ≤ i ≤ s, 0 ≤ j ≤ k + 1) where g(i)j are series in positive integral powers of

a parameter t which is transcendental over the field formed by adjoining theseries solution of Q to K〈η〉. Using this solution of Q′ ⋂Sk, one can obtain asolution αjyi = h

(i)j + η

(i)j (1 ≤ i ≤ s, 0 ≤ j ≤ k + 1) of Q

⋂Sk as follows. If

αjyi ∈ c∗0⋃

b∗0⋃ · · ·⋃ b∗k+1, we set h

(i)j = g

(i)j ; if αjyi /∈ c∗0

⋃b∗0

⋃ · · ·⋃ b∗k+1, then

h(i)j is defined by replacing each f

(µ)ν in the series for f

(i)j with g

(µ)ν . (Clearly,

h(i)j are series in positive integral powers of t.)

7.5. REVIEW OF FURTHER RESULTS 425

Let us show that g(i)j = h

(i)j for all i, j for which h

(i)j has been defined. Of

course, it is sufficient to consider i, j such that αjyi /∈ c∗0⋃

b∗0⋃ · · ·⋃ b∗k+1 (in

the case of inclusion our statement is trivial). Proceeding by induction on k weobserve, first, that if yl ∈ c∗0

⋃b∗0, then one can define an irreducible polynomial

Bl = 0 which lies in K[c∗0, b∗0, yl]. After substituting the g

(i)0 for the yi of Bl which

are in c∗0⋃

b∗0 we obtain a polynomial B′l in yl with power series coefficients. Since

∂Bl/∂yl is not annulled by the ηi, B′l has at most one solution for yl as the sum

of ηl and a series in positive integral powers of t (see Theorem 1.2.45). SinceBl ∈ Q

⋂Q′, these series is g

(l)0 and also h

(l)0 , so that g

(l)0 = h

(l)0 .

Suppose that it has been proved that for some integer m, 0 ≤ m ≤ k + 1,g(i)j = h

(i)j (i = 1, . . . , s; j = 0, . . . , m−1). We are going to show that g

(i)m = h

(i)m .

It is clear if yi ∈ b∗0. Suppose that yi ∈ c∗0. Then one can define the polyno-mial Ci considered in the first part of the proof, and this polynomial has theproperty that αm−1(Ci) ∈ K[c∗m−1, b

∗m−1, b

∗m, αmyi]. Replacing the elements of

c∗m−1

⋃b∗m−1

⋃b∗m in αm−1(Ci) with the corresponding g

(µ)ν , which are also h

(µ)ν

by the induction hypothesis and the previous case, we obtain a polynomial C ′i in

αmyi. It has at most one solution for αmyi as a sum of η(i)m and a series in positive

integral powers of t. Since αm−1(Ci) ∈ Q⋂

Q′, this solution must be g(i)m and also

h(i)m . Finally, if a

(i)0 ∈ b0

⋃c0, then Bi is defined and αmBi ∈ K[c∗m, b∗m, αmyi].

Replacing the elements of c∗m⋃

b∗m in αmBi with the corresponding g(µ)ν , which

are also h(µ)ν by the previous case, we obtain a polynomial B′

i in αmyi. Applyingthe uniqueness argument we obtain that g

(i)m = h

(i)m . Thus, g

(i)j = h

(i)j for all i, j

for which h(i)j has been defined. We have arrived at a contradiction, since g

(i)j

annul Q⋂

Sk, but these elements do not annul H. This completes the proof ofstatement (c).

The last statement of the theorem follows from the fact that no regularrealization of the kernel R can annul the σ-polynomial A.

Remark 7.4.5 The proof of part (b) of the last theorem shows that if R isa kernel of length 0 and η and ζ are two realizations of R generating twocompatible σ-field extensions of K, then η and ζ are specializations of the sameprincipal realization. In particular, if K is algebraically closed, then all principalrealizations of R are equivalent, so there can be no multiple realization. If Kis not algebraically closed, then distinct principal realizations of R generateincompatible field extensions of K. For example, 0 is a multiple realization ofthe kernel corresponding to the polynomial y2 + z2 over Q.

7.5 Review of Further Results on Varietiesof Ordinary Difference Polynomials

In this section we review some results on varieties of ordinary difference poly-nomials. Practically all of these results were obtained in R. Cohn’s works [28],[29], [31], [32], [35], [40], [41, Chapters 8 and 10], [42], [43], and [46], where onecan find the detailed proofs.


In [41, Chapter 8] R. Cohn showed that the solutions of any irreduciblevariety M over an ordinary difference field K can be obtained by rational op-erations, transforming, and the inverse of transforming from the solutions of aprincipal component N of the variety of an algebraically irreducible differencepolynomial. (N is a variety over K, but not over the same ring of differencepolynomials, as M.) More precisely, R. Cohn proved the following statement.

Theorem 7.5.1 Let K be an ordinary difference field with a basic set σ = α,Ky1, . . . , ys the ring of σ-polynomials in σ-indeterminates y1, . . . , ys over K,and M an irreducible variety over Ky1, . . . , ys. Suppose that K is completelyaperiodic or that dimM > 0, and also that M possesses a complete set of para-meters y1, . . . , yk such that rld (y1, . . . , yk)M = ld (y1, . . . , yk)M (We assumethat the σ-indeterminates are so numbered that the first k of them constitutethe complete set of parameters. Note also that the last equality always holds ifChar K = 0.) Then there exist:

(a) an irreducible variety N over the ring of σ-polynomials Ky1, . . . , yk;w(w is the (k + 1)-th σ-indetreminate of this ring);

(b) σ-polynomials Ak+1, . . . , As ∈ Ky1, . . . , yk, σ-polynomials Bk+1, . . . , Bs,C ∈ Fy1, . . . , yk;w, C /∈ N , and an integer t ≥ 0 such that

(i) N is a principal component of the variety of an algebraically irreducibleσ-polynomial of Ky1, . . . , yk;w, y1, . . . , yk constitute a complete set of para-meters of N , and Eord (y1, . . . , yk)N = Eord (y1, . . . , yk)M.

(ii) If (η1, . . . , ηs) is any solution in M, there is a solution in N with yi = ηi

for i = 1, . . . , k, and w given by the result of substituting ηj for yj ins∑

i=k+1

Aiyi.

(iii) If yi = ζi, i = 1, . . . , k, w = θ is a solution in N which does not annulC, then there is a solution in M with yi = ζi, i = 1, . . . , k and yj, k + 1 ≤j ≤ s, given by applying α−t to the result of substituting ζk+1, . . . , ζs, θ foryk+1, . . . , ys, w, respectively, in Bj/C.(iv) If D ∈ Ky1, . . . , ys \ Φ(M), then there exists a σ-polynomial E ∈

Fy1, . . . , yk;w \ Φ(N ) such that any solution in N not annulling E givesrise by a procedure described in (iii) to a solution in M not annulling D.

(v) The procedures of (ii) and (iii) carry generic zeros of M or N into genericzeros of N or M. Whenever (iii) is defined, these procedures, applied to elementsof M or N , are inverses of each other.

With the notation of the last theorem, W =s∑

i=k+1

Aiyi is called a resolvent

for M or for Φ(M), and Φ(N ) is called a resolvent ideal for M or for Φ(M).One can say that M is obtained from the solutions of its resolvent ideal by therelations αtyi = Bi/C (k + 1 ≤ i ≤ s) and the solutions of the resolvent idealare obtained from those of M by the relation w = W .

Let Ky1, . . . , ys be a ring of σ-polynomials in σ-indeterminates y1, . . . , ys

over an ordinary difference field K with a basic set σ = α. Let A be a σ-polynomial in Ky1, . . . , ys which contains at least one transform of some yi,


and let αmyi and αlyi be the transforms of yi of the highest and lowest or-ders, respectively, appearing in A. Then the separants of A with respect to yi

are defined to be the formal partial derivatives ∂A/∂(αmyi) and ∂A/∂(αlyi)(computed with the assumption that A is a polynomial over K in the set of in-determinates αiyj which appear in A). If A is written as a polynomial in αmyi,then the coefficient of the highest power of αmyi in A is said to be the initial ofA with respect to yi.

The following two theorems, proved in [41, Chapters 6 and 10], summarizebasic properties of a variety of one ordinary difference polynomial.

Theorem 7.5.2 Let K be an ordinary difference (σ-) field of zero character-istic, Ky1, . . . , ys a ring of σ-polynomials in σ-indeterminates y1, . . . , ys overK, and A an algebraically irreducible σ-polynomial in Ky1, . . . , ys. Then

(i) Every singular component of the variety M(A) and every solution com-mon to two principal components of M(A) annuls the separants of A.

(ii) It is possible to adjoin to K an arbitrarily large number of generic ze-ros of a principal component M of M(A), except in the case that s = 1 andEordA = 0.(iii) Let M(A) have only one principal component, let Φ ⊆ Ky1, . . . , ys be

the prime difference ideal of this component, and let I be an initial of A withrespect to some yk. Then a σ-polynomial B belongs to Φ if and only if there isan integer m > 0 and a product J of transforms of I such that (JB)m ∈ [A].(Therefore, every singular component of M(A) annuls the initials of A.)

Theorem 7.5.3 Let K be an ordinary difference (σ-) field, Ky1, . . . , ys aring of σ-polynomials over K, and A ∈ Ky1, . . . , ys \K an algebraically irre-ducible σ-polynomial. Then

(i) The variety M(A) consists of one or more principal components and (pos-sibly) of singular components. Each component of M(A) has dimension s − 1.

(ii) The relative effective orders and (with certain limitations) the relativeorders of principal components of M(A) are determined by the effective ordersand orders of A. Furthermore, it is sufficient for a component to have dimensions − 1 and one of these effective orders for it to be a principal component.(iii) Except in the case that s = 1 and A is of effective order 0, every principal

component of M(A) contains infinitely many generic zeros.(iv) The relative effective orders of the singular components of M(A) are

less by at least 2 than those of the principal components. More precisely, ifEordyi

A = ri (1 ≤ i ≤ s) and M is a principal component of M(A), theneither y1, . . . , yi−1, yi+1, . . . , ys do not constitute a set of parameters of M orthey do constitute such a set and Eord (y1, . . . , yi−1, yi+1, . . . , ys)M ≤ ri − 2.

(v) Any singular component of M(A) is itself a principal component of M(B)for some algebraically irreducible σ-polynomial B ∈ Ky1, . . . , ys. (As it followsfrom (iv), for each i, 1 ≤ i ≤ s, either B contains no transforms of yi orEordyi

B ≤ ri − 2.)


(vi) If s = 1 and A is a first order σ-polynomial in R, then M(A) has nosingular components.

Theorem 7.5.4 With the assumptions of Theorem 7.5.3, let B and C be σ-polynomials in Ky1, . . . , ys whose orders with respect to any σ-indeterminateyi (1 ≤ i ≤ s) do not exceed 1. If the variety M(A,B) is not empty, it has acomponent of dimension not less than s − 2.

COMPLETE SYSTEMS OF DIFFERENCE OVERFIELDS

Let K be a difference field with a basic set σ. A family E of σ-overfields ofK is said to be a complete system of difference (or σ-) overfields of K if distinctperfect σ-ideals of any ring of σ-polynomials over K have distinct E-varieties.As we have seen in Section 2.6, the universal system U(K) (see Definition 2.6.1)is complete. R. Cohn ([41, Chapter 8]) constructed a complete system U ′(K) ofdifference overfields of an ordinary difference field K whose members are trans-formally algebraic extensions of K. Furthermore, if the field K is algebraicallyclosed, then U ′(K) may be chosen to consist of a single difference field.

Another important result on complete systems of difference overfields is thefollowing criterion for complete systems (see [41, Chapter 8, Section 5]).

Theorem 7.5.5 Let K be an aperiodic ordinary difference field of zero char-acteristic with a basic set σ. Let Ky be the algebra of σ-polynomials in oneσ-indeterminate y over K and let E be a family of σ-overfields of K. Then E isa complete system of σ-overfields of K if and only if given any reflexive primeσ-ideal P of Ky and any σ-polynomial f ∈ Ky \ P , the E-variety ME(P )contains a solution not annulling f .

In what follows we consider an important for analytical applications exampleof an ordinary difference field of complex-valued functions of real argument. Sucha function f(x) is said to be a permitted function if it is defined for x ≥ 0 exceptat a set S(f) which has no limit points, is analytic in each of the intervals intowhich the non-negative real axis is divided by omission of the points of S(f), andis either identically 0, or is 0 at only finitely many points in any finite interval.A permitted difference ring is an ordinary difference ring R whose elements arepermitted functions and whose basic set consists of the isomorphism f(x) →f(x + 1), f(x) ∈ R. (More precisely, elements of R are equivalent classes ofpermitted functions, with f(x) equivalent to g(x) if and only if f(x) = g(x)for any x /∈ S(f) ∪ S(g).) If a permitted difference ring is a field, it is called apermitted difference field.

Let K0 be the difference field of rational functions with complex coefficientswhose domain is the set of non-negative real numbers and translation α is definedby α(f(x)) = f(x + 1), f(x) ∈ K0. Then K0 is a permitted difference field withbasic set σ = α. Let F be the set of all permitted σ-overfields of K0 and letF ′ = K ∈ F | there exists an infinite set of functions analytic on [0, 1] whichis algebraically independent over the field K[0,1] obtained by restricting thedomains of functions of K to [0, 1]. Considering functions with distinct isolated


essential singularities outside of [0, 1] one can easily obtain a non-enumerableset of functions analytic on [0, 1] which is algebraically independent over K0

(more precisely, over (K0)[0,1]). Therefore, every member of F which is at mostcountably generated over K0 is a member of F ′.

Theorem 7.5.6 With the above notation, if K is a σ-field in F ′, then F is acomplete system of σ-overfields of K.

Theorems 7.5.3 and 7.5.4 imply the following result.

Theorem 7.5.7 Let K ∈ F ′, let Ky be the ring of σ-polynomials in one σ-indeterminate y over K, and let P be a proper reflexive prime σ-ideal of Ky.Then P has a generic zero in one of the members of F .

Let a complex-valued function f(x) of real variable x be defined on the interval[0,∞) except at a set S(f) which has no limit points. Then f(x) is said to beessentially continuous if either it is identically 0 or it is continuous at every

point of [0,∞)\S(f) and limx→s

1f(x)

= 0 for every s ∈ S(f). The following result

describes a case when a prime difference ideal of K0y has a generic zero whosecoordinates are essentially continuous functions.

Theorem 7.5.8 With the notation introduced before Theorem 7.5.6, let K =K0 and let a reflexive prime σ-ideal P of K0y have order 0. Then P has ageneric zero in some difference field F ∈ F such that every function f(x) ∈ Fis continuous for all sufficiently large x and essentially continuous.

We conclude this brief review of properties of permitted functions with thefollowing “approximation theorem” obtained in [42]. Let U be a set of permittedfunctions and let g(x) be a permitted function. Then g(x) is said to adhereto U at a point a ∈ [0,∞) if the function g(x) is defined at the points a + i(i = 0, 1, 2, . . . ) and for every ε > 0 and positive integer t, there exists a functionh(x) ∈ U such that h(x) is defined at a, a+1, . . . , a+t and |g(a+i)−h(a+i)| < εfor i = 0, 1, . . . , t. Obviously, if a is a point of adherence, so is a + m for everym ∈ N.

Theorem 7.5.9 With the above notation, let K ∈ F and let Ky be the alge-bra of difference (σ-) polynomials in one σ-indeterminate y over K.

(i) Let J be a σ-ideal of Ky and let U be a set of solutions of J in σ-overfields of K belonging to F . Furthermore, let g(x) be a permitted functionsuch that the set of points of adherence of g(x) to U is dense in [0,∞). Theng(x) is a solution of J .

(ii) Suppose that K ∈ F ′, and let P be a prime reflexive σ-ideal of Ky.Furthermore, let g(x) be a solution of P in a member of F containing K and letV be the set of generic zeros of P in members of F containing K〈g(x)〉. Theng(x) adheres to V at a set of points dense in [0,∞).


WEIGHT CONDITIONS

The following results were obtained by R. Cohn in [32] and [41, Chapter 10].Let K be an ordinary difference field with a basic set σ and Kz the ring of

σ-polynomials in one σ-indeterminate z over K. By a weight function on Kzwe mean a function u → fu from the set of all terms u = czi0(αz)i1 . . . (αrz)ir

of Kz (r, i0, . . . , ir ∈ N, 0 = c ∈ K) to the ring of polynomials in onereal variable t with coefficients in N such that fu(t) = irt

r + · · · + i0. t iscalled the weight parameter. More generally, if Ky1, . . . , ys is the algebra of σ-polynomials in σ-indeterminates y1, . . . , ys over K, then every term u of this ring(that is, a power product of the σ-indeterminates and their transforms multi-plied by a nonzero coefficient from K) can be written as a product of termsu1, . . . , us where ui is a term in Kyi (1 ≤ i ≤ s). Then the weight function onKy1, . . . , ys assigns to u a function fu in s + 1 real variables k1, . . . , ks, t such

that fu(k1, . . . , ks, t) =s∑

i=1

kifui(t). In this case the variables k1, . . . , ks, t are

called weight parameters. If a1, . . . , as, b are real numbers, then fu(a1, . . . , as, b)is said to be the weight of the tern u for these values of the weight parameters.Clearly, fu(1, . . . , 1) is the total degree of the term u (treated as a monomialin variables αjyi, j ∈ N, 1 ≤ i ≤ s). Also, it is easy to see that if we considera formal symbol λ and define its transform α(λ) as λt, then the substitutionsyi = λki (1 ≤ i ≤ s) convert u to a power of λ whose exponent is fu(k1, . . . , ks, t).In what follows, for any σ-polynomial A ∈ Ky1, . . . , ys and any positive realnumbers a1, . . . , as, b as values of the weight parameters, l(A; a1, . . . , as, b) andh(A; a1, . . . , as, b) will denote the σ-polynomials consisting of the terms of leastand highest weights of A, respectively.

The proof of the following result can be found in [41, Chapter 10].

Theorem 7.5.10 With the above notation, let A and B be two σ-polynomials inKy1, . . . , ys. With positive real numbers a1, . . . , as, b as values of the weightparameters, if M(A) ⊆ M(B), then M(l(A; a1, . . . , as, b)) ⊆ M(l(B; a1, . . . ,as, b)) and M(h(A; a1, . . . , as, b)) ⊆ M(h(B; a1, . . . , as, b)).

Let A ∈ Ky1, . . . , ys and let M be an irreducible variety over Ky1, . . . , yssuch that M ⊆ M(A) and dimM = s − 1. Let η = (η1, . . . , ηs) be a genericzero of M and let B be an irreducible σ-polynomial such that M is a prin-cipal component of M(B). Without loss of generality we can assume thaty1, . . . , ys−1 form a complete set of parameters of M. Let K〈η〉z be the ringof σ-polynomials in one σ-indeterminate z over the σ-field K〈η〉 and let A andB be the σ-polynomials of this ring obtained from A and B, respectively, bysubstitutions yi = ηi (1 ≤ i ≤ s − 1), ys = ηs + z. Then A and B are not 0 andhave common solution z = 0.

Definition 7.5.11 With the above notation, we say that the σ-polynomial Asatisfies the low weight condition with respect to M for the parametric sety1, . . . , ys−1 if


(i) A contains a term whose weight is lower than that of any other term ofA for all values of the weight parameter t in the interval (0, 1).

(ii) A contains a term whose weight is lower than that of any other term ofA for all values of the weight parameter t in the interval (1,∞).(iii) M(A∗) ⊆ M(B∗) where A∗ and B∗ denote the σ-polynomials consisting

of the terms of least degree in A and B, respectively.

It is easy to see that the fact that A satisfies the low weight condition doesnot depend on the choice of B. (Of course, B is not uniquely determined: ifone replaces B by a σ-polynomial B1 which is a transform of B with a nonzerocoefficient from K, then M will still be a principal component of M(B1) =M(B).) At the same time, it is not known whether the low weight conditiondepends on the choice of a set of parameters of M.

Theorem 7.5.12 Let K be an ordinary difference field of zero characteristicwith a basic set σ and let Ky1, . . . , ys be the algebra of σ-polynomials in σ-indeterminates y1, . . . , ys over K. Let A ∈ Ky1, . . . , ys and let M be an ir-reducible variety over Ky1, . . . , ys such that M ⊆ M(A). Then a necessarycondition for M to be an irreducible component of M(A) is that A should satisfythe low weight condition with respect to M for every complete set of parametersof M.

In the case of σ-polynomials of one σ-indeterminate, the last theorem leadsto the following statement proved in [35] (see also [41, Chapter 10]).

Theorem 7.5.13 Let Ky be the algebra of difference polynomials in one dif-ference indeterminate y over an ordinary difference field K of zero characteris-tic. Let A be an irreducible difference polynomial in Ky of effective order atleast 1. Then

(i) A necessary condition for 0 to be a singular component of M(A) is thatA should contain a term which is of lower weight than any other term of A forevery value of the weight parameter in the interval (0,∞). If the σ-polynomialA consists of two terms, this condition is also sufficient.

(ii) Suppose that A admits the solution 0 and contains a term of first degree. Ifj is the order of the transform of y in this term, then a necessary and sufficientcondition for 0 to be an irreducible component of M(A) is that every termof A contain a transform of y of order not exceeding j and a transform of y oforder not less than j.

Another application of Theorem 7.5.12 is the following interesting result onsingular components of a variety of an irreducible difference polynomial.

Theorem 7.5.14 (R. Cohn, [32]) Let K be an ordinary difference field ofzero characteristic with a basic set σ = α, Ky1, . . . , ys the algebra of σ-polynomials in σ-indeterminates y1, . . . , ys over K, and A an irreducible σ-polynomial in this algebra. Let αiys and αjys be, respectively, the lowest and thehighest transforms of ys which appear in A. Then


(i) Every singular component of the variety M(A) annuls the formal partialderivatives ∂A/∂(αiys)k and ∂A/∂(αjys)k , k = 1, 2, . . . .

(ii) Let A be written as A = A0 + A1αiys + . . . Ap(αiys)p and A = B0 +

B1αjys + . . . Bq(αjys)q where Aν do not contain αiys, and Bµ do not contain

αjys (1 ≤ ν ≤ p, 1 ≤ µ ≤ q). Then every singular component of M(A) annulseach Aν and Bµ.

Let K be an ordinary difference field with a basic set σ, Ky1, . . . , ys thealgebra of σ-polynomials in σ-indeterminates y1, . . . , ys over K, and A an ir-reducible σ-polynomial in this algebra. Let M be an irreducible variety overKy1, . . . , ys contained in M(A) and let η = (η1, . . . , ηs) be a generic zero ofM. Furthermore, let ri = Eordyi

A (1 ≤ i ≤ s) and let A be the σ-polynomialover K〈η〉 obtained from A by replacing every yi by yi + ηi. We say that theσ-polynomial A satisfies the high order condition with respect to M if thereexist positive values a1, . . . , as; b of the weight parameters and an integer i,1 ≤ i ≤ s such that the polynomial A∗ consisting of the terms of A of leastweight (for these values of the weight parameters) has an irreducible factor B inK〈η〉y1, . . . , ys such that Eordyi

B > ri − 2. (If ri < 2, this condition meansthat some transform of yi appears in A.)

Theorem 7.5.15 With the above notation, suppose that Char K = 0 and theirreducible σ-polynomial A satisfies the high order condition with respect to M.Then M is contained in a principal component of M(A).

The low weight condition is not a sufficient condition for an irreducible va-riety to be an irreducible component of the variety of a difference polynomial.R. Cohn [35] presented the following example which justifies this assertion. LetQ be the field of rational numbers treated as an ordinary difference field whosebasis set σ consists of the identity automorphism α. Let K = Q〈a〉 where theelement a is a σ-transcendental over Q (a lies in some σ-overfield of Q), and letKy be the algebra of σ-polynomials in one σ-indeterminate y over K. R. Cohnproved that if one considers σ-polynomials

A = y(α2y)2 + (αy)2α3y − a(αy)(α2y)

andB = y(α2y)2 + (αy)2α3y − (αy)(α2y)

in Ky, then 0 is an irreducible component of M(A) but not of M(B). Since Aand B comprise the same power products, the example shows that the low weightcondition is not a sufficient one. Moreover, this example shows that no criteriondependent only on what power products appear in a difference polynomial is anecessary and sufficient condition for 0 to be an irreducible component of thevariety of the polynomial. The following theorem, together with Theorem 7.5.12,can be considered as a summary of most important known results on the lowweight condition.

7.6. RITT’S NUMBER. GREENSPAN’S AND JACOBI’S BOUNDS 433

Theorem 7.5.16 (R. Cohn, [40]) Let K be an ordinary difference field of zerocharacteristic with a basic set σ, Ky1, . . . , ys the algebra of σ-polynomialsin σ-indeterminates y1, . . . , ys over K, and A an irreducible σ-polynomial inthis algebra. Let M be an irreducible variety with parameters y1, . . . , ys−1 overKy1, . . . , ys such that M ⊆ M(A) and let r = Eord (y1, . . . , ys−1)M, k =Eordys

A − r. Then

(i) If k = 1, A does not satisfy the low weight condition with respect to Mfor the parametric set y1, . . . , ys−1.

(ii) If k = 2, the low weight condition for the parametric set y1, . . . , ys−1 isa necessary and sufficient condition for M to be an irreducible component ofM(A).(iii) If k > 2, the low weight condition for the parametric set y1, . . . , ys−1 is a

necessary but not a sufficient condition for M to be an irreducible componentof M(A).

7.6 Ritt’s Number. Greenspan’s and Jacobi’sBounds

Let K be an ordinary difference field with a basic set σ = α and letKy1, . . . , ys be the ring of σ-polynomials in a set of σ-indeterminatesY = y1, . . . , ys over K. Let Φ ⊆ Ky1, . . . , ys, Y ′ ⊆ Y , and let the or-ders of σ-polynomials of Φ in each yi ∈ Y \ Y ′ be bounded (in particular, Φmay be a finite family). For every yi ∈ Y \ Y ′, let ri denote the maximum ofthe orders of the σ-polynomials of Φ in yi. Then the number R(Y ′)Φ =

∑i

ri

(the summation extends over all values of the index i such that yi ∈ Y \ Y ′)is called the Ritt number of the system Φ associated with the set Y ′ ⊆ Y . If

Y ′ = ∅, we set R(Y ′)Φ =s∑

i=1

ri and also write RΦ for R(Y ′)Φ.

Let 0 = A ∈ Φ and let h(A) be the greatest nonnegative integer suchthat for each i with yi ∈ Y \ Y ′, αh(A)(A) is of order at most ri in yi. Leth = maxh(A)|0 = A ∈ Φ, that is, h is the greatest integer such that somepolynomial of Φ may be replaced by its hth transform without altering the Rittnumber of Φ. The number G(Y ′)Φ = R(Y ′)Φ−h is called the Greenspan numberof the system Φ associated with the set Y ′ ⊆ Y . If Y ′ = ∅, we write GΦ forG(Y ′)Φ.

Now, let the system of σ-polynomials Φ be finite, Φ = A1, . . . , Am, andlet rij = ordyj

Ai, 1 ≤ i ≤ m, 1 ≤ j ≤ s (rij is taken to be 0 if no transforms ofyj of order ≥ 1 appear in Ai). If m = s, then the number

J (Φ) = max

s∑

i=1

riji|(j1, . . . , js) is a permutation of 1, . . . , s


is called the Jacobi number of the system Φ. If m = s (it can be also applied tothe case m = s), the Jacobi number of the system Φ is defined as the Jacobi num-ber of the corresponding m × s-matrix (rij) (see the definition before Theorem1.2.47); this matrix will be denoted by AΦ. The following two theorems (wherewe use the above notation) gives some bounds on the effective orders with res-pect to Y ′ that involve the Ritt, Greenspan and Jacobi numbers. The proof ofthe statements of the first theorem is due to R. Cohn ([41, Chapter 8, TheoremsIX and X]); the result on the Jacobi bound (Theorem 7.6.7) was obtained by B.Lando in [116].

Theorem 7.6.1 (i) With the above notation, suppose that the Ritt numberR(Y ′)Φ for a system of σ-polynomials Φ (and a set Y ′ ⊆ Y ) is defined. If Mis a component of M(Φ) for which Y ′ contains a complete set of parameters,then Eord (Y ′)M ≤ R(Y ′)Φ.

(ii) If GΦ is defined, M is a component of M(Φ) and Y ′ contains a completeset of parameters of every component of M(Φ), then Eord (Y ′)M ≤ G(Y ′)Φ.

PROOF. (i) Let us show first that one can assume that Y ′ = ∅ in the firststatement of the theorem. Indeed, suppose that Y ′ is nonempty and let η be ageneric zero of M. Let η′ denote the subindexing of η whose coordinates corre-spond to elements of Y ′, and let η∗ and Y ∗ be the complimentary subindexingsto η′ in η and Y ′ in Y , respectively. Let Φ∗ be the set of σ-polynomials inK〈η′〉Y ∗ obtained from the σ-polynomials in Φ by replacing each coordinateof Y ′ by the corresponding coordinate of η′. It is easy to see that η∗ is a solutionof Φ∗ and a specialization of a generic zero η∗∗ of some component of M(Φ∗).(As in Section 5.3, if no confusion can result, we use term “specialization” for aσ-specialization over K.) It follows that η′, η∗ is a specialization of η′, η∗∗ overK. Since η′, η∗∗ is a solution of Φ, and η′, η∗ is a generic zero of a componentof M(Φ), this specialization is generic. Then there is a σ-K〈η′〉-isomorphismof K〈η′, η∗〉 onto K〈η′, η∗∗〉, hence η∗∗ is a generic zero of some componentM∗ of M(Φ∗). Clearly, the empty set is a complete set of parameters of M∗,EordM∗ = Eord(Y ′)M, and RΦ∗ = R(Y ′)Φ. Thus, it is sufficient to provethat EordM∗ ≤ RΦ∗ so from the very beginning we can assume that Y ′ = ∅.

Now we are going to show that one can assume that ri ≤ 1 for i = 1, . . . , s (weuse the notation introduced at the beginning of this section). Indeed, supposethat ri > 1 for some i, say r1 > 1. Let KY, z be the ring of σ-polynomialsin one σ-indeterminate z over KY = Ky1, . . . , ys. Let Φ′ be a family of σ-polynomials in KY, z consisting of z − αy1 and σ-polynomials resulting fromthose of Φ by the substitution of αjz for αj+1y1 (j ∈ N). If η is a generic zeroof a component M of M(Φ) such that dimM = 0, then (Y = η, z = α(η1)) is ageneric zero of a component M′ of Φ′. Furthermore, since generic zeros of M andM′ generate the same extensions of K, dimM′ = 0 and EordM′ = EordM.It follows that RΦ = RΦ′, so one may use Φ′ ⊆ KY, z instead of Φ ⊆ KY .Continuing such reductions we can replace Φ by a system Ψ of σ-polynomialsin some σ-polynomial ring KY, z1, . . . , zp that satisfies the condition ri ≤ 1


for i = 1, . . . , s + p. Thus, without loss of generality we can assume that ouroriginal system Φ satisfies the condition ri ≤ 1 for i = 1, . . . , s.

Before starting the main part of the proof we are going to show that it issufficient to prove our theorem in the case of inversive difference field K. Indeed,suppose that K is not inversive and let K∗ be its inversive closure. Let M bea component of M(Φ) with dimM = 0 and let η be a generic zero of M.Then there is a σ-overfield of K∗ containing all coordinates of η, hence η is aspecialization over K∗ of a generic zero η∗ of such a component. It follows thatη is a specialization of η∗ over K. Since η is the generic zero of a component ofM(Φ), there must be a σ-K-isomorphism of K〈η〉 onto K〈η∗〉 sending η to η∗.Clearly, this isomorphism can be extended to a σ-K∗-isomorphism of K∗〈η〉 ontoK∗〈η∗〉, hence η is a generic zero of a component M∗ of the variety of Φ overK∗. Since dimM∗ = 0 and EordM = EordM∗, one can consider M∗ insteadof M, so from the very beginning one can assume that the σ-field K is inversive.

Thus, from now on we assume that K is inversive, dimM = 0 and ri ≤ 1(1 ≤ i ≤ s). Then one can easily obtain the inequality EordM ≤ s. Indeed, letη be a generic zero of M and let K denote the kernel determined by K(η, α(η))Since Φ ⊆ K[Y, αY ], every realization of K annuls Φ. It is also obvious that η isa regular realization of K. By Theorem 7.4.4, η is a specialization of a principalrealization η of K. Since η is a solution of Φ, η is also a principal realization ofK, hence δK = 0 and ordM = trdegKK(η) ≤ s. Therefore, EordM ≤ s.

In order to prove the desired inequality EordM ≤ RΦ we proceed by in-duction on p where p denotes the number of coordinates of Y which occuronly to order 0 in the σ-polynomials of Φ. If p = 0, then RΦ = s, so thestatement is true. Suppose that p > 0 and the desired inequality holds for anysmaller number. Without loss of generality we can assume that ys is one of thecoordinates of Y appearing only to order 0.

Let us consider the ring of σ-polynomials Ky1, . . . , ys−1, z (where z isa σ-indeterminate over Ky1, . . . , ys−1), and let Φ∗ denote the family of σ-polynomials in this ring obtained by replacing ys by z+αz in the σ-polynomialsof Φ. Furthermore, let Y ∗ denote the s-tuple (y1, . . . , ys−1, z). Since ys appearsin at least one σ-polynomial in Φ (otherwise M(Φ) would not have a zero-dimensional component), αz appears in at least one σ-polynomial in Φ∗. Itfollows that

RΦ∗ = RΦ + 1 (7.6.1)

and Y ∗ has p − 1 coordinates which appear only to order 0 in Φ∗.Let ζ be a generic zero of the prime σ∗-ideal z+αz−ηs of the ring L〈η〉z

and let η∗ denote the s-tuple (η1, . . . , ηs−1, ζ). Then

EordK〈η∗〉/K = EordK〈η, ζ〉/K = EordK〈η, ζ〉/K〈η〉 + EordK〈η〉/K + 1.(7.6.2)

Let M∗ be a component of M(Φ∗) containing η∗. If η∗ = (η1, . . . , ηs−1, ζ) isa generic zero of M∗, then the s-tuple η = (η1, . . . , ηs−1, ζ + αζ) lies in M(Φ)and specializes to η. It follows that η ∈ M hence σ-trdegKK〈η〉 = 0. Since ζ isσ-algebraic over K〈η〉, one has σ-trdegKK〈η∗〉 = 0.


Thus, we can apply the inductive hypothesis to obtain the inequality

EordK〈η∗〉/K ≤ EordM∗ ≤ RΦ∗ , (7.6.3)

which, together with (7.6.1) and (7.6.2), implies that EordK〈η〉/K ≤ RΦ.(ii) Proceeding as at the beginning of the proof of part (i) we may assume

that Y ′ = ∅ provided that every component of the modified system Φ∗ is ofdimension 0. To show that it is really so, suppose that some component ofΦ∗ has positive dimension. Then there exists a solution η∗ of Φ∗ such thatσ-trdegK〈η′〉K〈η′, η∗〉 > 0 (we use the notation of the proof of (i)). Then σ-trdegKK〈η′, η∗〉 is greater than the number of coordinates of Y ′. Since η′, η∗

is a solution of Φ, this contradicts the assumption that Y ′ contains a completeset of parameters of every component of M(Φ). Applying similar additionalarguments to the corresponding part of the proof of part (i) one can also justifythe assumption that the field K is inversive.

Thus, from now on we suppose that K is inversive and Y ′ = ∅. As in the proofof part (i) we proceed with the reduction of Φ to a system of σ-polynomials Φ′

in a σ-polynomial ring Kz1, . . . , zt such that orders of elements of Y ′ do notexceed 1. Without loss of generality we may assume that the σ-indeterminatesz1, . . . , zt are arranged in such a way that αz1, . . . , αzr appear in elements of Y ′

while αzr+1, . . . , αzt do not. Note that r =s∑

i=1

ri = RΦ = RΦ′, and if ri > 0,

then each αjyi (j < ri) is equated by the newly introduced equations to some

zk, k ≤ r =s∑

i=1

ri. Distinct αjyi are equated to distinct zk.

Let M′ be the component of M(Φ′) corresponding to M, let θ = (θ1, . . . , θt)be a generic zero of M′, and let K be the kernel with isomorphism τ ob-tained in the usual way from K(θ, α(θ)). Since every realization of K is asolution of Φ′, δK = 0. Furthermore, since θ is a regular realization of K,trdegK(θ)K(θ, α(θ)) = 0, hence ordM′ = trdegKK〈θ〉 = trdegKK(θ). SinceEordM = EordM′ = ordM′, we obtain that

EordM = trdegKK(θ). (7.6.4)

We are going to show that trdegK(θ1,...,θr)K(θ) = 0. Indeed, by Corollary 1.6.41(with the fields K(θ1, . . . , θr), K(θ) and K(θ, α(θ1), . . . , α(θr)) = K(θ1, . . . , θt,α(θ1), . . . , α(θr)) instead of K, L and M in Corollary 1.6.41, respectively), thereexist elements λr+1, . . . , λt in some overfield of K(θ, α(θ1), . . . , α(θr)) with thefollowing properties:

(a) The contraction of τ to an isomorphism of the field K(θ1, . . . , θr) ontoK(α(θ1), . . . , α(θr)) extends to an isomorphism ρ of K(θ) into K(θ, λ) where λdenotes (α(θ1), . . . , α(θr), λr+1, . . . , λt).

(b) ρ(θi) = λi for i = r + 1, . . . , t.(c) trdegK(θ,α(θ1),...,α(θr))K(θ, λ) = trdegK(θ1,...,θr)K(θ).

Let K′ be the kernel formed by the field K(θ, λ) and the isomorphism ρ.Since Φ′ is free of αzi for i > r, every realization of K′ is a solution of Φ′.


Therefore, δK′ = 0 and 0 = trdegK(θ)K(θ, λ) ≥ trdegK(θ,α(θ1),...,α(θr))K(θ, λ) =trdegK(θ1,...,θr)K(θ). Thus, trdegK(θ1,...,θr)K(θ) = 0 and

trdegKK(θ) = trdegKK(θ1, . . . , θr). (7.6.5)

Let the integer h used in the definition of the Greenspan bound be positiveand let A be a nonzero σ-polynomial in Φ such that ordyi

αh(A) ≤ ri (1 ≤ i ≤ s).Suppose that some transform of some yjappears in A and let k = ordyj

A. WithA written as a polynomial in αkyj let Ij(A) denote the coefficient of the highestpower of αkyj in A.

With the above notation we perform the following procedure: if Ij(A) ∈Φ(M), we replace A by Ij(A); if Ij(A) /∈ Φ(M), we do not change A. Repeatingthis process we shall arrive at a nonzero σ-polynomial B ∈ Φ(M) such that

(I) B involves only those αjyi that appear in A.(II) There exists p ∈ 1, . . . , s such that if q = ordyp

B, then the coefficientof the highest power of αqyp in B does not belong to Φ(M).

Let υ = (υ1, . . . , υs) be a generic zero of M. Because of (II) the equationsαi(B)(υ) = 0 (0 ≤ i ≤ h − 1) show that the set Υ = αq(υp), . . . , αq+h−1(υp)is algebraically dependent over K on a subset Σ of the αj(υi) disjoint fromΥ and such that the inclusion αj(υi) ∈ Σ implies that αj(yi) appears in oneof the σ-polynomials B,α(B), . . . , αh−1(B) and, therefore (see (I)), in one ofA,α(A), . . . , αh−1(A). Obviously, for any such αj(yi) we have ri > 0 and j < ri.It follows that these αj(yi) correspond uniquely to distinct zk, 1 ≤ k ≤ r, as itis shown in the description of the reduction of Φ to Φ′.

We obtain that the elements of the sets Υ and Σ are in one-to-one correspon-dence with the elements of disjoint subsets Υ′ and Σ′ of θ1, . . . , θr such thatevery element of Υ′ is algebraic over K(Σ′). Since Card Υ′ = h, we have

trdegKK(θ1, . . . , θr) ≤ r − h = GΦ (7.6.6)

for every h ≥ 0 (for h = 0 this inequality is obvious). Combining (7.6.4), (7.6.5),and (7.6.6) we obtain the desired inequality EordM ≤ GΦ.

Remark 7.6.2 With the notation of the last theorem, let ki (1 ≤ i ≤ s) be thesmallest integer such that αkiyi appears in some σ-polynomial of Φ. Let Ψ bethe system obtained from Φ by the substitution of yi for αkiyi (i = 1, . . . , s).Then the components of M(Ψ) have the same relative effective orders as thecomponents of Φ. It follows that each ri can be replaced by r′i = ri − ki in thebounds in parts (i) and (ii) of Theorem 7.6.1.

Exercises 7.6.3 Let K be an ordinary difference field with a basic set σ = αand Ky1, y2 the algebra of σ-polynomials in two σ-indeterminates y1 and y2

over K. Let A1, A2 ∈ Ky1, y2 and let ordyjAi = rij (1 ≤ i, j ≤ 2).

1. Show that the Greenspan number for the system A1, A2 is equal tomaxr11 + r22, r12 + r21.


2. Prove that if A1 does not contain transforms of y2 (that is, A1 ∈ Ky1)and M is a component of the variety M(A1, A2) with dimM = 0, thenEordM ≤ r11 + r22.

The following three propositions give some estimations of the effective orderof a variety in terms of Ritt and Greenspan numbers.

Proposition 7.6.4 Let K be an ordinary difference field with a basic setσ = α and let Ky1, . . . , ys be a ring of σ-polynomials in the set ofσ-indeterminates Y = y1, . . . , ys over K. Let Φ be a system of nonzeroσ-polynomials in KY and let Y ′ be a proper subset of Y such that Φ′ =Φ ∩ KY ′ = ∅. Furthermore, suppose that every component of the varietyM(Φ′) (treated as a variety over KY ′) is of dimension 0. Then, for anycomponent M of M(Φ), we have EordM ≤ GΦ′ + G(Y ′)Φ.

PROOF. Let η be a generic zero of M and let η′ be a subindexing of ηwhose coordinates correspond to those of Y ′. Let M′ be a component of M(Φ′)with solution η′. Then

EordK〈η′〉/K = EordM′ ≤ GΦ′. (7.6.7)

Let Y ′′, η′′, and Φ′′ denote the complementary indexings or sets to Y ′ in Y , η′ inη, and Φ′ in Φ, respectively. Let Φ∗ be the set of σ-polynomials in K〈η′〉 obtainedby substituting η′ for Y ′ in the σ-polynomials of Φ′′. Then every component ofM(Φ∗) is of dimension 0. Indeed, if this were not so, then Φ∗ would have asolution η∗ with σ-trdegK〈η′〉K〈η′, η∗〉 > 0, and η′, η∗ would be a solution of Φ,which is impossible.

Applying Theorem 7.6.1(ii) we obtain that

EordK〈η′, η∗〉/K〈η′〉 ≤ GΦ∗ ≤ G(Y ′)Φ. (7.6.8)

Since EordM = EordK〈η′, η∗〉/K = EordK〈η′, η∗〉/K〈η′〉 + EordK〈η′〉/K,the statement of the proposition follows from the inequalities (7.6.7) and(7.6.8).

The proof of the following statement is similar to the proof of Proposition7.6.4.

Proposition 7.6.5 With the notation of Proposition 7.6.4, let M be a compo-nent of the variety M(Φ), and let M′ be a component of M(Φ′) admitting thesolution η′, where η′ is the subindexing of a generic zero η of M with the sameindices as Y ′. Then:

(i) If dimM = dimM′ = 0, then EordM ≤ RΦ′ + R(Y ′)Φ.(ii) If every component of M(Φ) is of dimension 0, and dimM′ = 0, then

EordM ≤ RΦ′ + G(Y ′)Φ.

Proposition 7.6.6 With the notation of Proposition 7.6.4 and its proof, sup-pose that the family Φ′′ consists of a single σ-polynomial A, and every compo-nent of M(Φ) is of dimension 0. Then dimM′ = 0, and therefore EordM ≤RΦ′ + G(Y ′)Φ.


PROOF. Let ζ ′ be a generic zero of M′ and let A′ be the σ-polynomial inK〈η′〉Y ′′ obtained by substituting η′ for Y ′ in A. Since A′ has the solution η′,A′ cannot be a nonzero element of K〈η′〉. At the same time A′ = 0 (otherwise,M(Φ) would contain an s-tuple η′, θ where the coordinate θ is σ-transcendentalover K〈η′〉). Therefore, A′ /∈ K〈η′〉.

Let A∗ be the σ-polynomial in K〈ζ ′〉Y ′ obtained by substituting ζ ′ for Y ′

in A. Since ζ ′ specializes to η′ over K, the coefficients of A∗ specialize to thoseof A′. Therefore, A∗ /∈ K〈ζ ′〉.

By Theorem 7.2.1, there is a solution ζ ′′ of A∗. Then ζ ′, ζ ′′ is a solution ofΦ, hence σ-trdegKK〈ζ ′, ζ ′′〉 = 0. It follows that σ-trdegKK〈ζ ′, ζ ′′〉 = 0, hencedimM′ = 0.

Theorem 7.6.7 Let K be an ordinary difference field with a basic set σ = αand let Ky1, . . . , ys be a ring of σ-polynomials in σ-indeterminates y1, . . . , ys

over K. Let Φ = A1, . . . , Am where A1, . . . , Am are first order σ-polynomialsin Ky1, . . . , ys (with the notation introduced before Theorem 7.6.1, it meansthat for every i = 1, . . . , m, rij ≤ 1 (1 ≤ j ≤ s) and at least one rij is equal to1). If M is an irreducible component of M(Φ) of dimension 0, then EordM ≤J (Φ).

PROOF. Note, first, that we may assume that K is inversive (to justify thisassumption one just needs to repeat the arguments of the first part of the proofof Theorem 7.6.1(i)). Let η = (η1, . . . , ηs) be a generic zero of the variety M.Then η is a principal realization of the kernel K with field K(η, η). Indeed, sinceη is a regular realization of K, it is a specialization of a principal realizationη′ of K (see Corollary 6.2.16). But η′ is a zero of the set A1, . . . , Am; thusη is itself a principal realization. Applying Theorem 5.2.10(i) we obtain thatδK = ordM = trdegKK(η).

Since (η, α(η)) is a zero of the ideal (A1, . . . , Am) in the polynomial ringK[y1, . . . , ys, αy1, . . . , αys], there is a generic zero (a, a1) = (a(1), . . . , a(s), a

(1)1 ,

. . . , a(s)1 ) of some component of the (non-difference) variety M(A1, . . . , Am)

such that (a, a1) specializes to (η, α(η)) over K. (We write this as (a, a1) −→K

(η, α(η)). Therefore, p = trdegKK(a) − trdegKK(η) and q = trdegKK(a1) −trdegKK(α(η)) are nonnegative integers.

By two applications of Proposition 1.6.66 one obtains s-tuples c and η1 overK such that

(a, a1) −→K

(c, α(η)) −→K

(η, η1) −→K

(η, α(η))

withtrdegKK(a, a1) − trdegKK(c, α(η)) ≤ q

andtrdegKK(c, α(η)) − trdegKK(η, η1) ≤ p.

It follows that

trdegKK(a, a1) − trdegKK(η, η1) ≤ trdegKK(a1)−trdegKK(α(η)) + trdegKK(a) − trdegKK(η). (7.6.9)


Since α(η) −→K

(η1) −→K

α(η), there is a K-isomorphism of the field K(η1) onto

K(α(η)). Furthermore, since K(η, α(η) forms a kernel, K(η, η1) also forms akernel K. By Theorem 6.2.14, there is a principal realization η of K whichspecializes to η over K.

Since (a, a1) −→K

(η, η1), η is a zero of the system A1, . . . , Am. Therefore,

the specialization η −→K

η is generic and the field K(η, η1) is K-isomorphic to

K(η, α(η)). This isomorphism, together with inequality (7.6.9), implies that

ordM = trdegKK(η) = trdegKK(α(η)) ≤ trdegKK(η, α(η)) − trdegKK(η)

+trdegKK(a1) − trdegKK(a, a1) + trdegKK(a) ≤ 0 + s − trdegK(a)K(a, a1).(7.6.10)

Let A′1, . . . , A

′m be the polynomials in the ring K(a)[αy1, . . . , αys] obtained

by substituting a(1), . . . , a(s) for y1, . . . , ys, respectively, in A1, . . . , Am. Let A′ =(r′ij) be the m × n-matrix with r′ij = 1 if αyj appears in A′

i, and r′ij = 0 if not.Then J (A′) ≤ J (AΦ) = J (Φ), since r′ij = 1 only if rij = 1.

By Theorem 1.2.47, every component of the variety M(A′1, . . . , A

′m) over

K(a) has dimension at least s − J (A′), and the last number is greater than orequal to s − J (AΦ). Furthermore, since a1 is a zero of A′

1, . . . , A′m, there is

a generic zero b of a component of M(A′1, . . . , A

′m) such that b −−−→

K(a)a1. But

(a, b) is a zero of Φ, and (a, a1) is a generic zero; therefore the specialization isgeneric and

trdegK(a)K(a, a1) ≥ s − J (AΦ). (7.6.11)

Combining (7.6.10) and (7.6.11) we obtain that ordM ≤ J (Φ).

A number of results on the Jacobi bound for systems of algebraic differentialequations (see, for example, [50], [51] and [110, Section 5.8]) give a hope thatthe last theorem can be essentially strengthen and generalized to the case ofpartial difference polynomials.

7.7 Dimension Polynomials and the Strengthof a System of Algebraic DifferenceEquations

Let K be an inversive difference field with a basic set σ = α1, . . . , αn,let Ky1, . . . , ys∗ be the ring of inversive difference (σ∗-) polynomials inσ∗-indeterminates y1, . . . , ys over K, and let Φ = fλ |λ ∈ Λ be a set ofσ∗-polynomials in in Ky1, . . . , ys∗. As we know, an s-tuple η = (η1, . . . , ηs)with coordinates in some σ∗-overfield of K is said to be a solution of the systemof algebraic difference (σ∗-) equations

fλ(y1, . . . , ys) = 0 (λ ∈ Λ) (7.7.1)

7.7. DIMENSION POLYNOMIALS AND THE STRENGTH 441

if Φ is contained in the kernel of the substitution of (η1, . . . , ηs) for (y1, . . . , ys)which is a difference homomorphism Ky1, . . . , ys∗ → K〈η1, . . . , ηs〉∗ sendingeach yi to ηi and leaving elements of K fixed. (Notice that every system ofalgebraic difference equations in the sense of Definition 2.2.3, that is, a systemof the form (7.7.1) with fλ from the ring of σ-polynomials Ky1, . . . , ys, canbe also viewed as a system of algebraic σ∗-equations.)

By Theorem 2.5.11, system (7.7.1) is equivalent to some its finite subsystem,that is, there exist finitely many σ∗-polynomials f1, . . . , fp ∈ Φ such that thevariety of the set Φ coincides with the variety of the set f1, . . . , fp. Thus, whilestudying systems of algebraic difference equations in s difference indeterminatesover K, one can consider just the systems of the form

fi(y1, . . . , ys) = 0 (i = 1, . . . , p) (7.7.2)

where f1, . . . , fp ∈ Ky1, . . . , ys∗.

Definition 7.7.1 A system of algebraic σ∗-equations (7.7.2) is called prime ifthe perfect σ-ideal f1, . . . , fp of the ring Ky1, . . . , ys∗ is prime.

Since a linear σ∗-ideal of the ring Ky1, . . . , ys∗ is prime (see Proposition2.4.9), every system of linear homogeneous σ∗-equations (that is, equations ofthe form

∑sj=1 ωijyj = 0 (0 ≤ i ≤ p) where all ωij are σ∗-operators over K) is

prime.

Definition 7.7.2 Let K be an inversive difference field with a basic set σ =α1, . . . , αn and let E be the ring of σ∗-operators over K. A σ∗-operator ω ∈ Eis called symmetric if it has the following property. Let ω be represented in thestandard form, that is, in the form ω = a1α

k111 . . . αk1n

n + · · · + arαkr11 . . . αkrn

n

where ai ∈ K, ai = 0, and (ki1, . . . , kin) = (kj1, . . . , kjn) if i = j. If thisexpression of ω contains a term aαl1

1 . . . αlnn (a ∈ K, a = 0), then it also contains

all terms of the form bα±l11 . . . α±ln

n with nonzero coefficients b ∈ K and allpossible distinct combinations of signs before l1, . . . , ln. (If there are m nonzeronumbers among l1, . . . , ln, then there are 2m such terms.)

For example, a σ∗-operator a1α21α2 + a2α

21α

−12 + a3α

−21 α2 + a4α

−21 α−1

2 withnonzero coefficients ai, 1 ≤ i ≤ 4, is symmetric.

Lemma 7.7.3 Let u and v be two σ∗-operators over an an inversive differ-ence field with a basic set σ = α1, . . . , αn. If at least one of the operators issymmetric, then ord(uv) = ord u + ord v.

PROOF. It is easy to see that if ω and ω′ are two σ∗-operators over K, thenord(ωω′) = ord(ω′ω). Therefore, without loss of generality we can assume thatu is symmetric. If ord u is equal to the order

∑ni=1 |li| of a monomial αl1

1 . . . αlnn

which appears in the standard form of u with a nonzero coefficient, then thisform of u contains all monomials α±l1

1 . . . α±lnn with nonzero coefficients (the

number of such monomials is 2m where m is the number of nonzero entries of


the n-tuple (l1, . . . , ln)). If t = aαk11 . . . αkn

n (a ∈ K, a = 0) is a term in thestandard form of v such that ord v = |k1| + · · · + |kn|, then the standard formof u contains a term t′ = bαε1l1

1 . . . αεnlnn (b ∈ K, b = 0, and εh is 1 or −1 for

h = 1, . . . , n) such that the n-tuples (ε1l1, . . . , εnln) and (k1, . . . , kn) belong tothe same ortant of Zn. (We refer to decomposition (1.5.5) of Zn into the unionof 2n ortants.) It is easy to see that ord(uv) is the order of the monomial in theproduct tt′, hence ord(uv) =

∑ni=1 |ki| +

∑ni=1 |li| = ord u + ord v.

With the above notation, let Γ denote the free commutative group generatedby the set σ. Then the ring of σ∗-polynomials R = Ky1, . . . , ys∗ coincides withthe polynomial ring K[γyj | γ ∈ Γ, 1 ≤ j ≤ s], and for any r ∈ N, we candefine a polynomial subring Rr = K[γ(ηj) | γ ∈ Γ(r), 1 ≤ j ≤ s] of R (asbefore, Γ(r) denotes the set γ ∈ Γ | ord γ ≤ r).

Let us consider a prime system of algebraic difference (σ∗-) equations of theform (7.7.2) and let P be a prime difference ideal of R generated (as a perfectdifference ideal) by σ∗-polynomials in the right-hand sides of the equations ofthe system. Furthermore, let ηi denote the canonical image of ηi in the factorring R/P (1 ≤ i ≤ s). As in the proof of Theorem 4.6.2, one can easily see thatfor every r ∈ N, P ∩ Rr is a prime ideal of the ring Rr and the quotient fieldsof the rings Rr/P ∩Rr and K[γ(ηj) | γ ∈ Γ(r), 1 ≤ j ≤ s] are isomorphic. ByTheorem 4.2.5, there exists a numerical polynomial ψP (t) in one variable t suchthat

ψP (t) = trdegKK[γ(ηj) | γ ∈ Γ(r), 1 ≤ j ≤ s] = trdegK(Rr/P ∩ Rr)

for all sufficiently large r ∈ Z, deg ψ(t) ≤ n and the polynomial ψP (t) can bewritten as

ψP (t) =2naP

n!tn + o(tn)

where aP = σ-trdegK(R/P ).

Definition 7.7.4 With the above notation, the numerical polynomial ψP (t) iscalled the difference dimension polynomial of the prime system of algebraic σ∗-equations.

It follows from Theorem 1.5.11 that the set all of numerical polynomialsassociated with prime systems of algebraic σ∗-equations is well-ordered withrespect to the order (recall that f(t) g(t) if and only if f(r)≤ g(r) for allsufficiently large r ∈ Z). Furthermore, by Proposition 4.2.22, if P and Q are twoprime σ∗-ideals of R such that P ⊇ Q, then ψP (t) ψQ(t), and the equalityψP (t) = ψQ(t) implies that P = Q.

The difference dimension polynomial of a prime system of algebraic difference(σ∗-) equations has an interesting interpretation as a measure of strength of asystem of such equations in the sense of A. Einstein. In his work [55] A. Einsteindefined the strength of a system of partial differential equations governing aphysical fields follows: “... the system of equations is to be chosen so that thefield quantities are determined as strongly as possible. In order to apply this


principle, we propose a method which gives a measure of strength of an equationsystem. We expand the field variables, in the neighborhood of a point P, intoa Taylor series (which presuppposes the analytic character of the field); thecoefficients of these series, which are the derivatives of the field variables at P,fall into sets according to the degree of differentiation. In every such degree thereappear, for the first time, a set of coefficients which would be free for arbitrarychoice if it were not that the field must satisfy a system of differential equations.Through this system of differential equations (and its derivatives with respect tothe coordinates) the number of coefficients is restricted, so that in each degree asmaller number of coefficients is left free for arbitrary choice. The set of numbersof “free” coefficients for all degrees of differentiation is then a measure of the“weakness” of the system of equations, and through this, also of its “strength”. ”

Considering a system of equations in finite differences over a field of of func-tions in several real variables, one can use the A. Einstein’s approach to definethe concept of strength of such a system as follows. Let

Ai(f1, . . . , fs) = 0 (i = 1, . . . , p) (7.7.3)

be a system of equations in finite differences with respect to s unknown grid func-tions f1, . . . , fs in n real variables x1, . . . , xn with coefficients in some functionalfield K. We also assume that the difference grid, whose nodes form the domain ofconsidered functions, has equal cells of dimension h1×· · ·×hn (h1, . . . , hn ∈ R)and fills the whole space Rn. As an example, one can consider a field K con-sisting of a zero function and fractions of the form u/v where u and v are gridfunctions defined almost everywhere and vanishing at a finite number of nodes.(As usual, we say that a grid function is defined almost everywhere if there areonly finitely many nodes where it is not defined.)

Let us fix some node P and say that a node Q has order i (with respect toP) if the shortest path from P to Q along the edges of the grid consists of isteps (by a step we mean a path from a node of the grid to a neighbor nodealong the edge between these two nodes). Say, the orders of the nodes in thetwo-dimensional case are as follows (a number near a node shows the order ofthis node).

P1 12 23 3

1

1

2 23 34 4

2

23

3

4

4

5

5

3 4 5

3 4 5

2 3 4

345

234

Let us consider the values of the unknown grid functions f1, . . . , fs at the nodeswhose order does not exceed r (r ∈ N). If f1, . . . , fs should not satisfy any


system of equations (or any other condition), their values at nodes of any ordercan be chosen arbitrarily. Because of the system in finite differences (and equa-tions obtained from the equations of the system by transformations of the formfj(x1, . . . , xs) → fj(x1 + k1h1, . . . , xs + knhn) with k1, . . . , kn ∈ Z, 1 ≤ j ≤ s),the number of independent values of the functions f1, . . . , fs at the nodes oforder ≤ r decreases. This number, which is a function of r, is considered asa “measure of strength” of the system in finite differences (in the sense of A.Einstein). We denote it by Sr.

With the above conventions, suppose that the transformations αj of the fieldof coefficients K defined by

αjf(x1, . . . , xn) = f(x1, . . . , xj−1, xj + hj , . . . , xn)

(1 ≤ j ≤ n) are automorphisms of this field. Then K can be considered asan inversive difference field with the basic set σ = α1, . . . , αn. Furthermore,assume that the replacement of the unknown functions fi by σ∗-indeterminatesyi (i = 1, . . . , s) in the ring Ky1, . . . , yn∗ leads to a a prime system of algebraicσ∗-equations (then the original system of equations in finite differences is alsocalled prime). The difference dimension polynomial ψ(t) of the latter systemis said to be the difference dimension polynomial of the given system in finitedifferences.

Clearly, ψ(r) = Sr for any r ∈ N, so the difference dimension polynomial ofa prime system of equations in finite differences is the measure of strength ofsuch a system in the sense of A. Einstein.

By Proposition 2.4.9, every system of homogeneous linear difference equa-tions is prime. Therefore, in order to determine the strength of a linear systemof equations in finite differences, one has to find the difference dimension poly-nomial of the linear σ∗-ideal P of Ky1, . . . , ys∗ generated by the left-handsides of the corresponding system of algebraic σ∗-equations. This problem, inturn, can be solved either by constructing a characteristic set of the ideal P (us-ing the procedure described before Theorem 2.4.13) and applying the formulain part (iv) of Theorem 4.2.5, or by the method based on Theorem 4.2.9 andformula (4.2.5). The latter approach requires the computation of the differencedimension polynomial of the module of differentials associated with the system.To realize this approach one should consider a generic zero η = (η1, . . . , ηs) ofthe σ∗-ideal P , a finitely generated σ∗-field extension L = K〈η1, . . . , ηs〉∗ ofK, and the corresponding E-module of differentials ΩL|K (E denotes the ringof σ∗-operators over L). As we have seen, the difference dimension polynomialof the given system of linear difference equations is the difference dimensionpolynomial of this module associated with the excellent filtration ((ΩL|K)r)r∈Z

where (ΩL|K)r is the vector L-space generated by the differentials d(γη) withγ ∈ Γ(r). (As before, Γ denotes the free multiplicative group generated by thebasic set σ and Γ(r) = γ ∈ Γ | ord γ ≤ r.) In order to find this polynomial, onecan construct a free resolution of filtered E-modules of type (4.2.4) and applyformula (4.2.5). Such a resolution has the form

0 → Mqdq−1−−−→ Mq−1 → . . .

d0−→ M0ρ−→ ΩL|K → 0


and can be constructed from the the following short exact sequences:

0 → Ker ρi0−→ M0

ρ−→ ΩL|K → 0,

0 → Ker π0i1−→ M1

π0−→ Ker ρ → 0,

0 → Ker π1ı2−→ M2

π0−→ Ker π0 → 0,

. . .

where i0, i1, . . . are inclusions and π0, π1, . . . are natural epimorphisms of filteredE-modules. The following diagram illustrates the construction of the resolution.

. . . d3 M3d2 M2

d1 M1 M0

d0 ρΩL|K

0

Ker π1 Ker ρ

Ker π0

000 0

0 0

π1 i1

i2 π0 i0

The next proposition describes the E-module Ker ρ for the system of homo-geneous linear difference equations of the form

s∑j=1

ωijyj = 0 (i = 1, . . . , p). (7.7.4)

(ωij are σ∗-operators with coefficients in an inversive difference field K witha basic set σ = α1, . . . , αn, and the left-hand sides of the equations are σ∗-polynomials in the ring Ky1, . . . , ys∗.)Proposition 7.7.5 With the above notation, let η = (η1, . . . , ηs) be a genericzero of the linear σ∗-ideal P of Ky1, . . . , ys∗ generated by the σ∗-polynomials

s∑j=1

ω1jyj , . . . ,

s∑j=1

ωpjyj. Let L = K〈η1, . . . , ηs〉∗, E the ring of σ∗-operators

over L, and ρ : M0 → ΩL|K the natural epimorphism of the free filtered E-module M0 = F 0

s with free generators f1, . . . , fs (we use the notation of Ex-

ample 3.5.4) onto the E-module of differentials ΩL|K =s∑

i=1

dηi (fi → dηi for

i = 1, . . . , s). Then Ker ρ is E-submodule of M0 generated by the elements

g1 =s∑

j=1

ω1jfj , . . . , gp =s∑

j=1

ωpjfj.


PROOF. Since d(ax) = adx and d(γx) = γdx for any a ∈ K, γ ∈ Γ, x ∈ L,

we obtain that ρ(gi) = ρ

⎛⎝ s∑j=1

ωijfj

⎞⎠ =s∑

j=1

ωijdηj = d

⎛⎝ s∑j=1

ωijηj

⎞⎠ = d0 = 0

for i = 1, . . . , p. Therefore,p∑

i=1

Egi ⊆ Ker ρ.

In order to show that Ker ρ ⊆p∑

i=1

Egi let us introduce, first, the following

terminology. If an element ω ∈ E is written in the irreducible form ω =l(ω)∑k=1

akγk

(l(ω) ∈ N, 0 = ak ∈ L, γk ∈ Γ (1 ≤ k ≤ l(ω)) and γi = γj for i = j), then the

number l(ω) will be called the length of ω. If g =q∑

k=1

ωkfk ∈ M0, then by the

length of g we mean the number l(g) =∑q

k=1 l(ωk).

Suppose that Ker ρ is not contained inp∑

i=1

Egi, and let g be an element

of minimal length in Ker ρ \p∑

i=1

Egi. Then g can be written as g =s∑

i=1

ukfk

where uk =lk∑

i=1

ckiγki ∈ E (cki ∈ L, γki ∈ Γ). Since ρ(g) =s∑

k=1

lk∑i=1

ckiγkidηk =

s∑k=1

lk∑i=1

ckid(γkiηk) = 0, the family γη = γkiηk | 1 ≤ k ≤ s, 1 ≤ i ≤ lkis algebraically dependent over K (see Proposition 1.7.13). Therefore, thereexists a difference polynomial A ∈ Ky1, . . . , ys, which is a polynomial in theset of indeterminates γkiyk | 1 ≤ k ≤ s, 1 ≤ i ≤ lk with coefficients in K,such that A(γη) = 0. Without loss of generality we may assume that A hasthe minimal total degree among all polynomials in the set of indeterminatesγkiyk | 1 ≤ k ≤ s, 1 ≤ i ≤ lk that vanish at γη. Then there exist k and i(1 ≤ k ≤ s, 1 ≤ i ≤ lk) such that

∂A

∂(γkiyk)(γη) =

∂A

∂(γkiyk)|yν=ην (1≤ν≤s) = 0.

Since A(γη) = 0, A lies in the σ-ideal

⎡⎣ s∑j=1

ω1jyj , . . . ,

s∑j=1

ωpjyj

⎤⎦ of the ring

Ky1, . . . , ys. Let ωµν =tµν∑r=1

bµνrγ′µνr where tµν ∈ N, bµνr ∈ K, and γ′

µνr ∈ Γ

(1 ≤ µ ≤ p, 1 ≤ ν ≤ s, 1 ≤ r ≤ tµν). Then the σ-polynomial A can be written as


A =p∑

µ=1

qµ∑λ=1

hµλγµλ

(s∑

ν=1

tµν∑r=1

bµνrγ′µνryν

)

=p∑

µ=1

qµ∑λ=1

hµλ

s∑ν=1

tµν∑r=1

γµλ(bµνr)γµλγ′µνryν , (7.7.5)

where qµ ∈ N, hµλ ∈ Ky1, . . . , ys, γµλ ∈ Γ (1 ≤ µ ≤ p, 1 ≤ λ ≤ qµ).Let Φ = γ′

abya | 1 ≤ a ≤ s, 1 ≤ b ≤ ma for some positive integersm1, . . . ,ms be the set of all terms γyi (γ ∈ Γ, 1 ≤ i ≤ s) which appear inA or in the last part of equation (7.7.5). (In particular, γkiyk ∈ Φ for anyk = 1, . . . , s; i = 1, . . . , lk such that γkiyk appears in A.)

Let A′ab =

∂A

∂(γ′abya)

(γη) for a = 1, . . . , s; b = 1, . . . , ma (it is easy to see that

if γ′abya is not equal to one of the γkiyk (1 ≤ k ≤ s, 1 ≤ i ≤ lk), then A′

ab = 0).The equality A(γη) = 0 implies dA(γη) = 0, that is,

s∑k=1

lk∑i=1

∂A

∂(γkiyk)(γη)d(γkiyk) =

s∑k=1

lk∑i=1

A′kiγkidηk =

s∑a=1

ma∑b=1

A′abγ

′abdηa = 0.

Let us show that

g0 =s∑

a=1

ma∑b=1

A′abγ

′abfa =

s∑k=1

lk∑i=1

A′kiγkifk ∈

p∑i=1

Egi.

Indeed, by (7.7.5) we have

g0 =s∑

a=1

ma∑b=1

⎧⎨⎩ ∂

∂(γ′abya)

⎡⎣ s∑µ=1

qµ∑λ=1

hµλ

s∑ν=1

tµν∑j=1

γµλ(bµνj)γµλγ′µνjyν

⎤⎦ (γη)

⎫⎬⎭ γ′abfa

=s∑

a=1

ma∑b=1

p∑µ=1

qµ∑λ=1

[(∂

∂(γ′abya)

hµλ

)γµλ

⎛⎝ s∑ν=1

tµν∑j=1

bµνjγ′µνjην

⎞⎠+ hµλ(γη)

s∑ν=1

tµν∑j=1

γµλ(bµνj)∂(γµλγ′

µνjyν)∂(γ′

abya)(γη)

]γ′

abfa

=p∑

µ=1

qµ∑λ=1

s∑ν=1

tµν∑j=1

hµλ(γη)γµλ(bµνj)s∑

a=1

ma∑b=1

∂(γµλγ′µνjyν)

∂(γ′abya)

(γη)γ′abfa

=p∑

µ=1

qµ∑λ=1

s∑ν=1

tµν∑j=1

hµλ(γη)γµλ(bµνj)γµλγ′µνjfν

=p∑

µ=1

qµ∑λ=1

hµλ(γη)γµλ

s∑ν=1

ωµνfν =p∑

µ=1

qµ∑λ=1

hµλ(γη)γµλgµ ∈p∑

i=1

Egi.


As we have seen, not all A′ki (1 ≤ k ≤ s, 1 ≤ i ≤ lk) are equal to zero. Without

loss of generality one can assume that A′11 = 0. Since g ∈ Ker ρ\

p∑i=1

Egi and g0 ∈p∑

i=1

Egi, we have g−c11(A′11)

−1g0 ∈ Ker ρ\∑pi=1 Egi and l(g−c11(A′

11)−1g0) <

l(g) (since the irreducible form of g0 contains only those elements of the formγfi (γ ∈ Γ, 1 ≤ i ≤ s) which appear in g, and g − c11(A′

11)−1g0 contains

fewer such elements than g). This contradiction with the choice of g shows that

Ker ρ =p∑

i=1

Egi.

Example 7.7.6 (Linear homogeneous symmetric σ∗-equation)Let K be an inversive difference field of zero characteristic with a basic set

σ = α1, . . . , αn and let E denote the ring of σ∗-operators over K. Consider adifference (σ∗-) equation

ω1y1 + · · · + ωsys = 0 (7.7.6)

where y1, . . . , ys are σ∗-indeterminates over K and ω1, . . . , ωs ∈ E are symmetricE-operators (see Definition 7.7.2). Let P denote the linear (and therefore prime)reflexive difference ideal generated by the σ∗-polynomial ω1y1 + · · · + ωsys andlet L denote the inversive difference field of quotients of Ky1, . . . , ys∗/P . Aftersetting ηi = yi + P ∈ L (1 ≤ i ≤ s) one may consider (7.7.6) as a defining σ∗-equation on the system of σ∗-generators η = (η1, . . . , ηs) of the σ∗-field extensionL/K (in the sense that the left-hand side of the equation generates the definingσ∗-ideal of L/K).

Let EL denote the ring of σ∗-operators over L. As we have seen, elementsdη1, . . . , dηs (where dηi denotes dL|Kηi) generate the module of differentialsΩL|K over the ring EL, so one can consider a natural homomorphism ρ of a freefiltered EL-module F 0

s with free generators f1, . . . , fs onto ΩL|K (ρ : fi → dηi,

1 ≤ i ≤ s). By Proposition 7.7.5, one has Ker ρ = ELg where g =s∑

i=1

ωifi.

The induced filtration on Ker ρ (with respect to which the embeddingKer ρ → F 0

s is a homomorphism of filtered EL-modules) is of the form(Ker ρ

⋂ s∑i=1

(EL)r fi

)r∈Z

. By Theorem 3.5.7, this filtration is excellent.

We are going to show that for all sufficiently large r ∈ Z,

Ker ρ⋂ s∑

i=1

(EL)r fi = (EL)r−k g (7.7.7)

where k = max1≤i≤sordωi.Indeed, since the σ∗-operators ω1, . . . , ωs are symmetric, ord(uωi) =

ord u+ordωi ≤ (r−k)+k = r for any u ∈ (EL)r−k, 1 ≤ i ≤ s (see Lemma 7.7.3).


Therefore, (EL)r−k g =

s∑

i=1

uωifi |u ∈ (EL)r−k

⊆ Ker ρ

⋂ s∑i=1

(EL)r fi.

Conversely, let A ∈ Ker ρ⋂ s∑

i=1

(EL)r fi Then

A =s∑

i=1

vifi = ωg =s∑

i=1

ωωifi

for some vi ∈ (EL)r (i = 1, . . . , s) and ω ∈ EL. Since f1, . . . , fs is basis ofthe free EL-module F 0

s , the last equality implies that vi = ωωi for i = 1, . . . , s.Since the σ∗-operators ωi are symmetric, one can apply Lemma 7.7.3 and obtainthat ordω = ord vi − ordωi (1 ≤ i ≤ s). It follows that there exists j ∈1, . . . , s such that ordω = ord vj − k ≤ r− k, so that ω ∈ (EL)r−k. Therefore,

Ker ρ⋂ s∑

i=1

(EL)r fi ⊆ (EL)r−k g, so that equality (7.7.7) is proved.

Let us consider the mapping π : F k1 → Ker ρ such that π(λh) = λg for every

λ ∈ EL (h denotes the element of the basis of F k1 ). It follows from (7.7.7) that

π is a homomorphism of filtered EL-modules and the sequence of such modulesF k

1π−→ Ker ρ

β−→ F 0s , where β is the injection, is exact in Ker ρ. Furthermore,

it is easy to see that Ker ρ = 0. (Indeed, if λh ∈ Ker π (λ ∈ EL), then λg =s∑

i=1

λωifi = 0. It follows that λ1ω1 = · · · = λsωs = 0 whence λ = 0.) Thus, we

obtain the following free resolution of the module ΩL|K :

0 → F k1

d0−→ F 0s

π−→ ΩL|K → 0. (7.7.8)

where d0 = β π. Applying to this resolution formulas (4.2.5) and (3.5.5), weobtain the dimension polynomial Ψη|K(t) of σ∗-equation (7.7.6):

Ψη|K(t) = s

n∑i=0

(−1)n−i2i

(n

i

)(t + i

i

)−

n∑i=0

(−1)n−i2i

(n

i

)(t − k + i

i

).

(7.7.9)

The last formula allows one to find the invariants σ∗-trdegKL, σ∗-typeKL,and σ∗-t.trdegKL of the σ∗-field extension L/K. These invariants coincide, re-spectively, with the characteristics σ∗-dimP , σ∗-type P , and σ∗-t.dimP of the

σ∗-ideal P =

[s∑

i=1

ωiyi

]∗: if s > 1, then σ∗-typeKL = σ∗-type P = n and

σ∗-trdegKL = σ∗-t.trdegKL = σ∗-dimP = σ∗-t.dimP = s − 1; if s = 1 andk ≥ 1 (the case k = 0 is trivial), then σ∗-typeKL = σ∗-type P = n − 1, σ∗-trdegKL = σ∗-dimP = 0, and σ∗-t.trdegKL = σ∗-t.dimP = 2k.

Let K be a field of functions of n real variables x1, . . . , xn, let h1, . . . , hn besome real numbers, and let the mappings αi : K → K defined by the equal-ity (αif)(x1, . . . , xn) = f(x1, . . . , xi−1, xi + hi, xi+1, . . . , xn) (f(x1, . . . , xn) ∈


K, 1 ≤ i ≤ n) be automorphisms of the field K. Then K can be treated asan inversive difference field with the basic set σ = α1, . . . , αn and the finitedifference approximation of linear differential equations produces algebraic σ∗-equations of the form (7.7.6). In what follows we determine dimension polyno-mials of such σ∗-equations associated with some classical differential equations.

Example 7.7.7 Using the five-point scheme for the finite difference approx-

imation of the two-dimension Laplace equation∂2y

∂x21

+∂2y

∂x22

= 0 we obtain a

σ∗-equation of the form[1h2

1

(α1 − α−11 − 2) +

1h2

2

(α2 − α−12 − 2)

]y = 0

where 0 = hl, h2 ∈ R. Applying (7.7.9) (with n = Card σ = 2), we see that thedimension polynomial for this equation is

Ψ(t) =2∑

i=0

(−1)2−i

(2i

)2i

[(t + i

i

)−

(t + i − 1

i

)]= 4t.

Note, that dimension polynomials of some other algebraic σ∗-equationswhich result from the finite difference approximations of classic differentialequations have the same form. For example, the dimension polynomial of theσ∗-equation[

12h1

(α1 − α−11 ) − δ

h2(α2 − α1 − α−1

1 − α−12 )

]y = 0 (h1, h2, δ ∈ R)

which results from the finite difference approximation of the simple diffusionequation by the scheme of Dufort and Frankel (see [160, Sect. 8.2, Table 8.1]),is also equal to 4t.

Example 7.7.8 The algebraic σ∗-equation obtained by the finite difference ap-proximation of two-dimensional Laplace equation via the nine-point scheme (see[77, Sect. 5.1]), has the form[

1h21(α1+α−1

1 − 2)+ 1h22(α2+α−1

2 − 2)+ h21+h2

212 (α1+α−1

1 − 2)(α2+α−12 −2)

]y=0

(h1, h2, δ ∈ R).Applying formula (7.7.9) we find the dimension polynomial Ψ(t) for this

equation (here, as in the previous case, n = Card σ = 2):

Ψ(t) =2∑

i=0

(−1)2−i2i

(2i

)[(t + i

i

)−

(t + i − 2

i

)]= 8t − 4.

Example 7.7.9 Consider the algebraic σ∗-equation obtained by finite differ-ence approximation of the Lorentz equation for the potentials of electromagnetic


field (see [174, App. 2 to Ch. 5, Sect. 2]) if every partial derivative is replacedby the corresponding central difference. This equation has the form

3∑i=1

1hi

(αi − α−1i )yi +

1τ

(α4 − α−14 )y4 = 0

where h1, h2, h3, τ ∈ R. The dimension polynomial of the equation is

Ψ(t) = 44∑

i=0

(−1)4−i2i

(4i

)(t + i

i

)−

4∑i=0

(−1)4−i2i

(4i

)(t + i − 1

i

)

= 48(

t + 44

)−80

(t + 3

3

)+40

(t + 2

2

)−5.

The two following examples deal with algebraic σ∗-equations obtained fromthe well-known systems of differential equations (see, for example, [150, Appen-dix, Sect. A4 and Sect. A15]) by a finite difference approximation where eachpartial derivative is replaced with the corresponding central difference. In bothof these examples K denotes an inversive difference field of coefficients, and αi

(1 ≤ i ≤ 4) are the elements of the basic set σ of K. The systems of algebraicσ∗-equations are treated as defining systems of equations on the generatorsη1, . . . , ηs of the σ∗-extension L = K〈η1, . . . , ηs〉∗. The ring of σ∗-operators overL is denoted by E .

In each example we write a free resolution of the module of differentialsΩL|K , compute the dimension polynomial Ψ(t) of the corresponding system,and find the invariants σ∗-trdeguKL, σ∗-typeKL, and σ∗-t.trdegKL.

Example 7.7.10 The finite difference approximation of the Dirac equation(with zero mass) produces the following system of linear σ∗-equations:⎧⎪⎪⎪⎨⎪⎪⎪⎩

a4(α4 − α−14 )y1 − a3(α3 − α−1

3 )y3 − [a1(α1 − α−11 ) + a2(α2 − α−1

2 )]y4 = 0,

a4(α4 − α−14 )y2 − [a1(α1 − α−1

1 ) − a2(α2 − α−12 )]y3 + a3(α3 − α−1

3 )y4 = 0,

a3(α3 − α−13 )y1 + [a1(α1 − α−1

1 ) + a2(α2 − α−12 )]y2 − a4(α4 − α−1

4 )y3 = 0,

[a1(α1 − α−11 ) − a2(α2 − α−1

2 )]y1 − a3(α3 − α−13 )y2 − a4(α4 − α−1

4 )y4 = 0

where the coefficients ai (1 ≤ i ≤ 4) belong to the field of constants C(K)of the σ∗-field K. Let ρ : F 0

4 → ΩL|K be the natural epimorphism of the freefiltered E-module F 0

4 with free generators f1, f2, f3, f4 onto ΩL|K (in our caseL = K(η1, η2, η3, η4)). By Proposition 7.7.5, we have Ker ρ =

∑4i=1 Egi where

g1 = a4(α4 − α−14 )f1 − a3(α3 − α−1

3 )f3 − [a1(α1 − α−11 ) + a2(α2 − α−1

2 )]f4,

g2 = a4(α4 − α−14 )f2 − [a1(α1 − α−1

1 ) − a2(α2 − α−12 )]f3 + a3(α3 − α−1

3 )f4,

g3 = a3(α3 − α−13 )f1 + [a1(α1 − α−1

1 ) + a2(α2 − α−12 )]f2 − a4(α4 − α−1

4 )f3,

g4 = [a1(α1 − α−11 ) − a2(α2 − α−1

2 )]f1 − a3(α3 − α−13 )f2 − a4(α4 − α−1

4 )f4.


Let us show that

Ker ρ ∩4∑

i=1

Erfi =4∑

i=1

Er−1gi. (7.7.10)

Indeed, the inclusion∑4

i=1 Er−1gi ⊆ Ker ρ is obvious (since σ∗-operator coef-ficients in the expressions of gi (1 ≤ i ≤ 4) are symmetric, ωgj ∈ ∑4

i=1 Erfi

for any ω ∈ Er−1). Conversely, let H =∑4

i=1 ωifi ∈ Ker ρ ∩ ∑4i=1 Erfi where

ωi ∈ Er (1 ≤ i ≤ 4). Then there exist λ1, λ2, λ3, λ4 ∈ E such that∑4

i=1 ωifi =∑4j=1 λjgj . Comparing the coefficients of fi 1 ≤ i ≤ 4) in the left- and right-

hand sides of the last equality, we obtain a system of four equations which, afterreducing to the row-echelon form, can be written as follows:

λ1a4(α4 − α−14 ) + λ3a3(α3 − α−1

3 ) + λ4[a1(α1 − α−11 ) − a2(α2 − α−1

2 )] = ω1,

λ2a4(α4 − α−14 ) + λ3[a1(α1 − α−1

1 ) − a2(α2 − α−12 )] − λ4a3(α3 − α−1

3 ) = ω2,

λ3u = ω′3,

λ4u = ω′4 (7.7.11)

where

u = a22(α2 − α−1

2 )2 + a24(α4 − α−1

4 )2 − a21(α1 − α−1

1 ) − a23(α3 − α−1

3 ),

ω′3 = ω3a4(α4 − α−1

4 ) − ω1a3(α3 − α−13 ) − ω2[a1(α1 − α−1

1 ) − a2(α2 − α−12 )],

ω′4 = ω4a4(α4 − α−1

4 ) − ω1[a1(α1 − α−11 ) − a2(α2 − α−1

2 )] + ω2a3(α3 − α−13 ).

Since the σ∗-operator u is symmetric and ω′3, ω

′4 ∈ Er+1, one has ord λ3 =

ordω′3 − ord u ≤ r− 1 and similarly ord λ4 ≤ r− 1. These inequalities, together

with the first two equations of (7.7.11), imply that ord(λ1a4(α4−α−14 )) ≤ r and

ord(λ2a4(α4−α−14 )) ≤ r, hence λ1, λ2 ∈ Er−1. It follows that H ∈ ∑4

i=1 Er−1gi,so the equality (7.7.10) is proved.

Let us consider the mapping π : F 14 → Ker ρ such that π

(∑4i=1 uihi

)=∑4

i=1 uigi where h1, h2, h3, h4 is a basis of the free filtered module F 14 over

E and u1, u2, u3, u4 ∈ E). Then equality (7.7.10) implies that the sequence of

filtered E-modules F 14

β π−−−→ F 04

ρ−→ ΩL|K → 0, where β is the injection Ker ρ →F 0

4 , is exact in F 04 .

If Q =∑4

i=1 µihi ∈ Ker π for some µ1, µ2, µ3, µ4 ∈ E , then∑4

i=1 µigi = 0.After replacing every gi (1 ≤ i ≤ 4) with its expression in terms of f1, . . . , f4 andequating the coefficients of every fj to zero we obtain that the σ∗-operators µj

(1 ≤ j ≤ 4) satisfy a system of equations which, after reducing to row-echelonform, has the form (7.7.11) with ω1 = ω2 = ω′

3 = ω′4 = 0 (and with µi = λi,

1 ≤ i ≤ 4). Since such a system has the unique solution µ1 = · · · = µ4 = 0, weobtain, that Q = 0, so that Ker π = 0. Thus, the free resolution of the moduleΩL|K has the form

0 → F 14

d0−→ F 04

ρ−→ ΩL|K → 0.


where d0 = β π. Applying formula (4.2.5) we obtain that

Ψ(t) = 44∑

i=1

(−1)4−i2i

(4i

)[(t + i

i

)−

(t + i − 1

i

)]

= 64(

t + 33

)− 128

(t + 2

2

)+ 96

(t + 1

1

)− 32.

In this case σ∗-trdegKL = 0, σ∗-typeKL = 3, σ∗-t.trdegKL = 8.

Exercise 7.7.11 Let K be an inversive difference field with a basic set σ =α1, α2, α3, α4 and let Ky1, y2, y3, y4∗ be the ring of σ∗-polynomials in σ∗-indeterminates y1, y2, y3, y4 over K. Let us consider the system of σ∗-equationsfrom Example 7.7.10 and denote by P the linear σ∗-ideal of Ky1, y2, y3, y4∗generated by the σ∗-polynomials

A1 = a4(α4 − α−14 )y1 − a3(α3 − α−1

3 )y3 − [a1(α1 − α−11 ) + a2(α2 − α−1

2 )]y4,

A2 = a4(α4 − α−14 )y2 − [a1(α1 − α−1

1 ) − a2(α2 − α−12 )]y3 + a3(α3 − α−1

3 )y4,

A3 = a3(α3 − α−13 )y1 + [a1(α1 − α−1

1 ) + a2(α2 − α−12 )]y2 − a4(α4 − α−1

4 )y3,

A4 = [a1(α1 − α−11 ) − a2(α2 − α−1

2 )]y1 − a3(α3 − α−13 )y2 − a4(α4 − α−1

4 )y4

in the left-hand side of this system.a) Show, that the σ∗-ideal P has a characteristic set consisting of the fol-

lowing eighteen σ∗-polynomials: A1, A2, A3, A4,

A5 = [−a21(α1 −α−1

1 )2 + a22(α2 −α−1

2 )2 − a23(α3 −α−1

3 )2 + a24(α4 −α−1

4 )2]y1,

A6 = [−a21(α1 −α−1

1 )2 + a22(α2 −α−1

2 )2 − a23(α3 −α−1

3 )2 + a24(α4 −α−1

4 )2]y2,

A7 = [−a3(α3 − α−13 )A1 + [a1(α1 − α−1

1 ) + a2(α2 − α−12 )]A2,

A8 = α−11 A1, A9 = α−1

1 α−12 A1, A10 = α−1

3 A2, A11 = α−14 A3,

A12 = α−14 A4, A13 = α−1

1 A5, A14 = α−11 α−1

2 A5, A15 = α−11 A6,

A16 = α−11 α−1

2 A6, A17 = α−11 A7, A18 = α−1

1 α−22 A7.

Consider the leaders of the σ∗-polynomials A1, . . . , A18 and use Theorems4.2.5 and 1.5.14 to find the the dimension polynomial Ψ(t) of the linear σ∗-ideal P .

Example 7.7.12 The system of linear σ∗-equations obtained by the finite dif-ference approximation of the system of Lame equations has the form[

Λ − 1h2

1

(α21 + α−2

1 − 2)]

y1 − 1h1h2

(α1α2 − α1α−12 + α−1

1 α−12 )y2

− 1h1h3

(α1α3 − α1α−13 + α−1

1 α−13 )y3 = 0,

− 1h1h2

(α1α2 − α1α−12 + α−1

1 α−12 )y1 +

[Λ − 1

h21

(α21 + α−2

1 − 2)]

y2


− 1h2h3

(α2α3 − α2α−13 + α−1

2 α−13 )y3 = 0,

− 1h1h3

(α1α3 − α1α−13 + α−1

1 α−13 )y1 − 1

h2h3(α2α3 − α2α

−13 + α−1

2 α−13 )y2

+[Λ − 1

h23

(α23 + α−2

3 − 2)]

y3 = 0 (7.7.12)

where Λ = − 1h2

4

(α24 + α−2

4 − 2) − a

4∑i=1

1h2

4

(α2i + α−2

i − 2) (a and hi (1 ≤ i ≤ 4)

are constants of the σ∗-field K).

Proceeding as in the previous example, we obtain that the free resolution ofthe module of differentials in this case has the form

0 → F 23

d0−→ F 03

π−→ ΩL|K → 0.

Using this resolution we arrive at the following expression for the dimensionpolynomial of system (7.7.12):

Ψ(t) = 34∑

i=1

(−1)4−i2i

(4i

)[(t + i

i

)−

(t + i − 2

i

)]= 96

(t + 3

3

)− 240

(t + 2

2

)+ 240

(t + 1

1

)− 120.

The invariants of the σ∗-field extension L/K defined by our system are as fol-lows: σ∗-trdegKL = 0, σ∗-typeKL = 3, and σ∗-t.trdegKL = 12.

The natural generalization of the concept of the strength of a system of dif-ference equations arises in the case when one evaluates the maximal numberof values of unknown grid functions that can be chosen arbitrarily in a regionwhich is not symmetric with respect to a fixed node P. More precisely, using theprevious settings (where algebraic σ∗-equations are considered over an inversivedifference field with a basic set σ = α1, . . . , αn), let us denote the automor-phisms α−1

i by αn+i (1 ≤ i ≤ n), add to the system (7.7.2) ns new equations(αiαn+i − 1)fj = 0 (1 ≤ i ≤ n, 1 ≤ j ≤ s) and divide the set σ = α1, . . . , α2ninto p disjoint subsets σ1, . . . , σp. Then for any r1, . . . , rp ∈ N, the transcendencedegree of the field

K

(αk1

1 . . . αk2n2n fj | k1, . . . , k2n ∈ N, 1 ≤ j ≤ s,

∑ν∈σi

ki ≤ ri for i = 1, . . . , p

)

over K is a function of r1, . . . , rp which can be naturally treated as a generalizedstrength of the given system of equations in finite differences. Theorem 4.2.16shows that this characteristic is a polynomial function of r1, . . . , rp. The compu-tation of the corresponding difference dimension polynomial in p variables canbe performed with the use of the technique of characteristic sets with respect to

7.8. DIMENSION POLYNOMIALS FOR TWO TRANSLATIONS 455

several orderings developed in section 4.2 (see the proof of Theorems 4.2.16 and4.2.17). However, this process is quite lengthy even for simple nonlinear differ-ence equations. In the case of linear difference equations one can use the obviousanalogue of Theorem 4.2.9 for multidimensional filtrations and Proposition 7.7.5to reduce the problem to the computation of the (σ1, . . . , σp)-dimension poly-nomial of a finitely generated σ-K-module with the basic set σ = α1, . . . , α2nand the corresponding fixed partition σ = σ1 ∪ · · · ∪ σp of this basic set.

Example 7.7.13 Let us consider the algebraic difference equation

α1y1 − α2y2 = 0 (7.7.13)

over an inversive difference field with a basic set σ = α1, α2. (The σ∗-indeterminates y1 and y2 represent unknown grid functions.)

Using the language of the A. Einstein’s approach, let us determine the num-ber S(r1, r2) of values of the grid functions that can be chosen freely (thatis, algebraically independently) in the nodes with coordinates (a, b) such that|a| ≤ r1, |b| ≤ r2. (We fix the origin at some node P of the grid and assign twocoordinates to every node of our two-dimensional grid in the natural way.)

By Theorem 4.2.17, there exists a polynomial Ψ(t1, t2) in two variables t1and t2 such that S(r1, r2) = Ψ(r1, r2) for all sufficiently large (r1, r2) ∈ Z2.Using the remark before this example, one can find Ψ(r1, r2) as the (σ1, σ2)∗-dimension polynomial of the σ∗-K-module M with two generators h1 and h2

and one defining relation α1h1 − α2h2 = 0. Using the result of Example 3.5.41,where this polynomial was computed with the use of a generalized Grobner basismethod developed in section 3.3, we obtain that

Ψ(t1, t2) = 4t1t2 + 4t1 + 4t2 + 2.

It follows from Theorem 4.2.17 that the degree 2 of the polynomial Ψ(t1, t2)and the coefficient 4 of the term t1t2 are the invariants of the inversive differencefield extension determined by equation (7.7.13).

7.8 Computation of Difference DimensionPolynomials in the Case of Two Translations

Let K be an inversive difference field of zero characteristic with a basic setσ = α, β, Γ a free commutative group generated by elements α and β, andfor every k ∈ N let

Γ(k) = γ = αiβj ∈ Γ | ord γ = |i| + |j| ≤ k.

Furthermore, let Ky∗ be the ring of σ∗-polynomials in one σ∗-indeterminatey over K, and for every σ∗-polynomial A ∈ Ky∗ let τ(A) denote the set of allpoints (p, q) of the real plane R2 such that the term αpβqy appears in A. Finally,


let πA denote the closed rectangle of the smallest area in the set of all closedrectangles which contain τ(A) and whose sides are perpendicular to the vectors〈1, 1〉 and 〈1,−1〉.

The following example illustrates the introduced concepts.

Example 7.8.1 Let us consider the σ∗-polynomial

A = (αβ−1y)(α−1βy)2 + (αy)(β−1y) + 2α2y + 2β2y − β−1y + y2.

Then the set τ(A) consists of the points (1,−1), (−1, 1), (1, 0), (0,−1), (2, 0),(0, 2), and (0, 0). In this case the rectangle πA is as follows.

•

•

• • •

•

•

(0,−1)

(0, 0) (1, 0)

(−1, 1)

(1,−1)

(0, 2)

(2, 0)

M1

M2

M3

M4

Exercise 7.8.2 Let M(x1, x2) and M ′(x′1, x

′2) be two adjacent vertices of the

rectangle πA (A ∈ Ky). Prove that |x′1 − x1| + |x′

2 − x2| ∈ N.

Exercise 7.8.3 Let A be a σ∗-polynomial in Ky and let d be a real numbersuch that |p| + |q| ≤ d for every term αpβq which appears in A. Prove that|x1| + |x2| ≤ d for any point M(x1, x2) of the rectangle πA.

The following definition generalizes the concept of effective order of an ordi-nary difference polynomial introduced in Section 2.4.

Definition 7.8.4 Let ρ denote a metric on R2 such that ρ(M,M ′) = |x′1 −

x1| + |x′2 − x2| for any two points M(x1, x2), M ′(x′

1, x′2) ∈ R2. With the above

notation, the rectangle πA of a σ∗-polynomial A ∈ Ky is called the basicrectangle of A; the perimeter of this rectangle in the metric ρ is called the σ∗-order of the σ∗-polynomial A and is denoted by σ∗-ordA.


It is easy to see that if A is the σ∗-polynomial considered in Example 7.8.1, thenσ∗-ordA = 16.

Lemma 7.8.5 With the above notation, σ∗-ordA = σ∗-ord γA for any σ∗-polynomial A ∈ Ky∗ and for any γ ∈ Γ.

PROOF. Let γ = αpβq where p, q ∈ Z. Then the basic rectangle πγA isthe result of shifting of the rectangle πA by the vector 〈p, q〉. Obviously thisgeometric transformation does not change the perimeter of the rectangle.

Theorem 7.8.6 Let K be an inversive difference field with a basic set σ =α, β and let Ky∗ be the ring of σ∗-polynomials of one σ∗-indeterminatey over K. Let A be a linear σ∗-polynomial in the ring Ky∗, A /∈ K, and letP = [A]∗ be the σ∗-ideal of the ring Ky∗ generated by A. Then σ∗-type P = 1,σ∗-t.dimP = (σ∗-ordA)/4, and the dimension polynomial ΨP (t) of the linear

σ∗-ideal P has the form ΨP (t) =σ∗-ordA

2t + a0, where a0 ∈ Z.

PROOF. By Corollary 2.4.12, one can choose a characteristic set A of theideal P as the set of minimal elements of the set γA | γ ∈ Γ with respectto the preorder on the ring Ky∗ considered in Corollary 2.4.12. Let us

represent Z2 in the form Z2 =4⋃

j=1

Zj where Z1 = N × N, Z2 = Z− × N,

Z3 = Z− × Z−, and Z4 = N × Z− (we use the notation of section 1.5). SettingΓj = γ = αpβq | (p, q) ∈ Zj and Yj = γy | γ ∈ Zj (1 ≤ j ≤ 4) we obtain the

corresponding representations Γ =4⋃

j=1

Γj and Y =4⋃

j=1

Yj of the group Γ and

the set of terms Y = γy | γ ∈ Γ, respectively.Since P = [γ(A)]∗ for any γ ∈ Γ, without loss of generality we may assume

that A belongs to the described characteristic set of the σ∗-ideal P and has thefollowing properties:

(i) The leader uA belongs to Y1.(ii) The sides of the basic rectangle πA, which are parallel to the vector 〈1, 1〉,

are not shorter than the other sides of the rectangle (with respect to the met-ric ρ).(iii) The rectangle πA contains the origin (0, 0).

Let us denote the sides of the basic rectangle by l1, . . . , l4 where the numer-ation goes in the counterclockwise direction starting with the side l1 which isparallel to the vector 〈−1, 1〉 and intersects the first quadrant. Furthermore, letdi denote the distance (in metric ρ) from the origin to the side li (1 ≤ i ≤ 4).It is easy to see that σ∗-ordA =

∑4i=1 di and if uA = αiβjy, then i + j = d1.

For every γ = αpβq ∈ Γ, the set τ(γA) is obtained by the shift of theset τ(A) by the vector 〈p, q〉. Therefore, in order to determine a characteristicset of the ideal [A]∗ we can consider only points of the set τ(A) which lie on


the sides li (1 ≤ i ≤ 4) of the basic rectangle πA. Actually, all leaders of σ∗-polynomials γ(A) (γ ∈ Γ), which lie in the same quadrant, are defined by one ofthe monomials of the σ∗-polynomial A: the leaders uγA from the jth quadrant(1 ≤ j ≤ 4) can be obtained by multiplication of certain term vj in the σ∗-polynomial A by some elements γ′ ∈ Γ. Clearly, v1 = uA and if γ ∈ Γ1, thenuγA = γv1. It follows that γA /∈ A for every γ ∈ Γ1, γ = 1.

In order to find elements of the characteristic set A, whose leaders lie in thesecond quadrant, we will consider polynomials γA with γ ∈ Γ2. Let be anorder on Γ such that αiβj αpβq if and only if the 3-tuple (|i|+ |j|, i, j) is lessthan (|p|+ |q|, p, q) with respect to the lexicographic order on Z3. Furthermore,for every set Λ ⊆ Γ, let µ(Λ) denote the minimal element of the set Λ withrespect to the order . Consider a sequence Λ0,Λ1,Λ2, . . . of subsets of Γ suchthat

Λ0 = Γ2,

Λ2k+1 = ∅ if uµ(Λ2k)A ∈ Y2,

α−1Λ2k otherwise

(k = 0, 1, . . . ),

Λ2k = ∅ if uµ(Λ2k−1)A ∈ Y2,

βΛ2k−1 otherwise.

(k = 1, 2, . . . ).Note that µ(Λ2k) = α−kβk and µ(Λ2k+1) = α−k−1βk (k ∈ N), so that if

γ′l = µ(Λl) (l ∈ N), one can write γ′

l = α−[ l+12 ]β[ l

2 ] (as usual, [r] denotes theinteger part of a real number r).

Let m be the minimal integer for which Λm = ∅. Then the σ∗-polynomialsγ′

kA (k = 0, . . . , m) belong to the characteristic set A, the leader uγ′mA = γ′v2

lies in the second quadrant, and the leaders uγ′lA

(0 ≤ l ≤ m− 1) lie in the firstquadrant if l is even, and in the third quadrant if l is odd.

Similarly, if we replace α by β and β by α, we will find a number n ∈ N andelements γ′′

0 , . . . , γ′′n ∈ Γ such that σ∗-polynomials γ′′

i A (0 ≤ i ≤ n) lie in thecharacteristic set A, and the leader uγ′′

nA = γ′′nv4 of the σ∗-polynomial γ′′

nA liesin the fourth quadrant.

Let us now consider two cases.Case I. Suppose that at least one of the following three conditions holds:

(a) max(m,n) > 1;(b) v2 = v3;(c)) v3 = v4.

In this case, we have a characteristic set A = γ′lA | 0 ≤ l ≤ m⋃γ′′

k A | 0 ≤k ≤ n of the σ∗-ideal P . The set of all leaders of σ∗-polynomials in A, whichwe denote by UA, is as follows:

UA = γ′2kv1 | 0 ≤ 2k < m

⋃γ′′

2lv1 | 0 ≤ 2l < n⋃

γ′2p+1v3 | 0 ≤ 2p + 1 < m⋃

γ′′2q+1v3 | 0 ≤ 2q + 1 < n

⋃γ′′

nv4. (7.8.1)


Case II. Suppose that none of the conditions (a) - (c) of Case I hold. Thenwe have a characteristic set

A = γ′lA | 0 ≤ l ≤ m

⋃γ′′

k A | 0 ≤ k ≤ n⋃

α−1β−1Aof the ideal P and the corresponding set UA, consisting of all leaders of σ∗-polynomials of A, is obtained by adjoining the term α−1β−1v3 to the set (7.8.1).

By Theorem 4.2.17, the dimension polynomial of the σ∗-ideal P coincideswith the standard Z-dimension polynomial φB(t) of the set B = τ(u) |u ∈UA ⊆ Z2. In Case I, when elements vi (1 ≤ i ≤ 4) are pairwise distinct, theset B does not contain points with zero coordinates. Setting Bi = B ∩ Z(2)

i weobtain that Bi = τ(u) ∈ B |u = γvi for some γ ∈ Γ (1 ≤ i ≤ 4).

By Theorem 1.5.23, the leading coefficient of the dimension polynomialφB(t) = a1t + a0 is equal to

∑4j=1(bj1 + bj2)− 4 where bjk = minτ(u)k | τ(u) ∈

Bk (k = 1, 2; 1 ≤ j ≤ 4). Note that if maxm,n > 1, then b11 = |τ(v1)1| −[m − 1

2

], b12 = |τ(v1)2| −

[n − 1

2

], b21 = |τ(γ′

mv2)1|, b22 = |τ(γ′mv2)2|, b31 =

|τ(v3)1| −[

m

2+ 1

], b32 = |τ(v3)2| −

[n

2+ 1

], b41 = |τ(γ′′

nv4)1|, and b42 =

|τ(γ′′nv4)2|.If m = n = 1, then b11 = |τ(v1)1|, b12 = |τ(v1)2|, b21 = |τ(v2)1| + 1, b22 =

|τ(v2)2|, b31 = |τ(v3)1|+ 1, b32 = |τ(v3)2|+ 1, b41 = |τ(v4)1|, b42 = |τ(v4)2|+ 1.In both cases the leading coefficient of the polynomial φB(t) is of the form

a1 =4∑

i=1

di =σ∗-ordA

2.

Now let us consider the case v1 = v2, v3 = v4, and n > 1. Then B containsone point with a zero coordinate, hence a1 =

∑4j=1(bj1 + bj2)− 3 (see Theorem

1.5.23). Therefore, in this case we also obtain that a1 =σ∗-ordA

2. It is easy to

see that every other case, where some of the terms v1, . . . , v4 are equal, leads tothe same expression for a1 (we leave the details to the reader as an exercise).

Thus, the σ∗-dimension polynomial of the σ∗-ideal P = [A]∗ is of the form

ΨP (t) =σ∗-ordA

2t + a0, where a0 ∈ Z. It follows that σ∗-type P = 1 and

σ∗-t.dimP =σ∗-ordA

4.

Corollary 7.8.7 Let K be an inversive difference field with a basic set σ =α, β and let P be a σ∗-ideal of the ring Ky∗, generated by one linear σ∗-polynomial A ∈ Ky∗\K. Let k be the minimal element of the set of all numbersl ∈ N such that the ring K[γy | γ ∈ Γ(l)] contains some element of the formγ′A where γ′ ∈ Γ. Then σ∗-t.dimP ≤ k.

PROOF. Since [A]∗ = [γA]∗ for every element γ ∈ Γ, without loss of genera-lity we may assume that A ∈ K[γy | γ ∈ Γ(k)]. In this case every point (p, q)


of the basic rectangle πA of the σ∗-polynomial A satisfies the condition |p| +|q| ≤ k, whence σ∗-ordA ≤ 4k. Applying Theorem 7.8.6 we obtain that σ∗-

t.dimP =σ∗-ordA

4≤ k.

Corollary 7.8.8 Let K be an inversive difference field with a basic set σ =α, β, Ky1, . . . , ys∗ the ring of σ∗-polynomials in σ-indeterminates y1, . . . , ys

over K and P a σ∗-ideal of this ring generated by one linear σ∗-polynomialA ∈ Ky1, . . . , ys∗ \ K. Let B denote the σ∗-polynomial obtained by replacingevery term γyj (γ ∈ Γ, 2 ≤ j ≤ s) of the polynomial A with γy1. Then theσ∗-dimension polynomial ΨP (t) of P has the form

ΨP (t) = 4(s − 1)(

t + 22

)+

(σ∗-ordB

2− 4s + 4

)(t + 1

1

)+ b (7.8.2)

where b ∈ Z.

PROOF. By Corollary 2.4.12, the set A consisting of the minimal elementsof the set γA | γ ∈ Γ with respect to the preorder is a characteristic set of theσ∗-ideal [P ]∗. (Recall that if A and B are two σ∗-polynomials in Ky1, . . . , ys∗,then A B if and only if the leader uB is a transform of uA.)

For every j = 1, . . . , s, let Bj = (p, q) ∈ Z2 |αpβqyj is a leader of someσ∗-polynomial in A. Then Theorem 4.2.5(iv) shows that ΨP (t) =

∑sj=1 φBj

(t)where φBj

(t) is the standard Z-dimension polynomial of the set B (1 ≤ j ≤ s).

Now, let us represent each set Bj (1 ≤ j ≤ s) in the form Bj =4⋃

i=1

Bji where

Bji is the family of all points of the set Bj that lie in ith quadrant (1 ≤ i ≤ 4).It is easy to see (as in the corresponding part of the proof of Theorem 7.8.6)that all leaders αpβqyj of σ∗-polynomials of A, for which the correspondingpoints (p, q) lie in the same quadrant, are determined by one of the terms of theσ∗-polynomial A. Therefore, if we fix i (1 ≤ i ≤ s), then Bji is not empty foronly one index j (1 ≤ j ≤ s). Thus, if Q is a σ∗-ideal of the ring Ky1, . . . , ys∗generated by the σ∗-polynomial B, then ΨP (t) = ΨQ(t) =

∑sj=1 φB′

j(t) where

B′j = (p, q) ∈ Z2 | the term αpβqyj is the leader of some σ∗-polynomial in the

characteristic set of Q described in Corollary 2.4.12 (1 ≤ j ≤ s).Since B ∈ Ky1∗, B′

j = ∅ for j = 2, . . . , s. Therefore (see Theorem

1.5.17(iv)), φB′j(t) = 4

(t + 2

2

)− 4

(t + 1

1

)+ 1 (2 ≤ j ≤ s). Since φB′

1(t) =

σ∗-ordB

2

(t + 1

1

)+ b′ with b′ ∈ Z (see Theorem 4.2.5), we obtain formula

(7.8.2).

We complete this section with several examples of computation of dimensionpolynomials associated with systems of linear σ∗-polynomials. In each of theexamples we construct a characteristic set of the corresponding linear σ∗-idealof the ring of σ∗-polynomials and then apply the formula in the last part of


Theorem 4.2.5. The advantage of this method (in comparison with the methodbased on the construction of a free resolution of the corresponding module ofdifferentials) is that it can be applied to any (not necessarily symmetric) systemof linear σ∗-equations.

Example 7.8.9 Let K be an inversive difference field of zero characteristicwith a basic set σ = α1, α2 and let Ky∗ be the ring of σ∗-polynomials inone σ∗-indeterminate y over K. Let us consider the σ∗-equation

[a1(α1 + α−11 − 2) + a2(α2 + α−1

2 − 2)]y = 0 (7.8.3)

where a1 and a2 are constants of the field K. Note that the dimension polynomialof such an equation was computed in Example 7.7.7 (see also Example 7.7.6)with the use of the free resolution of the σ∗-L-module of differentials ΩL|K whereL is the σ∗-field of quotients of the factor ring Ky∗/[A]∗, A = [a1(α1 +α−1

1 −2) + a2(α2 + α−1

2 − 2)]y.It is easy to see that the basic rectangle πA of the σ∗-polynomial A is a square

with the vertices (1, 0), (0, 1), (−1, 0) and (0,−1). It follows that σ∗-ordA = 8.Applying Theorem 7.8.6 we obtain that the σ∗-dimension polynomial of our σ∗-equation (i.e., the σ∗-dimension polynomial of the σ∗-ideal [A]∗) has the formΨ(t) = 4t + b where b ∈ Z. Therefore, σ∗-dim [A]∗ = 0, σ∗-type [A]∗ = 1, andσ∗-t.dim [A]∗ = 2.

Furthermore, Corollary 2.4.12 shows that there is a characteristic set of theσ∗-ideal [A]∗ consisting of the σ∗-polynomials A, α−1

1 A = [a1(1+α−21 −2α−1

1 )+a2(α−1

1 α2+α−11 α−1

2 −2α−11 )]y, and α−1

1 α−12 A = [a1(α−1

2 +α−21 α−1

2 −2α−11 α−1

2 )+a2(α−1

1 + α−11 α−2

2 − 2α−11 α−1

2 )]y whose leaders are the terms α1y, α−11 α2y, and

α−11 α−2

2 y, respectively.Thus, the σ∗-dimension polynomial Ψ(t) of the σ∗-equation (7.8.3) coin-

cides with the standard Z-dimension polynomial of the set B = (1, 0), (−1, 1),(−1,−2) ⊆ Z2. Applying Theorem 1.5.14(iii) and formula (1.5.4) for the com-putation of Kolchin polynomials of subsets of Nm we obtain that Ψ(t) =φB(t) = 4t.

Exercise 7.8.10 Let K be an inversive difference field of zero characteristicwith a basic set σ = α1, α2 and Ky∗ the ring of σ∗-polynomials in oneσ∗-indeterminate y over K. Let us consider a σ∗-equation

[a1(α1 − 1) − a2(α1α2 + α1α−12 + α2 + α−1

2 − 2α1 − 2)]y = 0 (7.8.4)

where a1 and a2 are constants in K. (This equation is obtained by the finitedifference approximation of the diffusion equation using the Krank and Nicolsonscheme, see [160, Chapter 8, Section 2]).

a) Find the basic rectangle of the σ∗-polynomial A = [a1(α1−1)−a2(α1α2+α1α

−12 + α2 + α−1

2 − 2α1 − 2)]y and use Theorem 7.8.6 to show that the σ∗-dimension polynomial of equation (7.8.4) is of the form Ψ(t) = 6t + b. Findσ∗-dim [A]∗, σ∗-type [A]∗, and σ∗-t.dim [A]∗.


b) Use Corollary 2.4.12 to show that the σ∗-polynomials A, α−11 A, α−1

2 A, andα−1

1 α−12 A form a characteristic set of the σ∗-ideal [A]∗. Conclude that Ψ(t) =

φB(t), where B = (1, 1), (−1, 1), (1,−2), (−1,−2) ⊆ Z2, and use Theorem1.5.14(iii) and formula (1.5.4) to show that Ψ(t) = 6t − 1.

Example 7.8.11 Let K be an inversive difference field of zero characteristicwith a basic set σ = α1, α2 and let Ky1, y2∗ be the ring of σ∗-polynomialsin one σ∗-indeterminates y1 and y2 over K. The finite difference approximationof the wave equation leads to the system of σ∗-equations of the form

a1(α1 − 1)y1 − a2(α2 − α−12 )y2 = 0,

a2(α2 − α−12 )y1 − a1(α1 − 1)y2 = 0

where a1 and a2 are constants in K.Let P denote the σ∗-ideal [A1, A2]∗ where A1 = a1(α1−1)y1−a2(α2−α−1

2 )y2

and A2 = a2(α2 − α−12 )y1 − a1(α1 − 1)y2. The application of Corollary 2.4.12

leads to a characteristic set of the ideal P consisting of the σ∗-polynomials A1,A2, A3 = a1(α1 − 1)A1 − a2(α2 − α−1

2 )A2 = [a21(α1 − 1)2 − a2

2(α2 − α−12 )2]y1,

A4 = α−12 A1 = a1(α1α

−12 − α−1

2 )y1 − a2(1 − α−22 )y2,

A5 = α−11 A2 = a2(α−1

1 α2 − α−11 α−1

2 )y1 − a1(1 − α−11 )y2,

A6 = α−11 α−1

2 A2 = a2(α−11 − α−1

1 α−12 )y1 − a1(α−1

2 − α−11 α−1

2 )y2.The leaders of these σ∗-polynomials are α2y2, α1y2, α2

1y1, α−22 y2, α−1

1 α2y1,and α−1

1 α−22 y1, respectively.

By Theorem 4.2.5, the σ∗-dimension polynomial Ψ(t) of our system ofalgebraic difference equations can be represented as a sum of the standardZ-dimension polynomials φF1(t) and φF2(t) of the subsets F1 = (2, 0), (−1, 1),(−1,−2) and F2 = (0, 1), (1, 0), (0,−2) of Z2, respectively. ApplyingTheorem 1.5.14 and Corollary 1.5.8 (formulas (1.5.6) and (1.5.4)) we obtainthat φF1(t) = 6t − 1 and φF2(t) = 2t + 1. Therefore, Ψ(t) = 8t.

Chapter 8

Elements of the DifferenceGalois Theory

In this chapter we consider some basic aspects of the difference Galois theory.The first section is devoted to the study of Galois groups of normal and separable(but not necessarily finite) difference field extensions and the application ofthe results this study to the problems of compatibility and monadicity. Thecorresponding theory was developed by P. Evanovich in [60]. The other twosections of the chapter present a review of fundamentals of two approaches tothe Picard-Vessiot theory of ordinary difference field extensions. The originalversion of this theory, which adjusts the main ideas of the the Picard-Vessiottheory of differential fields to difference case, is due to C. Franke. In section8.2 we give a review of this approach omitting proofs of most of the results(we prove just a few fundamental statements on Picard-Vessiot extensions). Thedetail exposition of the theory can be found in the fundamental C. Franke’s work[67], in his further papers [68]–[73], and also in the works by R. Infante [88] and[90]. Section 8.3 provides a review of the basics of a Galois theory of differenceequations developed by M. Singer and M. van der Put. This theory, which isbased on the study of Picard-Vessiot difference rings, is perfectly presented inthe monograph [159] where the reader can also find interesting applicationsof the results on difference Galois groups to the analytic theory of differenceequations.

8.1 Galois Correspondence for Difference FieldExtensions

Let K be an inversive difference field with a basic set σ = α1, . . . , αn and L aσ∗-overfield of K. As before, Gal(L/K) and Galσ(L/K) will denote the Galoisgroup and difference (or σ-) Galois group of the extension L/K, that is, thegroups of all K-automorphisms and all difference (σ-) K-automorphisms of the

463

464 CHAPTER 8. ELEMENTS OF THE DIFFERENCE GALOIS THEORY

field L, respectively. We will also consider induced automorphisms αi (1 ≤ i ≤ n)of the group Gal(L/K) (αi : θ → α−1

i θαi for any θ ∈ Gal(L/K) ). Recall thata subgroup B of Gal(L/K) is said to be σ-stable if αi(B) = B (i = 1, . . . , n)and σ-invariant if αi(b) = b for every b ∈ B,αi ∈ σ. As we have seen, if F/Kis a σ∗-field subextension of L/K, then Gal(L/F ) = θ ∈ Gal(L/K) | θ(a) = afor every a ∈ F is a σ-stable subgroup of Gal(L/K). Also, if B is a σ-stablesubgroup of Gal(L/K), then a ∈ L|θ(a) = a for every θ ∈ B is a σ∗-overfieldof K. This field will be denoted by L(B).

In what follows, the group Gal(L/K) is considered as a topological groupwith the Krull topology. Recall that a fundamental system of neighborhoods ofthe identity in this topology is the set of all groups Gal(L/F ) ⊆ Gal(L/K) suchthat F is a subfield of L which is a Galois extension of K of finite degree. (WhenL is of finite degree over K, the Krull topology is discrete.) The topologicalgroup G = Gal(L/K) is compact, Hausdorff, and has a basis at the identityconsisting of the collection of invariant, open (and hence closed and of finiteindex in G) subgroups of G (see Proposition 1.6.53). If H is a closed normalσ-stable subgroup of G, then each αi induces a topological automorphism βi

on G/H such that βi(gH) = αi(g)H for any g ∈ G. Now one can obtain thefollowing theorem on the Galois correspondence of difference fields in the sameway as its classical analog, Theorem 1.6.54 (see the proof in [144, Section 17]).

Theorem 8.1.1 Let L,K, and G = Gal(L/K) be as above. Then(i) The mapping F → Gal(L/F ) establishes a 1-1 correspondence between the

set of σ∗-field subextensions of L/K and the set of closed σ-stable subgroups ofG. If F/K is a σ∗-field subextension of L/K, then L(Gal(L/F )) = F , and if His a closed σ-stable subgroup of G, then Gal(L/L(H)) = H.

(ii) Let F/K be a σ∗-field subextension of L/K such that F is normal overK. Let γ1, . . . , γn be the automorphisms of the group Gal(F/K) induced byα1, . . . , αn (treated as automorphisms of F/K), respectively. Then there existsa natural isomorphism of topological groups φ : G/Gal(L/F ) → Gal(F/K) suchthat φβi = γiφ (1 ≤ i ≤ n) where βi denotes the automorphism of G/Gal(L/F )induced by αi.

In what follows, while dealing with extensions of inversive difference fields weuse the notation of section 2.1. If K is an inversive difference field with a basicset σ = α1, . . . , αn, M a σ∗-field extension of K and Γ the free commutativegroup generated by σ, then for any γ ∈ Γ, the automorphism g → γ−1gγ of theGalois group Gal(M/K) will be denoted by γ. (Clearly, if γ = αk1

1 . . . αknn with

k1, . . . , kn ∈ Z, then γ = αk11 . . . αkn

n .) The material presented below is based onthe results of P. Evanovich [60].

Theorem 8.1.2 Let K be an inversive difference field with a basic set σ =α1, . . . , αn and let M be a σ∗-overfield of K such that the field extensionM/K is normal and separable. Let G = Gal(M/K) and let α1, . . . , αn be auto-morphisms of G induced by α1, . . . , αn, respectively. Furthermore, let Γ denotethe free commutative group generated by α1, . . . , αn. Then:

8.1. GALOIS CORRESPONDENCE 465

(i) An intermediate σ∗-field L of M/K is a finitely generated σ∗-field ex-tension of K if and only if there exists an open subgroup H of G such that⋂γ∈Γ

γ(H) = Gal(M/L).

(ii) Let L be an intermediate σ∗-field of M/K such that L/K is normal andL is a finitely generated σ∗-field extension of K. Then there exists a normalsubgroup H of G such that

⋂γ∈Γ

γ(H) = Gal(M/L).

PROOF. Suppose that L = K〈S〉∗ for some finite set S ⊆ L, and let A =Gal(M/K(S)). Then A is a closed subgroup of G and since K(S) : K < ∞, wehave G : A < ∞. Let B =

⋂g∈G

g−1Ag. Then B is a closed normal subgroup of

finite index in G. Since G is compact, the subgroup B is open.Denoting the group Gal(M/L) by C we obtain that

C = Gal(M/K(⋃γ∈Γ

γ(S))) =⋂γ∈Γ

Gal(M/K(γ(S))) =⋂γ∈Γ

γ(A)

⊇⋂γ∈Γ

γ(BC) ⊇ C.

It follows that H = BC is and open subgroup of G and⋂γ∈Γ

γ(H) = C. If

the field extension L/K is normal, then C is a σ-invariant subgroup of G andH is normal.

Conversely, suppose that H is an open subgroup of G such that Gal(M/L) =⋂γ∈Γ

γ(H). Then L(H) : K < ∞ and since Gal(M/L) ⊆ H, L(H) = K(S) for

some finite set S ⊆ L. Since L is the fixed field of the group⋂γ∈Γ

γ(H), we have

L = K(⋃γ∈Γ

γ(S)), so that L = K〈S〉∗. This completes the proof of both parts

of our theorem.

Exercise 8.1.3 With the notation of Theorem 8.1.2, suppose that n > 1. Showthat if H is a σ-invariant subgroup of the Galois group G = Gal(M/K), L =∩αk1

1 . . . αkn−1n−1 | k1, . . . , kn−1 ∈ Z, and σ′ = α1, . . . , σn−1, then for every

k ∈ Z, the field αn(L) is σ′-stable.

Exercise 8.1.4 With the notation of Theorem 8.1.2, show that M is a finitelygenerated σ∗-field extension of K if and only if G has an open invariant subgroupH such that ∩γ∈Γγ(H) = e where e is the unity of the group G.

Definition 8.1.5 Let M be a difference field with a basic set σ and N aσ-overfield of M . The extension N/M is said to be universally compatibleif given a σ-field extension Q/M , there exist a σ-field extension R/M andσ-homomorphisms φ : N/M → R/M and ψ : Q/M → R/M .


The following three statements are immediate consequences of the defini-tions. We leave the proofs to the reader as an exercise.

Proposition 8.1.6 If a difference (σ-) field extension L/K is universally com-patible, then any its σ-field subextension F/K is universally compatible.

Proposition 8.1.7 Let K be a difference (σ-) field, L a σ-overfield of K, andM a σ-overfield of L. If the extensions M/L and L/K are universally compat-ible, then M/K is also universally compatible.

Proposition 8.1.8 Let K be an inversive ordinary difference field with a basicset σ = α and let L be a σ-overfield of K such that the field extension L/K isnormal and separable. If H is an open normal subgroup of G such that α(H) ⊆H, then H is σ-stable.

In what follows we concentrate on the questions of universal compatibilityand monadicity of ordinary difference field extensions which are normal andseparable. The corresponding results are due to P. Evanovich (see [60]).

Proposition 8.1.9 Let K be an ordinary inversive difference field with a basicset σ = α and L a σ∗-overfield of K such that L/K is a Galois extension.Furthermore, let G = Gal(L/K), H = Galσ(L/K), and λ the mapping of thegroup G into itself defined by λ(θ) = θ−1α(θ) (θ ∈ G). Then

(i) λ is a continuous function.(ii) H is a closed subgroup of G.(iii) The extension L/K is universally compatible if and only if λ maps G

onto itself.

PROOF. It is easy to see that λ is the composition of the continuous maps

µ : G → G × G and ν : G × G → G

defined by µ(θ) = (θ−1, α(θ) ) and ν(θ1, θ2) = θ1θ2, respectively. Therefore, λ iscontinuous. Furthermore, H = λ−1(e), where e is the identity of the group G,hence H is a closed subgroup in G. The last statement is the direct consequenceof Theorem 5.6.31.

Theorem 8.1.10 Let K be an ordinary inversive difference field with a basicset σ = α and L a σ∗-overfield of K such that L/K is a Galois extension.Then L/K is universally compatible if and only if every σ-subextension of L/K,which is a finite Galois extension of K, is universally compatible.

PROOF. Let G = Gal(L/K) and let λ be the mapping of G into itselfdefined in Proposition 8.1.9 (λ : θ → θ−1α(θ)). Suppose that L/K is not uni-versally compatible. Then there exists an element θ ∈ G such that θ /∈ λ(G)(see Proposition 8.1.9(iii) ). Since the group G is compact and Hausdorff, andthe map λ is continuous, θ−1λ(G) is a closed subset of G. Furthermore, the


identity e of the group G does not lie in θ−1λ(G), hence there exists an opennormal subgroup N of G such that θ−1λ(G)

⋂N = ∅.

Let H be maximal among all open normal subgroups N of G which aredisjoint with θ−1λ(G). We are going to show that the group H is σ-stable.Taking into account Proposition 8.1.8, one just needs to show that α(H) ⊆ H.Assuming that this is not so, we obtain that Hα(H) is a proper open normalsubgroup of G properly containing H. Therefore, θ−1λ(G)

⋂Hα(H) = ∅, so

there exist elements γ ∈ G and h1, h2 ∈ H such that θ−1λ(γ) = h1α(h2), henceθ−1γ−1α(γ)α(h−1

2 ) = h1. Since the subgroup H is normal, there exists g ∈ Hsuch that θ−1h−1

2 = gθ−1. Therefore, gθ−1h2γ−1α(γh−1

2 ) = gθ−1λ(γh−12 ) =

h1θ−1λ(γh2) = g−1h1 ∈ θ−1λ(G)

⋂H = ∅, a contradiction. Thus, α(H) ⊆ H,

so that the subgroup H is σ-stable. Clearly, L(H)/K is a finite Galois σ-fieldsubextension of L/K.

Notice now, that there is no γ ∈ G such that θH = λ(γ)H. (Indeed, ifθH = λ(γ)H, then θ−1λ(γ) ∈ H

⋂θ−1λ(G). ) It follows that the finite Galois

extension L(H)/K is not universally compatible.The converse statement is a direct consequence of Proposition 8.1.6. In what follows K denotes an inversive ordinary difference field with a basic

set σ = α and L is a σ∗-overfield of K such that L/K is a Galois (that is,normal and separable) field extension. Furthermore, G will denote the Galoisgroup Gal(L/K) and e will denote the unity of G. We say that the group G is offinite type if the σ∗-field extension L/K is finitely generated (that is, L = K〈S〉∗for a finite set S).

Definition 8.1.11 The intersection of all open σ-stable normal subgroups of Gis called the core of the group G. It is denoted by C(G).

It is easy to see that C(G) is a closed σ-stable normal subgroup of G and Gis the intersection of all open σ-stable subgroups of G.

Exercise 8.1.12 Prove that L(C(G)) is the core of the extension L/K in thesense of Definition 4.3.17.

Proposition 8.1.13 If G is of finite type and C(G) = e, then the group Gis finite.

PROOF. Let G be the set of all open σ-stable normal subgroups of G. Let

us choose an open subgroup N of G such that∞⋂

k=0

αk(N) = e. Since the

group G is compact and the elements of G are closed, there exist subgroups

H1, . . . , Hn ∈ G such that H =n⋂

j=0

Hj ⊆ N . Clearly, H is σ-stable, hence

e = C(G) ⊆ H ⊆∞⋂

k=0

αk(N) = e. Since H is open in G, we obtain that

CardG = (G : H) < ∞.


Corollary 8.1.14 If the group G is of finite type, then(i) (G : C(G)) < ∞.(ii) C(C(G)) = C(G).

PROOF. It follows from Theorem 4.4.1 that the factor group G/C(G) is offinite type. Furthermore, one can easily see that the core of G/C(G) with respectto the automorphism induced on G/C(G) by α is C(G). Applying Proposition8.1.13 we obtain that (G : C(G)) < ∞.

Statement (ii) follows from the fact that if H is an open normal subgroup ofC(G), then (G : H) = (G : C(G))(C(G) : H) < ∞, hence H = C(G).

Corollary 8.1.15 Even without the requirement that G is a group of finite type,one has C(C(G)) = C(G).

PROOF. By Corollary 8.1.14, if G is a group of finite type, then C(C(G)) =C(G).

Let us consider the general case (G is not necessarily a group of finite type).Let C = C(G) and let C contain a proper open σ-stable normal subgroup H.Then there exists an open normal subgroup N of G such that A = N

⋂C ⊆ H.

Since H is σ-stable, αk(A) ⊆ H for every k ∈ Z. Let A′ be the subgroup of H

generated by the set⋃k∈Z

αk(A) (clearly, this is a σ-stable normal subgroup of G)

and let B denote the closure of A′ in G. Since the closure of a σ-stable subgroupis obviously σ-stable, B is a closed σ-stable normal subgroup of G contained inH. Furthermore, (C : B) < ∞.

It is easy to see that C(G/B) = C/B = B and one can replace G with G/Band α by the automorphism induced on G/B by α to reduce our considerationsto the case where C is a nontrivial finite subgroup of G.

Under this reduction, e is an open subgroup of G and there is an opennormal subgroup D of G such that D

⋂C = e. Then P =

⋃k∈Z

αk(D) is a

closed σ-stable normal subgroup of G and G/P is of finite type.We are going to show that C(G/P ) = CP/P . Obviously, CP/P ⊆ C(G/P ).

If CP = G, then G/P = CP/P ∼= C/C⋂

P = C ⊆ e, hence (G : P ) <∞ (that is, P is open in G). Since C is the intersection of all open σ-stablesubgroups of G, we obtain that C ⊆ P , hence e = P

⋂C = C contradicting

the fact that C = e.Thus, CP G, so there is an element θ ∈ G \ CP . Then θP

⋂C = ∅.

Furthermore, it follows from the definition of C and compactness of the groupG that there exist a finite number of open σ-stable subgroups N1, . . . , Nq of

G such that θP⋂ q⋂

i=1

Ni = ∅. Let E =q⋂

i=1

Ni. Then EP/P is an open σ-

stable subgroup of G/P not containing θP . Therefore, CP/P ⊇ C(G/P ), henceCP/P = C(G/P ).


Thus, G/P is a G/P is a group of finite type with core CP/P ∼= C/C⋂

P =C which is finite. Then P/P is an open σ-stable normal subgroup of C(G/P )that contradicts Corollary 8.1.14. Theorem 8.1.16 With the assumptions made before Definition 8.1.11, theextension L/K is universally compatible if and only if the σ∗-field extensionL(C(G))/K is universally compatible.

PROOF. Let C = C(G). The result of Corollary 8.1.15 implies that the ex-tension L/L(C) has no finite σ∗-field subextensions. Applying Theorem 8.1.10we obtain that the extension L/L(C) is universally compatible. Now our state-ment follows from Proposition 8.1.7.

It follows from the definition of a monadic extension (see Definition 5.6.1)that any Galois (and therefore normal) difference (σ-) field extension L/K ismonadic if and only if Galσ(L/K) is the identity group. This observation impliesthe following result.

Theorem 8.1.17 With the above notation and assumptions (L/K is a Galoisσ∗-field extension of an inversive ordinary difference field with a basic set σ =α, G = Gal(L/K), and λ is a mapping of G into itself such that λ(θ) =θ−1α(θ) = θ−1α−1θα for any θ ∈ G), the extension L/K is monadic if andonly if λ is injective. Corollary 8.1.18 With the assumptions of the last theorem, if the extensionL/K is monadic, then any its σ∗-field subextension F/K is monadic.

PROOF. The statement immediately follows from Theorem 8.1.17, since ifthe map λ is one-to-one on the group G = Gal(L/K), then λ is one-to-one onany subgroup of G. Lemma 8.1.19 With the assumptions of Theorem 8.1.17, let N be an opennormal subgroup of G = Gal(L/K) such that

⋂k∈Z

αk(N) = e. The the natural

group homomorphism π : Galσ(L/K) → G/N is injective (and therefore, if L/Kis a finitely generated σ∗-field extension, then Galσ(L/K) is a finite group).

PROOF. If θ ∈ Ker π, then θ ∈ N hence θ = αk ∈ αk(N) for all k ∈ Z. Itfollows that θ ∈

⋂k∈Z

αk(N) = e. Thus, π is injective.

Proposition 8.1.20 Keeping the previous notation and assumptions, supposethat a σ∗-field extension L/K is finitely generated. Then the extension L/K ismonadic if and only if every its σ∗-field subextension F/K with L : F < ∞ ismonadic.

PROOF. Suppose that any σ∗-field subextension F/K of L/K with L :F < ∞ is monadic. If L/K is not monadic, then Galσ(L/K) = e. By Lemma8.1.19, the group H = Galσ(L/K) is finite and L : L(H) < ∞. Since L/L(H)is not monadic, we obtain a contradiction with our assumption. The conversestatement follows from Corollary 8.1.18.


Proposition 8.1.21 Let L/K be as above and let F/K be a σ∗-field subexten-sion of L/K such that F/K is Galois. If the extension L/K is monadic andL/F is universally compatible, then F/K is monadic.

PROOF. Let G = Gal(L/K) and H = Gal(L/F ). Then the map λ : G → Gdefined in Proposition 8.1.9 is injective and maps H onto itself (see Theorem8.1.17 and Proposition 8.1.9(iii) ). Let us show that the map λ′ : G/H → G/H,defined by λ′(θH) = λ(θ)H for any θ ∈ G, is injective. Then Theorem 8.1.17will imply that F/K is monadic.

Let θH ∈ Ker λ′, that is, λ(θ) ∈ H. Since λ maps H onto itself, there existsh ∈ H such that λ(θ) = λ(h). Since λ is injective on G, θ = h, so θH = H.Thus, λ′ is injective, hence F/K is monadic.

Theorem 8.1.22 Let K be an ordinary inversive difference field with a basicset σ = α and L a σ∗-overfield of K such that L/K is a Galois extension. LetG = Gal(L/K) and C = C(G), the core of G. If the extension L/K is monadicand L(C)/K is a finitely generated σ∗-field extension, then L/K is universallycompatible.

PROOF. As in the proof of Theorem 8.1.16 we obtain that the extensionL/L(C) is universally compatible. By Proposition 8.1.21, the extension L(C)/Kis monadic. Therefore, the mapping λ′ : H → G/H defined in the proof ofProposition 1.8.21 (with H = G/Gal(L/L(C))) is surjective, hence the extensionL(C)/K is universally compatible. Applying Theorem 8.1.16 we obtain thatL/K is also universally compatible.

Corollary 8.1.23 With the notation and assumptions of the last theorem, ifthe σ∗-field extension L/K is finitely generated and monadic, then L/K is uni-versally compatible.

PROOF. By Theorem 4.4.1, the extension L(C)/K is finitely generated.Now Theorem 8.1.22 implies that L/K is universally compatible.

With the above notation, if an extension L/K is not finitely generated,then the monadicity of L/K does not imply the universal compatibility of thisextension. The following example of a monadic Galois difference field extension,which is not universally compatible, is due to P. Evanovich [60].

Example 8.1.24 Let us consider Q as an inversive ordinary difference fieldwhose basic set σ consists of the identity automorphism α. Let ω1 = −1 andfor n = 2, 3, . . . , let ωn denotes the primitive 2nth root of unity such thatω2

n = ωn−1. Then ωn−1ωn is also a primitive 2nth root of unity.For every n ∈ N, n ≥ 1, let us define inductively an automorphism αn

of the field Q(ωn) such that and αn(ωn) = ωn−1ωn and the restriction of αn

on Q(ωn−1) is αn−1 (α1 = α). Let us consider the field K =∞⋃

n=1

Q(ωn) as a

σ∗-ovefield of Q whose translation (also denoted by α) is defined by the conditionα = αn on Q(ωn) (n = 1, 2, . . . ).


Let η1 be a root of the polynomial X2 − 5. Then it is easy to see thatη /∈ K, K

⋂Q(η1) = Q, and the field extensions K/Q and Q(η1)/Q are normal.

Therefore (see Theorem 1.6.24), the extensions K/Q and Q(η1)/Q are linearlydisjoint. Since there exists an automorphism β of Q(η1)/Q such that β(η1) = η1,it follows from Theorem 1.6.43 that there exists an automorphism γ1 of thefield K(η1) such that γ1(η1) = η1 and γ1 extends α. Let L1 be the inversivedifference field field (K(η1), γ1) treated as a σ∗-overfield of K. (As before, wewill denote the underlying field of L1 by L1 and write L1 = (L1, γ1) where γ1

is the translation of L1.)The extension L1/K is not universally compatible, since it is finitely gener-

ated and not monadic. In particular, one has a nontrivial difference automor-phism φ of L1/K defined by φ(η1) = −η1. By Theorem 8.1.10, any σ∗-fieldextension of L1 is not a universally compatible extension of K. Let us induc-tively define a sequence of inversive difference fields Ln |n = 1, 2, . . . in whichevery Ln is a difference field extension of Ln−1 such that Ln = Ln−1(ηn) whereηn is a zero of the polynomial X2n − 5, η2

n = ηn−1, and the translation γn of Ln

is defined by the condition γn(ηn) = ωpnηn with p = 2 or p = 2n−1 + 2.

Let L =∞⋃

n=1

Ln (that is, L =∞⋃

n=1

Ln and the translation of L, coincide with

γn on Ln). Then L/K is a Galois inversive difference field extension and by theabove remark it is not universally compatible. Let us show that L/K is monadic.Let χ be a difference automorphism of L/K. Then for every n = 1, 2, . . . , therestriction of χ on Ln is a difference automorphism of Ln/K. Let χ(ηn) = ωj

nηn

where j ∈ 1, 2, . . . , 2n. Then χγn(ηn) = χ(ωpnηn) = ωp

nωjnηn where p = 2 or

p = 2n−1 + 2. Furthermore, γnχ(ηn) = γn(ωjnηn) = (ωn−1ωn)jωp

nηn.Since χγn = γnχ, one has ωp+j

n = ωp+3jn , hence ω2j

n = 1. It follows thatχ(ηn−1) = χ(η2

n) = (ωjnηn)2 = ηn−1, so the restriction of χ on Ln−1 is the

identity automorphism. Since n is arbitrary, χ is the identity automorphism ofL. Thus, the extension L/K is monadic.

Theorem 8.1.25 Let K be an ordinary inversive difference field with a basicset σ = α and L a σ∗-overfield of K such that L/K is a Galois extension.Let F/K be a Galois σ∗-field subextension of L/K. Then

(i) Galσ(L/F ) is a closed σ-stable normal subgroup of Galσ(L/K).(ii) There exists a natural monomorphism

ψ : Galσ(L/K) /Galσ(L/F ) → Galσ(F/K).

If L/F is universally compatible then ψ is an isomorphism.(iii) If L/K is universally compatible and ψ is an isomorphism, then L/F is

universally compatible.

PROOF. Let G = Gal(L/K), H = Gal(L/F ), D1 = Galσ(L/K), D2 =Galσ(L/F ), and D3 = Galσ(F/K).

Statement (i) follows from the fact that H is a closed normal subgroup of Gand D2 = D1

⋂H.


To prove (ii) let us identify D3 with the subgroup of G/H consisting ofthe cosets θH for which λ(g) = e (λ is the mapping of the group G into itselfdefined in Proposition 8.1.9, e is the unity of G). If π : G → G/H is the naturalhomomorphism and θ ∈ D1, then λ(θ) = e ∈ H. It follows that the restrictionof π on D1 is a homomorphism of D1 into D3 whose kernel is D1

⋂H = D2.

Let ψ denote the monomorphism of D1/D2 into D3 induced by π. Then ψ is acontinuous and closed map. If the extension L/F is universally compatible, sothat λ(H) = H and θH ∈ D3 (θ ∈ G), λ(θ) ∈ H and there exists h ∈ H suchthat λ(h) = λ(θ). It follows that λ(θh−1) = e, θh−1 ∈ D1, and ψ(θh−1D2) =θH. Thus, ψ is an isomorphism of topological groups, so (ii) is proved.

Suppose that ψ is an isomorphism and λ(G) = G (that is, L/K is universallycompatible). Let h ∈ H. Then there exists θ ∈ G such that λ(θ) = h, gH ∈ D3.Since ψ is an isomorphism, there exists θ1 ∈ D1 such that θ1 = θh1 for some ele-ment h1 ∈ H. Then θh1 = θ1 = α(θ1) = α(θh1) = θhα((h1), hence λ(h−1

1 ) = h.Thus, λ(H) = H, so the extension L/F is universally compatible.

The following result was obtained in [60] with the use of the theory of limitgroups of ordinary difference fields developed by P.Evanovich. We refer to thework [60] for the proof.

Theorem 8.1.26 Let K be an ordinary inversive difference field with a basic setσ = α and let L/K be a finitely generated monadic Galois σ∗-field extensionof K. Then the group Gal(L/K) is finite.

8.2 Picard-Vessiot Theory of LinearHomogeneous Difference Equations

Most of the results of this section are due to C. Franke and contained in hisworks [67] - [70]. In what follows we assume that all fields have characteristiczero and all difference fields are inversive and ordinary. If K is such a differencefield with a basic set σ = α, then its field of constants c ∈ K|α(c) = cwill be denoted by CK . All topological statements of this section will refer tothe Zariski topology. As before, Ky will denote the ring of σ-polynomials inone σ-variable y over K. Furthermore, for any n-tuple b = (b1, . . . , bn), C∗(b) orC∗(b1, . . . , bn) will denote the determinant of the matrix (αibj)0≤i≤n−1,1≤j≤n.This determinant is called the Casorati determinant of b1, . . . , bn.

Lemma 8.2.1 With the above notation, the following two conditions are equiv-alent:

(i) Elements b1, . . . , bn are linearly dependent over the constant field of anydifference field containing them.

(ii) C∗(b) = 0.

8.2. PICARD-VESSIOT THEORY 473

PROOF. Ifn∑

i=1

cibi = 0 for some constants c1, . . . , cn, the rows of C∗(b) are

linearly dependent, hence C∗(b) = 0.Conversely, suppose that C∗(b) = 0. We are going to show property (i) by

induction on n. Since the case n = 1 is trivial, it is sufficient to show that ifb1, . . . , bn are elements of a difference field K such that C∗(b) = 0 and no propersubset of b1, . . . , bn has Casorati determinant zero, then b1, . . . , bn are linearlydependent over the constant field of any difference field containing them. LetAi denote the cofactor of αn−1bi in C∗(b) (i = 1, . . . , n). By our assumption,all Ai are different from zero, so the rank of the matrix of C∗(b) is n − 1.Furthermore, it is easy to see that (−1)n−1αAi is a cofactor of bi (αAi denotesthe determinant obtained by applying α to every element of the determinantAi). Since C∗(b) = 0, we obtain that

A1(αjb1) + · · · + An(αjbn) = 0,

αA1(αjb1) + · · · + αAn(αjbn) = 0 (j = 0, . . . , n − 1). (8.2.1)

Since the rank of the matrix of C∗(b) is n − 1, these equations imply thatαAi/αA1 = Ai/A1 for i = 2, . . . , n. Setting ci = Ai/A1 (i = 2, . . . , n) and

c1 = 1 we obtain constants c1, . . . , cn ∈ K〈b1, . . . , bn〉 such thatn∑

i=1

cibi = 0.

This completes the proof.

Let us consider a linear homogeneous difference equation of order n over K,that is, an algebraic difference equation of the form

αny + an−1αn−1y + · · · + a0y = 0 (8.2.2)

where a0, . . . , an−1 ∈ K (n > 0) and a0 = 0.

Definition 8.2.2 A σ∗-overfield M of K is said to be a solution field overK for equation (8.2.2) or a solution field over K for the σ-polynomial f(y) =αny +an−1α

n−1y + · · ·+a0y, if M = K〈b〉∗ for an n-tuple b = (b1, . . . , bn) suchthat f(bj) = 0 for j = 1, . . . , n and C∗(b) = 0. Any such n-tuple b is said to bea basis of M/K or a fundamental system of solutions of (8.2.2). If, in addition,CK is algebraically closed and CM = CK , then M is said to be a Picard-Vessiotextension (PVE) of K. (By Lemma 8.2.1, the elements b1, . . . , bn are linearlyindependent over the constant field of any difference field containing them.)

Proposition 8.2.3 With the above notation, let M be a σ∗-overfield of K andR ⊆ CM . Then

(i) A subset of R linearly (algebraically) dependent over K is linearly (respec-tively, algebraically) dependent over CK .

(ii) If N is a σ∗-overfield of K with CN = CK , then N and K(R) are linearlydisjoint over K.(iii) CK(R) = CK(R).


PROOF. (i) Suppose that the first statement of our proposition is false andW ⊆ R is a minimal linearly dependent set over K which is linearly independent

over CK . Then there exist w0, w1, . . . , wm ∈ W such that w0 =m∑

i=1

kiwi for

some k1, . . . , km ∈ K. Since 0 = α(w0)−w0 =m∑

i=1

(α(ki)− ki)wi, we arrive at a

contradiction with the minimality of W .Let a set U ⊆ R be algebraically dependent over K and let V be a basis of K

treated as a vector CK-space. If f(u1, . . . , up) = 0 for some u1, . . . , up ∈ U andsome polynomial in p variables f ∈ K[X1, . . . , Xp], then every coefficient of fcan be written as a linear combination of elements o V . Therefore, there are somepolynomials h1, . . . , hr ∈ CK [X1, . . . , Xp] and elements v1, . . . , vr ∈ V such that

f =r∑

j=1

hjvj . Since the elements v1, . . . , vr are linearly independent over CK ,

their Casorati determinant is not zero, so v1, . . . , vr are linearly independent overCK〈u1,...,up〉 (see Lemma 8.2.1). Therefore, hj(u1, . . . , up) = 0 for j = 1, . . . , r,so u1, . . . , up are algebraically dependent over CK .

(ii) Clearly, the set of all power products of elements of R contains a basis Bof the vector K-space K[R]. Since elements of B are constants, statement (i)implies that the set B is linearly independent over N . By Theorem 1.6.24, thefields N and K(R) are linearly disjoint over K.(iii) It is easy to see that if the last statement of the proposition is false, it is

false for a finite set R. By induction it is sufficient to consider the case whereR consists of a single element u. If u is algebraic over K, then every element

w ∈ CK(R) ⊆ K(R) = K(u) can be written uniquely as w =d∑

i=0

aiui for some

a0, a1, . . . , ad ∈ K (we assume d ≥ 1, since the case R ⊆ K is trivial). Then

0 = α(w) − w =d∑

i=0

(α(ai) − ai)ui hence ai ∈ CK for i = 0, . . . , d. Therefore,

CK(R) ⊆ CK(R).

If u is transcendental over K, the arbitrary element v ∈ CK(R) ⊆ K(u) can be

written uniquely as a ratio of two relatively prime polynomials in u: v =f(u)g(u)

.

Then 0 = α(v)−v =fα(u)gα(u)

− f(u)g(u)

(as usual, fα denotes a polynomial obtained

by applying α to every coefficient of the polynomial f), hence fα(u)g(u) =f(u)gα(u). Since the polynomials f(u) and g(u) (as well as fα(u) and gα(u)) arerelatively prime, fα(u) = f(u) and gα(u) = g(u), so that all coefficients of f andg are constants. Thus, in this case we also have the inclusion CK(R) ⊆ CK(R).Since the opposite inclusion is obvious, this completes the proof.


Proposition 8.2.4 Let K be a difference field with a basic set σ = α,Ky the ring of σ-polynomials in one σ-variable y over K, and f = αny +an−1α

n−1y + · · · + a0y a homogeneous linear difference polynomial in Ky(a0, . . . , an−1 ∈ K, a0 = 0). Then equation (8.2.2) (whose left-hand side is f)has n distinct solutions u1, . . . , un such that

(i) The elements u1, . . . , un are linearly independent over the field of constantof K〈u1, . . . , un〉.

(ii) If v is any solution of equation (8.2.2) lying in some difference overfield ofK〈u1, . . . , un〉, then v can be written as v = c1u1 + · · · + cnun where c1, . . . , cn

are constants of the field K〈u1, . . . , un, v〉.(iii) The σ-field extension K〈u1, . . . , un〉/K is compatible with every σ-field

extension of K.

PROOF. Clearly, the variety M(f) is irreducible and ordM = n. Let u1

be a generic zero of this variety. Then we define ui (i = 2, . . . n) inductively asa generic zero of the variety of f over K〈u1, . . . , ui−1〉.

It is easy to see that the set αjui | 1 ≤ i ≤ n, 0 ≤ j ≤ n−1 is algebraicallyindependent over K, so the Casorati determinant of the elements of this set isnot zero. Applying Lemma 8.2.1 we obtain the first statement of our proposition.

If v is any solution of (8.2.2) in a σ-overfield of K〈u1, . . . , un〉, then

αnui + an−1αn−1ui + · · · + a0ui = 0 (i = 1, . . . , n),

αnv + an−1αn−1v + · · · + a0v = 0. (8.2.3)

It follows that the Casorati determinant of u1, . . . , un, v is zero, so that theseelements are linearly dependent over the field of constants of K〈u1, . . . , un, v〉.Since u1, . . . , un are linearly independent over CK〈u1,...,un,v〉, this completes theproof of statement (ii).

By Proposition 5.1.12, for every i, K〈u1, . . . , ui〉 is a purely transcendentalfield extension of K〈u1, . . . , ui−1〉 and, therefore, of K. Applying Theorem 5.1.6we obtain that K〈u1, . . . , un〉/K is compatible with every σ-field extension ofthe field K.

The last proposition implies that if M is a solution field for equation (8.2.2)over K with basis b = (b1, . . . , bn) and b′ is any solution of (8.2.2) in a σ∗-

overfield N of M , then b′ =n∑

i=1

cjbj for some elements c1, . . . , cn ∈ CN . It

follows that a σ-homomorphism h of Kb/K into a σ∗-overfield N of M de-

termines an n × n-matrix (cij) over CN by the equations h(bi) =n∑

i=1

cijbj .

The following theorem and corollary show that the matrices corresponding toσ-homomorphisms satisfy a set of algebraic equations over CM , and, in the caseof a PVE, form an algebraic matrix group.


Theorem 8.2.5 If M/K is a solution field with basis b = (b1, . . . , bn), thenthere is a set Sb in the polynomial ring CM [Xij | 1 ≤ i, j ≤ n] such that if Nis a σ∗-overfield of M then the following hold.

(i) A σ-homomorphism of Kb/K to N/K (that is, a difference K-homomorphism of Kb to N) determines a matrix in CN that annuls everypolynomial of Sb. (In the last case we say “the matrix satisfies Sb”).

(ii) A matrix in CN satisfying Sb defines a σ-homomorphism of Kb/K toN/K.(iii) If CM = CK then a σ-homomorphism of Kb/K to N/K determines a

σ-isomorphism if and only if its matrix is nonsingular.

PROOF. Let Ky1, . . . , yn be the ring of σ-polynomials in σ-indeterminatesy1, . . . , yn over K and let P be a prime σ∗-ideal of Ky1, . . . , yn with genericzero b. Let φ be a ring homomorphism from Ky1, . . . , yn to M [Xij | 1 ≤ i, j ≤n] defined by φ(yi) =

n∑j=1

bjXij and let J = φ(P ). Then every polynomial h ∈ J

can be written ash =

∑gkvk (8.2.4)

where gk ∈ CM [Xij | 1 ≤ i, j ≤ n] and vk are elements of a basis V of Mover CM (when M is treated as a vector CM -space). We define Sb to be the setg ∈ CM [Xij | 1 ≤ i, j ≤ n] | g appears as some gk in representation (8.2.4) ofsome polynomial h ∈ J.

(i) If θ is a difference K-homomorphism of Kb to N with matrix (cij)≤i,j≤n,

then the mappings defined by yi → bi → θ(bi) and yi →n∑

j=1

bjXij →n∑

j=1

bjcij

are identical. Therefore, the latter mapping sends P to zero, and every polyno-mial in J vanishes for Xij = cij . Since the linear independence of V over CM

carries over to CN , all the polynomials of Sb vanish at cij .(ii) If (cij)1≤i,j≤n is a matrix over CN satisfying Sb, then the mapping de-

fined by yi →n∑

j=1

bjXij →n∑

j=1

bjcij is a difference K-homomorphism from

Ky to N whose kernel contains P . This homomorphism induces a differenceK-homomorphism of Kb to N .(iii) To prove the last statement of the theorem it is sufficient to show that

if φ is the σ-homomorphism of Kb/K to N/K defined by a nonsingular ma-trix (cij)1≤i,j≤n over CN (as it is shown in the proof of (ii)), then φ is in-jective. Suppose that this is not so. Then Proposition 1.6.32(v) implies thattrdekKK〈b1, . . . , bn〉 > trdegKK〈φ(b1), . . . , φ(bn)〉. (The transcendence degreesare finite, since each bi satisfies (8.2.2).)

Denoting the sets b1, . . . , bn, φ(b1), . . . , φ(bn) and cij | 1 ≤ i, j ≤ n byb, φ(b) and c, respectively, and using the additive property of transcendencedegree we obtain that

trdegK〈b〉K〈b, φ(b)〉 < trdekK〈φ(b)〉K〈b, φ(b)〉.


Furthermore, denoting the field CK = CM by C we obtain that

trdegK〈b〉K〈b, φ(b)〉 = trdegK〈b〉K〈b, c〉 = trdegCC(c) .

(The last equality follows from Proposition 8.2.3(i).) Similarly,

trdegK〈φ(b)〉K〈b, φ(b)〉 = trdegC′C ′(c)

where C ′ denotes the field of constants of K〈φ(b)〉. We have arrived at a contra-diction with the obvious inequality trdegC′C ′(c) ≤ trdegCC(c). This completesthe proof of the theorem.

Part (iii) of the last theorem immediately implies the following statement.

Corollary 8.2.6 If M/K is a PVE then the difference Galois group Galσ(M/K) is an algebraic matrix group over CK .

If M is a solution field for equation (8.2.2) and b = (b1, . . . , bn) a basisof M/K, then Sb will denote the set of polynomials in Theorem 8.2.5 (Sb ⊆CM [Xij | 1 ≤ i, j ≤ n] ), and Tb will denote the variety of Sb over the alge-braic closure of CM . The following example shows that a matrix in Tb may notcorrespond to a difference homomorphism of Kb.Example 8.2.7 ([Franke, [67]). Let b be a solution of the difference equationαy + y = 0 which is transcendental over K. Then the constant field of K(b)contains CK(b2), Sb = 0, and Tb contains the algebraic closure of CK(b2).Since no σ∗-overfield of K〈b〉∗ contains b in its constant field, Theorem 8.2.5does not apply to the matrix (b). The algebraic isomorphism h : Kb → Kbdefined by h(b) = b2, is not a σ-homomorphism.

Let L be an intermediate σ∗-field of a difference (σ∗-) field extension M/K(M is a solution field for equation (8.2.2) over K). A σ∗-overfield N of M issaid to be a universal extension of M for L if every σ-isomorphism of L overK can be extended to a σ-isomorphism of M into N . It follows from Theorem5.1.10 that if L is algebraically closed in M , then universal extensions of M forL exist. At the same time, one can show (see [70, Example 4.2]) that even if Litself is a solution field over K, M need not be a universal σ∗-field extensionfor L.

Theorem 8.2.8 Let M be a solution field for equation (8.2.2) over an ordinarydifference field K with a basic set σ = α and let b be a basis of M/K. Ifthe field K is algebraically closed in M , then the variety Tb is irreducible anddimTb = trdegKM .

PROOF. Let us define the prime σ∗-ideal P of the ring of σ-polynomialsKy1, . . . , yn (this ring will be also denoted by Ky) and the ring homomor-phism φ from Ky to the polynomial ring M [Xij | 1 ≤ i, j ≤ n] (also denotedby M [X]) as it is done in the proof of Theorem 8.2.5. Furthermore, let f denotethe linear difference polynomial in the left-hand side of (8.2.2).


Let J = φ(P ) and let S′b denote the ideal generated by Sb in the polynomial

ring CM [Xij | 1 ≤ i, j ≤ n] (this ring will be denoted by CM [X]). If P ′ isthe perfect ideal generated by P in My (that is, in the ring of σ-polynomialsMy1, . . . , yn), then P ′ is a prime σ∗-ideal consisting of linear combinations ofelements of P with coefficients in M (see Corollary 7.1.19). Let J ′ = φ(P ′) andlet Q′ be the set of all polynomials g ∈ CM [X] which appear when each h ∈ J ′ iswritten as a finite sum h =

∑k

gkvk for a vector space basis vk of M over CM .

Note that φ maps My onto M [X], since the equations φ(αkyi) =∑

j

Xijαkbj

can be solved for the Xij showing that each Xij is in the range of φ.Let b′ be a generic zero of P ′. Then f(b′) = 0, hence there are constants

c′ij in a difference overfield N of M such that b′i =∑

j

c′ijbj . If h ∈ M [X]

and h = φ(u), then h(Xij) = φ(h(yi)) = u(∑

j

Xijbj), hence h(c′ij) = u(bi). It

follows that h(c′ij) = 0 if and only if u ∈ P ′ and J ′ is a prime ideal in M [X] withgeneric zero (c′ij). Since vk is linearly independent over the field of constantsCN , Q′ is a prime ideal in CM [X] with generic zero (c′ij).

If p ∈ P ′, then p =∑

j

pjmj for some pj ∈ P and mj ∈ M . Therefore,

any element in J ′ can be written as φ(p) =∑

j

φ(pj)mj where φ(pj) ∈ J . If

h ∈ Q′, then there exists z ∈ J ′ such that z = hv1 +∑

j

hjvj . Furthermore,

z =∑

j

zjmj =∑i,j

gijvimj for some gij ∈ Sb. The last inequality implies that

z =∑i,j,k

gijdijkvijk for some dijk ∈ CM . Since every element has a unique

expression as a linear combination of elements of a vector space basis, h =∑dijkgij where the sum is taken over all i, j, k with vijk = v1. Therefore,

h ∈ S′b and since S′

b ⊆ Q′, we obtain that S′b = Q′. Since Q′ is a prime ideal,

the variety Tb is irreducible.Since M/K is compatible with the extension formed by a generic zero of P ,

we have ordP = ordP ′, hence trdegKM = ordP = ordP ′ = trdegMM〈b′〉 =trdegMM(c′ij) = trdegCM

CM (c′ij) = dimQ′ = dimTb. This completes theproof.

Let M be a σ∗-overfield of a difference field K with a basic set σ. We saythat the extension M/K is σ-normal if for every x ∈ M \ K, there exists aσ-automorphism φ of M such that φ(x) = x and φ(a) = a for every a ∈ K. (Notethat C. Franke [67] called such extensions “normal” while similar differentialfield extensions are called “weekly normal”.) The existence of proper monadicalgebraic difference extensions suggests the existence of solution field that arenot σ-normal extensions.


In what follows we present some further results of the Franke’s version of thePicard-Vessiot theory of difference equations. We refer the reader to the works[67] - [73] for the proofs.

The next statement is a version of the fundamental Galois theorem for PVE.As usual, primes indicate the Galois correspondence: if H is a subgroup ofGal(M/K), then H ′ is the fixed field of H, and if L is an intermediate fieldof M/K, then L′ is a subgroup of Gal(M/K) whose elements fix all elementsof the field L.

Theorem 8.2.9 (Franke, [67]). Let M/K be a difference (σ∗-) PVE, G =Galσ(M/K), L an intermediate σ∗-field of M/K, and H an algebraic subgroupof G. Then

(i) L′ is an algebraic matrix group.(ii) H ′′ = H.(iii) If L is algebraically closed in M , then M is σ-normal over L and L′′ = L.(iv) There is a one-to-one correspondence between intermediate σ∗-fields of

M/K that are algebraically closed in M and connected algebraic subgroups ofthe group G.

(v) Let K denote the algebraic closure of K in M . If H is a connected normalsubgroup of G, then G/H is the full group of H ′ over K (that is, G/H isisomorphic to Galσ(H ′/K)) and H ′ is σ-normal over K.(vi) If L is algebraically closed in M and σ-normal over K, then L′ is a normal

subgroup of G and G/L′ is the full group of L over K.

Let M be a σ∗-overfield of a difference (σ∗-) field K and H a subgroupof Galσ(M/K). If L is an intermediate σ∗-field of M/K, then L′

H will denotethe group h ∈ H|h(a) = a for all a ∈ L ⊆ H. A subgroup A of H is said tobe Galois closed in H if (A′)′H = A. An intermediate field N of M/K is said tobe Galois closed with respect to H if (N ′

H)′ = N .The results of the following theorem were obtained in [67] and [69].

Theorem 8.2.10 Let M be a solution field for difference equation (8.2.2) over adifference (σ-) field K. Let b be a basis of M/K and H a subgroup of Galσ(M/K)which is naturally isomorphic to the set of matrices Tb corresponding to H.Then:

(i) Algebraic subgroups of H are Galois closed in H.(ii) Connected subgroups of H correspond to intermediate σ∗-fields of M/K

algebraically closed in M .(iii) Let L be an intermediate σ∗-field of M/K which is algebraically closed

in M . Let TLb be the variety obtained by considering M as a solution field over

L with basis b. If L′H is dense in TL

b , then (L′H)′ = L and L′

H is connected.(iv) If the algebraic closure of K(CM ) in M coincides with K, then M/K

is a σ-normal extension. In this case, there is a one-to-one correspondence be-tween connected algebraic subgroups of Galσ(M/K) and intermediate σ∗-fieldsof M/K algebraically closed in M .


Some generalization of the last theorem was obtained in [70]. Let K and M beas in Theorem 8.2.10, L an intermediate σ∗-field of M/K, and N a σ∗-overfieldof M . Furthermore, let IL denote the set of all σ-isomorphisms of M into Nleaving L fixed.

Proposition 8.2.11 If L is algebraically closed in M , then IL is a connectedalgebraic matrix group. Furthermore, Galσ(M/L) is dense in IL and IL is iso-morphic to Galσ(M(CN )/L(CN )).

Theorem 8.2.12 Let K and M be as in Theorem 8.2.10, H an algebraic sub-group of Galσ(M/K), and L an intermediate σ∗-field of M/K. Then

(i) H ′′ = H.(ii) If the field L is algebraically closed in M , then L′′ = L and M is σ-normal

over L.(iii) There is a one-to-one correspondence between connected algebraic sub-

groups of Galσ(M/K) and intermediate σ∗-fields of M/K algebraically closedin M .(iv) Assume that H is connected and L = H ′. In this case(a) H is a normal subgroup of Galσ(M/K) if and only if L is σ-normal over

the field K.(b) If H is a normal subgroup of Galσ(M/K) and N is any universal

extension of M for L, then IL is a normal subgroup of IK . The homomor-phisms defined by restriction and extension determine natural isomorphisms

Galσ(M/K)/H → Galσ(L/K) → IK/IL

and the image of Galσ(M/K)/H is dense in IK/IL.

In general a full difference Galois group G = Galσ(M/K) is not naturallyisomorphic to a matrix group (if g, h ∈ G, then the matrix of the composite of gand h is the matrix of g times the matrix obtained by applying g to the entriesof the matrix of h). However, if we adjoin CM to K and consider M as a solutionfield over K(CM ), we obtain a group D which is naturally isomorphic to a groupof matrices contained in an algebraic variety T . Theorem 8.2.5 implies that Tconsists only of isomorphisms and singular matrices. The Galois correspondencegiven in Theorem 8.2.10 for D and fields between K(CM ) and M depends inpart on whether a subgroup of D is dense in a variety containing it. Exampleswhere this is not the case are not known.

If M is a solution field for (8.2.2) over K with a basis b, then the subsets of Tb

and D consisting of nonsingular matrices with entries in CK are automorphismgroups. The following two results obtained in [67] deal with these groups.

Proposition 8.2.13 Let M be a solution field for difference equation (8.2.2)over a difference (σ-) field K. Let b be a basis of M/K, Λ a subfield of CM , andSb the subset of the polynomial ring CM [xij ] (1 ≤ i, j ≤ n) whose existence isestablished by Theorem 8.2.5. Let us write the polynomials of Sb as f =

∑k

fkvk


where vk is a basis of CM as vector space over Λ and fk ∈ Λ[xij ]. Let S′b be

the set of all such fk. Then

(i) Every solution of the set S′b is a solution of Sb.

(ii) Every solution of Sb that lies in Λ is a solution of S′b.

(iii) If Λ is algebraically closed and contained in K, then the variety of S′b

over Λ is an algebraic matrix group of automorphisms of M/K plus singularmatrices.

Note that Theorem 8.2.10 can be applied to any group G(1)b obtained by

deleting the singular matrices from a variety T(1)b determined as in Proposition

8.2.13 by a basis b and a subfield Λ.

Proposition 8.2.14 Let K, M and b be as in Proposition 8.2.13, and let Λ bean algebraically closed field of constants of K. Let G

(1)b be the group determined

by b and Λ as in Proposition 8.2.13 and let C(1)b be the component of the identity

of G(1)b . Finally, let Cb be the irreducible subvariety of Tb determined by K (the

algebraic closure of K in M). The following are equivalent and imply that K isGalois closed with respect to C

(1)b .

(i) C(1)b is dense in Cb.

(ii) dimCb = dimC(1)b .

(iii) There is a basis for the ideal of Cb in the polynomial ring C[xij ] (1 ≤i, j ≤ n).

Let M be a solution field for difference equation (8.2.2) over a difference (σ-)field K. M/K is called a generalized Picard-Vesiot extension (GPVE) if thereis a basis b of M/K and an algebraically closed subfield Λ of CK such that C

(1)b

is dense in Cb. M is said to be a generic solution field for equation (8.2.2) iftrdegKM = n2 (n is the order of the difference equation).

Proposition 8.2.15 Every linear homogeneous difference equation over a dif-ference field K has a generic solution field M . Therefore, if CK contains analgebraically closed subfield, then every linear homogeneous difference equationover K has a solution field which is a GPVE.

Theorem 8.2.16 If L = K〈a〉∗ and M = K〈b〉∗ are solution fields of equa-tion (8.2.2) over a difference (σ-) field K, then trdegK(CL)L = trdegK(CM )M .Furthermore, if L/K and M/K are compatible, then

(i) There is a difference (σ-) field M1 isomorphic to M and a set of constantsR such that L(R) = M1(R).

(ii) If L is a PVE, then there is a specialization b → b′ with L = K〈b′〉∗.(iii) If L and M are PVE of K, then L and M are σ-isomorphic over K.

As in the corresponding theory for differential case, three types of extensionsare used in constructing solution fields for linear homogeneous difference equa-tions over a difference field K: solution fields for difference equations αy = Ay


or αy − y = B (A,B ∈ K), and algebraic extensions. The following is a briefaccount of this approach (the proofs can be found in [67]).

Proposition 8.2.17 Let K be a difference field with a basic set σ = α, 0 =B ∈ K, and let a be a solution of the difference equation

αy − y = B. (8.2.5)

Then M = K(a) is a corresponding solution field over K with basis b = (a, 1).If there is no solution of equation (8.2.5) in K, then a is transcendental over

K, K(a) has no new constants, and there are no intermediate σ∗-fields differentfrom K and K(a).

If there is a solution f ∈ K, then K(a) is an extension of K generatedby a constant which may be either transcendental or algebraic over K. If a is

transcendental, then Tb is the set of matrices(

1 c0 1

)where c lies in the algebraic

closure of CM . If CM = CK , then the full Galois group of M/K is isomorphicto the additive group of CK .

Proposition 8.2.18 Let K be as before and let a be a nonzero solution of thedifference equation

αy − Ay = 0 (8.2.6)(A ∈ K). Let us consider the equation

αy − Any = 0 (8.2.7)

where n ∈ N, n > 0. Then(i) If there is no nonzero solution in K of the equation (8.2.7), then a is

transcendental over K and K(a) has no new constants.(ii) If L is an intermediate σ∗-field, then L = K(an) for some n ∈ N.(iii) If equation (8.2.7) has a solution in K for some n ∈ N, n > 0, then

K(a) is obtained from K by an extension by a constant, which may be eithertranscendental or algebraic over K, followed by an algebraic extension. If a istranscendental over K, the variety Ta is the full set of all constants. If CK(a) =CK , then the full Galois group of K(a)/K is a multiplicative subgroup of CK .

Definition 8.2.19 Let K be a difference field with a basic set σ and N a σ∗-overfield of K. N/K is said to be a Liouvillian extension (LE) if there exists achain

K = K0 ⊆ K1 ⊆ · · · ⊆ Kt = N, Kj+1 = Kj〈aj〉∗ (j = 0, . . . , t − 1) (8.2.8)

where aj is one of the following.(a) A solution of an equation (8.2.5) where B ∈ Kj and there is no solution

of (8.2.5) in the field Kj.(b) A solution of an equation (8.2.6) where A ∈ Kj and for any n ∈ N, n > 0,

there is no nonzero solution of (8.2.7) in the field Kj.(c) An algebraic element over Kj.


More generally, N/K is said to be a generalized Liouvillian extension (GLE)if there exists a chain (8.2.8) where aj is either a solution of equation (8.2.5)with B ∈ Kj or a solution of equation (8.2.6) with A ∈ Kj, or an algebraicelement over Kj.

It follows from the definition that if a difference field K has an algebraicallyclosed field of constants and a solution field M for a difference equation (8.2.2)is contained in a Liouvillian extension N of K, then M is a PVE of K.

The following results connect the solvability of a difference equation withthe solvability of a matrix group.

Theorem 8.2.20 Let M be a solution field for difference equation (8.2.2) overa difference (σ-) field K. Let H be a connected group of automorphisms of M/Kwith matrix entries with respect to some basis b in an algebraically closed subfieldof CM . (It need not be isomorphic to the set of matrices corresponding to H.)

(i) If H is solvable, then M/H ′ is a GLE.(ii) If H is reducible to diagonal form, then M/H ′ can be obtained by solving

equations of the type (8.2.6).(iii) If H is reducible to special triangular form, then M/H ′ can be obtained

by solving equations of the type (8.2.5).

Theorem 8.2.21 Let K be a difference field with a basic set σ = α and Ma σ∗-overfield of K.

(i) If M/K is a PVE, then M/K is a GLE if and only if the component ofidentity of the Galois group is solvable.

(ii) If M is a solution field of a difference equation (8.2.2) contained in a GLEN/K, then the component of identity of Gal(M/K(CM )) is solvable. Further-more, if M/K(CM ) is a GPVE, then M/K is a GLE.(iii) Suppose that M and L are solution fields of a difference equation (8.2.2)

over K. If L is contained in a GLE N of K and M/K is compaitible with N/K,then M is contained in a GLE of K.(iv) If N is a generic solution field for (8.2.2) over K and a solution field L

for (8.2.2) is contained in a GLE of K, then N is contained in a GLE of K.

The following example indicates that it is not satisfactory to consider equa-tion (8.2.2) to be “solvable by elementary operations” only if its solution fieldis contained in a GLE.

Example 8.2.22 With the notation of Theorem 8.2.21, suppose that K con-tains an element j with α(j) = j and α2(j) = j, and an element u with the

following property. If uk = vα(v) or uk =α(v)

vfor some v ∈ K, k ∈ N, then

k = 0.If η is any nonzero solution of the difference equation α2y−uy = 0, then M =

K〈η〉∗ is a solution field for this equation with basis (η, α(η)), trdegKM = 2,


CM = CK and Galσ(M/K) is commutative. However, as it is shown in [68,Example 1], M is not a GLE of K.

In what follows we consider some results by C. Franke (see [68] and [73])that characterize the solvability of a difference equation of the form (8.2.2)“by elementary operations”. Throughout the rest of the section K denotes aninversive difference field with a basic set σ = α. If L is a σ∗-overfield of K,then KL will denote the algebraic closure of K(CL) in L.

Definition 8.2.23 Let N be a σ∗-overfield of K, and q a positive integer. Aq-chain from K to N is a sequence of σ∗-fields K = K1 ⊆ K1 ⊆ · · · ⊆ Kt = N ,Ki+1 = Ki〈ηi〉∗ where ηi is one of the following.

(a) An element algebraic over Ki−1.(b) A solution of an equation αqy = y + B for some B ∈ Ki.(c) A solution of an equation αqy = Ay for some A ∈ Ki.

If there is a q-chain from K to N , then N is called a qLE of K.

Let K(q) denote the field K treated as an inversive difference field witha basic set σq = αq and let N (q) be a σ∗-overfield N of K treated as aσ∗

q -overfield of K(q). In this case N is a qLE of K if and only if N (q) is a GLEof K(q) (see [68, Proposition 2.1]).

Theorem 8.2.24 If M is a σ-normal σ∗-overfield of K such that K = KM

and the group Galσ(M/K) is solvable, then M is contained in a qLE of K. If,in addition, M/K is a σ∗-field extension generated by a fundamental system ofsolutions of a difference equation (8.2.2), then M is contained in a GLE of K.

Theorem 8.2.25 Let N be a qLE of K and L an intermediate σ∗-field of N/K.Then Galσ(L/KL) is solvable.

Let M be σ∗-field extension of K. We say that difference equation (8.2.2) issolvable by elementary operations in M over K if M is a solution field for (8.2.2)over K and M is contained in a qLE of K. This concept is independent of thesolution field M , as it follows from the second statement of the next theorem.

Theorem 8.2.26 Let M be a σ∗-overfield of K.(i) Equation (8.2.2) is solvable by elementary operations in M over K if and

only if the group Galσ(M/K) has a subnormal series whose factors are eitherfinite or commutative.

(ii) If (8.2.2) is solvable by elementary operations in M over K and N isanother solution field for (8.2.2) over K, then (8.2.2) is solvable by elementaryoperations in N over K. (This property allows one to say that (8.2.2) is solvableby elementary operations over K if it is solvable by elementary operations insome solution field M ⊇ K.)(iii) If (8.2.2) is solvable by elementary operations over K and L a σ∗-overfield

of K, then (8.2.2) is solvable by elementary operations over L.


A number of results that specify the results of this section for the case ofsecond order difference equations were obtained in [67] and [68]. C. Franke [67]also showed that the properties of having algebraically closed field of constantsand having full sets of solutions of difference equations can be incompatible.Indeed, if K is a difference field with a basic set σ = α (Char K = 2) suchthat CK is algebraically closed, then the difference equation αy + y = 0 has nononzero solution in K. (If b is such a solution, then α(b2) = b2, so b2 ∈ CK .Since CK is algebraically closed, b ∈ CK contradicting the fact that α(b) = −b.)

This observation and the fact that one could not associate a Picard-Vessiot-type extension to every difference equations have led to a different approach tothe Galois theory of difference equations. This approach, based on the study ofsimple difference rings rather than difference fields, was realized by M. van derPut and M. F. Singer in their monograph [159]. In the next section we give anoutline of the fundamentals of the corresponding theory.

We conclude this section with one more theorem on Galois correspondencefor difference fields (see Theorem 8.2.30 below). This result is due to R. Infantewho developed the theory of strongly normal difference field extensions (see[86] - [90]). Under some natural assumptions, the class of such extensions ofa difference field K includes, in particular, the class of solution fields of linearhomogeneous difference equations over K. We refer the reader to works [87] and[89] for the proofs.

Let K be an ordinary inversive difference field of zero characteristic with abasic set σ = α. Let M be a finitely generated σ∗-overfield of K such thatK is algebraically closed in M and CM = CK = C. As above, KM will denotethe algebraic closure of K(CM ) in M . Furthermore, for any σ-isomorphism φ ofM/K into a σ∗-overfield of M , Cφ will denote the field of constants of M〈φM〉∗.

Definition 8.2.27 With the above conventions, M is said to be a strongly nor-mal extension of K if for every σ-isomorphism φ of M/K into a σ∗-overfield ofM , M〈Cφ〉∗ = M〈φM〉∗ = φM〈Cφ〉∗.

Proposition 8.2.28 Let M be a solution field of a difference equation of theform (8.2.2) over K. Then M is a strongly normal extension of KM .

Proposition 8.2.29 Let M be a strongly normal σ∗-field extension of K. Then(i) ld (M/K) = 1.(ii) If φ is any σ-isomorphism of M/K into a σ∗-overfield of M , then Cφ is

a finitely generated extension of C and trdegMM〈φM〉∗ = trdegCCφ.

Theorem 8.2.30 If M is a strongly normal σ∗-field extension of K, then thereis a connected algebraic group G defined over CM such that the connected al-gebraic subgroups of G are in one-to-one correspondence with the intermediateσ∗-fields of M/K algebraically closed in M . Furthermore, there is a field of con-stants C ′ such that C ′-rational points of G are all the σ-isomorphisms of M/Kinto M〈C ′〉∗ and this set is dense in G.


8.3 Picard-Vessiot Rings and the Galois Theoryof Difference Equations

In this section we discuss some basic results of the Galois theory of differenceequations based on the study of simple difference rings associated with suchequations. The complete theory is presented in [159] where one can find theproofs of all statements of this section.

All difference rings and fields considered below are supposed to be ordinaryand inversive. The basic set of a difference ring will be always denoted by σ andthe only element of σ will be denoted by φ (we follow the notation of [159]).As usual, GLn(R) will denote the set of all nonsingular n × n-matrices over aring R.

Let R be a difference ring, A ∈ GLn(R), and Y a column vector (y1, . . . , yn)T

whose coordinates are σ∗-indeterminates over R. (The corresponding ring ofσ∗-polynomials is still denoted by Ry1, . . . , yn∗.) In what follows, we willstudy systems of difference equations of the form φY = AY where φY =(φy1, . . . , φyn)T . Notice that an nth order linear difference equation φny + · · ·+a1φy+a0y = 0 (a0, a1, · · · ∈ R and y is a σ∗-indeterminate over R) is equivalentto such a system with yi = φi−1y (i = 1, . . . , n) and

A =

⎛⎜⎜⎝0 1 0 . . . 00 0 1 . . . 0

. . . . . . . . . . . . . . .−a0 −a1 −a2 . . . −an−1

⎞⎟⎟⎠ .

Clearly, A ∈ GLn(R) if and only if a0 = 0.With the above notation, a fundamental matrix with entries in R for φY =

AY is a matrix U ∈ GLn(R) such that φU = AU (φU is the matrix obtainedby applying φ to every entry of U). If U and V are fundamental matrices forφY = AY , then V = UM for some M ∈ GLn(CR) since U−1V is left fixed byφ. (As in section 8.2, CR denotes the ring of constants of R, that is, CR = a ∈R|φa = a.)

Definition 8.3.1 Let K be a difference field. A K-algebra R is called a Picard-Vessiot ring (PVR) for an equation

φY = AY (A ∈ GLn(K)) (8.3.1)

if it satisfies the following conditions.(i) R is a σ∗-K-algebra (as usual, the automorphism of R which extends φ is

denoted by the same letter).(ii) R is a simple difference ring, that is, the only difference ideals of R are

(0) and R.(iii) There exists a fundamental matrix X for φY = AY having entries in R

such that R = K[X, (det X)−1].

8.3. PICARD-VESSIOT RINGS AND THE GALOIS THEORY 487

The following example is due to M. Singer and M. van der Put (see [159,Examples 1.3 and 1.6]).

Example 8.3.2 Let C be an algebraically closed field, Char C = 2. Let usdefine an equivalence relation on the set of all sequences a = (a0, a1, . . . ) ofelements of C by saying that a is equivalent to b = (b0, b1, . . . ) if there existsN ∈ N such that an = bn for all n > N . With the coordinatewise addition andmultiplication, the set of all equivalence classes forms a ring S. This ring canbe treated as a difference ring with respect to its automorphism φ that mapsan equivalent class of (a0, a1, . . . ) to the equivalent class of (a1, a2, . . . ). (It iseasy to check that φ is well-defined.) To simplify notation we shall identify asequence a with its equivalence class.

Let R be the difference subring of S generated by C and j = (1,−1, 1,−1, . . . ),that is, R = C[j]∗. The 1 × 1-matrix whose only entry is j is the fundamentalmatrix of the equation φy = −y. This ring is isomorphic to C[X]/(X2−1) (C[X]is a polynomial ring in one indetrminate X over C) whose only non-trivial idealsare generated by the cosets of X − 1 and X + 1. Since the ideals generated inR by j + 1 and j − 1 are not difference ideals, R is a simple difference ring.Therefore, R is a PVR for φy = −y over C. Note that R is reduced but it is notan integral domain.

Proposition 8.3.3 Let K be a difference (σ∗-) field with an algebraically closedfield of constants CK .

(i) If a σ∗-K-algebra R is a simple difference ring finitely generated as aK-algebra, then CR = CK .

(ii) If R1 and R2 are two PVR for a difference equation (8.3.1), then thereexists a σ-isomorphism between R1 and R2 that leaves the field K fixed.

To form a PVR for a difference equation (8.3.1) one can use the followingprocedure suggested in [159, Chapter 1]. Let (Xij) denote an n × n-matrix ofindeterminates over K and det denote the determinate of this matrix. Thenone can extend φ to an automorphism of the K-algebra K[Xij ,

1det

] (we write1

detfor det−1) by setting (φXij) = A(Xij). If I is a maximal difference ideal of

K[Xij ,1

det] then K[Xij ,

1det

]/I is a PVR for (8.3.1), it satisfies all conditions of

Definition 8.3.1. (It is easy to see that I is a radical σ∗-ideal and K[Xij ,1

det]/I

is a reduced prime difference ring.) Moreover, any PVR for difference equation(8.3.1) will be of this form.

Let K denote the algebraic closure of K and let D = K [Xij ,1

det]. Then

the automorphism φ extends to an automorphism of K which, in turn, extendsto an automorphism of D such that (φXij) = A(Xij) (the extensions of φ arealso denoted by φ). It is easy to see that every maximal ideal M of D has


the form (X11 − b11, . . . , Xnn − bnn) and corresponds to a matrix B = (bij) ∈GLn(K). Then φ(M) is a maximal ideal of D that corresponds to the matrixA−1φ(B) where φ(B) = (φ(bij)). Thus, the action of φ on D induces a map τ onGLn(K) such that τ(B) = A−1φ(B). The elements f ∈ D are seen as functionson GLn(K). For any f ∈ D,B ∈ GLn(K), we have (φf)(τ(B)) = φ(f(B)).

Furthermore, if J is an ideal of K[Xij ,1

det] such that φ(J) ⊆ J , then φ(J) = J .

Also, for reduced algebraic subsets Z of GLn(K), the condition τ(Z) ⊆ Z impliesτ(Z) = Z.

Proposition 8.3.4 ([159, Lemma 1.10]). The ideal J of a reduced subset Z ofGLn(K) satisfies φ(J) = J if and only if Z(K) satisfies τZ(K) = Z(K).

An ideal I maximal among the φ-invariant ideals corresponds to a minimal(reduced) algebraic subset Z of GLn(K) such that τZ(K) = Z(K). Such a setis called a minimal τ -invariant reduced set.

Let Z be a minimal τ -invariant reduced subset of GLn(K) with an ideal

I ⊆ K[Xij ,1

det] and let O(Z) = K[Xij ,

1det

]/I. Let us denote the image of Xij

in O(Z) by xij and consider the rings

K

[Xij ,

1det

]⊆ O(Z)

⊗K

K

[Xij ,

1det(Xij)

]

= O(Z)⊗C

C

[Yij ,

1det(Yij)

]⊇ C

[Yij ,

1det(Yij)

](8.3.2)

Let (I) denote the ideal of O(Z)⊗

K K[Xij ,1

det] generated by I and let J =

(I)⋂

C[Yij ,1

det]. The ideal (I) is φ-invariant, the set of constants of O(Z) is C,

and J generates the ideal (I) in O(Z)⊗

K K[Xij ,1

det]. Furthermore, one has

natural mappings

O(Z) → O(Z)⊗K

O(Z)

= O(Z)⊗C

(C

[Yij ,

1det(Yij)

]/J

)← C

[Yij ,

1det(Yij)

]/J . (8.3.3)

Suppose that O(Z) is a separable extension of K (for example, Char K = 0 or Kis perfect). One can show (see [159, Section 1.2]) that O(Z)

⊗K O(Z) is reduced.

Therefore, C[Yij ,1

det(Yij)]/J is reduced and J is a radical ideal. Furthermore,

the following considerations imply that J is the ideal of an algebraic subgroupof GLn(C).

Let A ∈ GLn(C) and let δA denote the action on the terms of (8.3.2) definedby (δAXij) = (Xij)A and (δAYij) = (Yij)A. Then the following eight propertiesare equivalent:


(1) ZA = Z;(2) ZA

⋂Z = ∅;

(3) δAI = I;

(4) I + δAI is not the unit ideal of K[Xij ,1

det];

(5) δA(I) = (I);

(6) (I) + δA(I) is not the unit ideal of O(Z)⊗

K[Xij ,1

det];

(7) δAJ = J ;

(8) J + δAJ is not the unit ideal of O(Z)⊗

C[Yij ,1

det].

The set of all matrices A ∈ GLn(C) satisfying the equivalent conditions(1) - (8) form a group.

Proposition 8.3.5 (see [159, Lemma 1.12]). Let O(Z) be a separable extensionof K. With the above notation, A satisfies the equivalent conditions (1) - (8) ifand only if A lies in the reduced subspace V of GLn(C) defined by J . Therefore,the set of such A is an algebraic group.

Let G denote the group of all automorphisms of O(Z) over K which commutewith the action of φ. The group G is called the difference Galois group of theequation φ(Y ) = AY over the field K.

If δ ∈ G, then (δxij) = (xij)A where A ∈ GLn(C) is such that δA (as definedabove) satisfies δAI = I. Therefore, one can identify G and the subspace Vin the last proposition. Denoting the ring C[Yij ,

1det ]/J by O(G) and setting

O(Gk) = O(G)⊗

C k, GK = spec(O(Gk)), one can use (8.3.3) to obtain thesequence

O(Z) → O(Z)⊗K

O(Z) = O(Z)⊗C

O(G) = O(Z)⊗K

O(GK) (8.3.4)

The first embedding of rings corresponds to the morphism Z × GK → Z givenby (z, g) → zg. The identification

O(Z)⊗K

O(Z) = O(Z)⊗C

O(G) = O(Z)⊗K

O(GK)

corresponds to the fact that the morphism Z ×GK → Z × Z given by (z, g) →(zg, z) is an isomorphism. Thus, Z is a K-homogeneous space for GK , that isZ/K is a G-torsor. The following result (proved in [159, Section 1.2]) shows thata PVR is the coordinate ring of a torsor for its difference Galois group.

Theorem 8.3.6 Let R be a separable PVR over K, a difference field with analgebraically closed field of constants C. Let G denote the group of the K-algebraautomorphisms of R which commute with φ. Then

(i) G has a natural structure as reduced linear algebraic group over C and theaffine scheme Z over K has the structure of a G-torsor over K.


(ii) The set of G-invariant elements of R is K and R has no proper, nontrivialG-invariant ideals.(iii) There exist idempotents e0, . . . , et−1 ∈ R (t ≥ 1) such that

a) R = R0

⊕ · · ·⊕Rt−1 where Ri = eiR for i = 0, . . . , t − 1.b) φ(ei) = ei+1(mod t) and so φ maps Ri isomorphically onto Ri+1(mod t)

and φt leaves each Ri invariant.c) For each i, Ri is a domain and is a Picard-Vessiot extension of eiK with

respect to φt.

Let K be a difference field with an algebraically closed field of constantsC and R a PVR for an equation φ(Y ) = AY over K. Let δ = δA and letR = R0

⊕ · · ·⊕Rt−1 (Ri = eiR for i = 0, . . . , t − 1) be as in the last theorem.Then δ : Ri → Ri+1 is an isomorphism and R0 is a PVR over K with respectto the automorphism δt. Let us define two mappings

Γ : Gal(R0/K) → Gal(R/K) and ∆ : Gal(R/K) → Z/tZ

as follows. For any ψ ∈ Gal(R0/K), let Γ(ψ) = χ where for r = (r0, . . . , rt−1) ∈R, χ(r0, . . . , rt−1) = (ψ(r0), δψδ−1(r1), . . . , δt−1ψδ1−t(rt−1)). In order to define∆, notice that if χ ∈ Gal(R/K), then χ permutes with each ei. If χ(e0) = ej ,we define ∆(χ) = j.

Proposition 8.3.7 Let R be a separable PVR over K, a difference field withan algebraically closed field of constants C.

(i) With the above notation, we have the exact sequence

0 → Gal(R0/K) Γ−→ Gal(R/K) ∆−→ Z/tZ → 0.

(ii) Let G denote the difference Galois group of R over K. If H1(Gal(K/K),G(K)) = 0, then Z = spec(R) is G-isomorphic to the G-torsor GK and soR = C[G]

⊗K.

In what follows we consider a characterization of the difference Galois groupof a PVR over the field of rational functions C(z) in one variable z over analgebraically closed field C of zero characteristic (one can assume C = C).We fix a ∈ C(z), a = 0 and consider C(z) as an ordinary difference field withthe basic automorphism φa : z → z + a (φa leaves the field C fixed). Thisdifference field will be denoted by K. Note that φa does not extend to anyproper finite field extension of K (see [159, Lemma 1.19]).

Theorem 8.3.8 Let K = C(z) be as above and let G be an algebraic subgroupof GLn(C). Let φ(Y ) = AY be a system of difference equations with A ∈ G(K).Then

(i) The Galois group of φ(Y ) = AY over K is subgroup of GC .(ii) Any minimal element in the set of C-subgroups H of G for which there

exists a B ∈ Gln(K) with B−1A−1φ(B) ∈ H(K) is the difference Galois groupof φ(Y ) = AY over K.


(iii) The difference Galois group of φ(Y ) = AY over K is G if and only if forany B ∈ G(K) and any proper C-subgroup H of G, one has B−1A−1φ(B) /∈H(K).

Definition 8.3.9 Let K be an ordinary difference field with a basic set σ = φand let A ∈ Gln(K). A difference overring L of K is said to be the total Picard-Vessiot ring (TPVR) of the equation φ(Y ) = AY over K if L is the total ringof fractions of the PVR R of the equation.

As we have seen, a PVR R is a direct sum of domains: R = R0

⊕ · · ·⊕Rt−1

where each Ri is invariant under the action of φt. The automorphism φ of Rpermutes R0, . . . , Rt−1 in a cyclic way (that is, φ(Ri) = Ri+1 for i = 1, . . . , t−2and φ(Rt−1) = R0). It follows that the TPVR L is the direct sum of fields:L = L0

⊕ · · ·⊕Lt−1 where each Li is the field of fractions of Ri, and φ permutesL0, . . . , Lt−1 in a cyclic way.

Proposition 8.3.10 With the above notation, let K be a perfect differencefield with an algebraically closed field of constants C and let φ(Y ) = BY bea difference equation over K (B ∈ Gln(K)). Let a difference ring extensionK ′ ⊇ K have the following properties:

(i) The ring K ′ has no nilpotent elements and every non-zero divisor of K ′

is invertible.(ii) The set of constants of K ′ is C.(iii) There is a fundamental matrix F for the equation with entries in K ′.(iv) K ′ is minimal with respect to (i), (ii), and (iii).

Then K ′ is K-isomorphic as a difference ring to the TPVR of the equation.

Corollary 8.3.11 Let K be as in the last proposition and let φ(Y ) = AY bea difference equation over K (A ∈ Gln(K)). Let a difference overring R ⊆ Khave the following properties.

(i) R has no nilpotent elements.(ii) The set of constants of the total quotient ring of R is C.(iii) There is a fundamental matrix F for the equation with entries in R.(iv) R is minimal with respect to (i), (ii), and (iii).Then R is a PVR of the equation.

With the above notation, let R = R0

⊕ · · ·⊕Rt−1 be the PVR of theequation φ(Y ) = AY (A ∈ Gln(K)). Let us consider the difference field (K,φt)(that is, the field K treated as a difference field with the basic set σt = φt)and the difference equation φt(Y ) = AtY with At = φt−1(A) . . . φ2(A)φ(A)A.

Proposition 8.3.12 (i) Each component Ri of R is a PVR for the equationφt(Y ) = AtY over the difference field (K,φt).

(ii) Let d ≥ 1 be a divisor of t. Using cyclic notation for the indices

0, . . . , t − 1, we consider subrings(t/d)−1⊕

m=0

Ri+md of R = R0

⊕ · · ·⊕Rt−1.


Then each of these subrings is a PVR for the equation φd(Y ) = AdY over thedifference field (K,φd).

Proposition 8.3.13 Let L be the TPVR of the equation φ(Y ) = AY (A ∈Gln(K)) over a perfect difference field K whose field of constants C = CK isalgebraically closed. Let G denote the difference Galois group of the equationand let H be an algebraic subgroup of G. Then G acts on L and moreover:

(i) LG, the set of G-invariant elements of L, is equal to K.(ii) If LH = K, then H = G.

The following result describes the Galois correspondence for total Picard-Vessiot rings. As it is noticed in [159, Section 1.3], one cannot expect a similartheorem for Picard-Vessiot rings. Indeed, let K be as in the last proposition andR = K

⊗C C[G] where C[G] is the ring of regular functions on an algebraic

group G defined over C. For an algebraic subgroup H of G, the ring of invariantsRH is the ring of regular functions on (G/H)K . In some cases, e. g., G = Gln(C)and H a Borel subgroup, the space G/H is a connected projective variety andso the ring of regular functions on (G/H)K is just K.

Theorem 8.3.14 Let K be a difference field of zero characteristic with a basicset σ = φ. Let A ∈ GLn(K) and let L be a TPVR of the equation φ(Y ) =AY over K. Let F denote the set of intermediate difference rings F such thatK ⊆ F ⊆ L and every non-zero divisor of F is a unit of F . Furthermore, let Gdenote the set of algebraic subgroups of G.

(i) For any F ∈ F , the subgroup G(L/F ) ⊆ G of the elements of G which fixF pointwise, is an algebraic subgroup of G.

(ii) For any algebraic subgroup H of G, the ring LH belongs to F .(iii) Let α : F → G and β : G → F denote the maps F → G(L/F ) and

H → LH , respectively. Then α and β are each other’s inverses.

Corollary 8.3.15 With the notation of Theorem 8.3.14, a group H ∈ G isa normal subgroup of G if and only if the difference ring F = LH has theproperty that for every z ∈ F \ K, there is an automorphism δ of F/K whichcommutes with φ and satisfies δz = z. If H ∈ G is normal, then the group of allautomorphisms δ of F/K which commute with φ is isomorphic to G/H.

Corollary 8.3.16 With the above notation, suppose that an algebraic groupH ⊆ G contains G0, the component of the identity of G. Then the differencering RH (R is a PVR for the equation φ(Y ) = AY over K) is a finite dimensionvector space over K with dimension equal to G : H.

A number of applications of the above-mentioned results on ring-theoreticaldifference Galois theory to various types of algebraic difference equations can befound in [78] - [81] and [159]. Using the technique of difference Galois groups,the monograph [159] also develops the analytic theory of ordinary differenceequations over the fields C(z) and C(z−1).

We conclude this section with a fundamental result on the inverse problemof the ring-theoretical difference Galois theory.


Theorem 8.3.17 ([159, Theorem 3.1]). Let K = C(z) be the field of fractions ofone variable z over an algebraically closed field C of zero characteristic. ConsiderK as an ordinary difference field with respect to the automorphism φ that leavesthe field C fixed and maps z to z + 1. Then any connected algebraic subgroup Gof Gln(C) is the difference Galois group of a difference equation φ(Y ) = AY ,A ∈ Gln(K).

Bibliography

[1] Adams, W.; Loustaunau, P. An Introduction to Grobner Bases. Amer.Math. Soc., Providence, 1994.

[2] Aranda-Bricaire, E.; Kotta, U.; Moog, C. H. Linearization of discrete-timesystems. SIAM J. Control Optim., 34 (1996), 1999 - 2023.

[3] Atiyah, M. F.; Macdonald, I. G. Introduction to Commutative Algebra.Addison-Wesley, Reading, MA, 1969.

[4] Babbitt, A. E. Finitely generated pathological extensions of differencefields. Trans. Amer. Math. Soc., 102 (1962), no. 1, 63-81.

[5] Balaba, I. N. Dimension polynomials of extensions of difference fields.Vestnik Moskov. Univ., Ser. I, Mat. Mekh., 1984, no. 2, 31-35. (Russian)

[6] Balaba, I. N. Calculation of the dimension polynomial of a principal differ-ence ideal. Vestnik Moskov. Univ., Ser. I, Mat. Mekh., 1985, no. 2, 16-20.(Russian)

[7] Balaba, I. N. Finitely generated extensions of difference fields. VINITI(Moscow, Russia), 1987, no. 6632-87. (Russian)

[8] Bastida, Julio R. Field extensions and Galois Theory. Encyclopedia ofMathematics and its Applications, Vol. 22. Addison-Wesley, MA 1984.

[9] Becker, T.; Weispfenning, V. Grobner bases. A computational approachto commutative algebra. Springer-Verlag, New York, 1993.

[10] Bentsen, Irving. The existence of solutions of abstract partial differencepolynomials. Trans. Amer. Math. Soc., 158 (1971), no. 2, 373-397.

[11] Bialynicki-Birula, A. On Galois theory of fields with operators. Amer. J.Math., 84 (1962), 89-109.

[12] Birkhoff, G. D. General theory of linear difference equations. Trans. Amer.Math. Soc., 12 (1911), 243-284.

[13] Birkhoff, G. D. The generalized Riemann problem for linear differentialequations and the allied problem for difference and q-difference equations.Proc. Nat. Acad. Sci., 49 (1913), 521-568.

495

496 BIBLIOGRAPHY

[14] Birkhoff, G. D. Note on linear difference and differential equations. Proc.Nat. Acad. Sci., 27 (1941), 65-67.

[15] Borceux, F. Categories and Structures. Handbook of Categorical Algebra.Vol. I, II. Cambridge Univ. Press, Cambridge 1994.

[16] Bronstein, M. On solutions of linear ordinary difference equations in theircoefficient field. J. Symbolic Comput., 29 (2000), no. 6, 841-877.

[17] Brualdi, R. A. Introductory Combinatorics, 2nd ed. Prentice Hall, Engle-wood Cliffs, NJ, 1992.

[18] Buchberger, B. Ein algorithmus zum auffinden der basiselemente des restk-lassenringes nach einem nulldimensionalen polynomideal. Ph. D. Thesis,University of Innsbruck, Institute for Mathematics, 1965.

[19] Bucur, I., Deleanu, A. Introduction to the Theory of Categories and Func-tors. J. Wiley and Sons, New York, 1968.

[20] Casorati, F. Il calcolo delle differenze finite. Annali di Matematica Puraed Applicata, Series II, 10 (1880-1882), 10-43.

[21] Chatzidakis, Z. A survey on the model theory of difference fields. ModelTheory, Algebra and Geometry , MSRI Publications, 39 (2000), 65-96.

[22] Chatzidakis, Z. Difference fields: model theory and applications to num-ber theory. European Congress of Mathematics, Vol. I (Barcelona, 2000),275-287.

[23] Chatzidakis, Z.; Hrushovski, E. Model theory of difference fields. Trans.Amer. Math Soc., 351 (1999), no. 8, 2997-3071.

[24] Chatzidakis, Z.; Hrushovski, E.; Peterzil, Y. Model theory of differencefields. II. Periodic ideals and the trichotomy in all characteristics. Proc.London Math. Soc., 85 (2002), no. 2, 257-311.

[25] Chevalley, C. Introduction to the theory of algebraic functions of onevariable. Mathematics Surveys, no. 6. Amer. Math. Soc., Providence,R. I., 1951.

[26] Cohn, P. M. Free rings and there relations. Academic Press, London, NewYork, 1971.

[27] Cohn, P. M. Skew fields. Theory of general division rings. Cambridge Univ.Press, Cambridge, 1995.

[28] Cohn, R. M. Manifolds of difference polynomials. Trans. Amer. Math.Soc., 64 (1948), 133-172.

[29] Cohn, R. M. A note on the singular manifolds of a difference polynomial.Bulletin of the Amer. Math. Soc., 54 (1948), 917-922.

BIBLIOGRAPHY 497

[30] Cohn, R. M. A theorem on difference polynomials. Bulletin of the Amer.Math. Soc., 55 (1949), 595-597.

[31] Cohn, R. M. Inversive difference fields. Bull. Amer. Math. Soc., 55 (1949),597-603.

[32] Cohn, R. M. Singular manifolds of difference polynomials. Ann. Math., 53(1951), 445-463.

[33] Cohn, R. M. Extensions of difference fields. Amer. J. Math., 74 (1952),507-530.

[34] Cohn, R. M. On extensions of difference fields and the resolvents of primedifference ideals. Proc. Amer. Math. Soc, 3 (1952), 178-182.

[35] Cohn, R. M. Essential singular manifolds of difference polynomials. Ann.Math., 57 (1953), 524-530.

[36] Cohn, R. M. Finitely generated extensions of difference fields. Proc. Amer.Math. Soc., 6 (1955), 3-5.

[37] Cohn, R. M. On the intersections of components of a difference polyno-mial. Proc. Amer. Math. Soc, 6 (1955), 42-45.

[38] Cohn, R. M. Specializations over difference fields. Pacific J. Math., 5,Suppl. 2 (1955), 887-905.

[39] Cohn, R. M. An invariant of difference field extensions. Proc. Amer. Math.Soc., 7 (1956), 656-661.

[40] Cohn, R. M. An improved result concerning singular manifolds of differ-ence polynomials. Canadian J. Math., 11 (1959), 222-234.

[41] Cohn, R. M. Difference Algebra. Interscience, New York, 1965.

[42] Cohn, R. M. An existence theorem for difference polynomials. Proc. Amer.Math. Soc., 17 (1966), 254-261.

[43] Cohn, R. M. Errata to “An existence theorem for difference polynomials”.Proc. Amer. Math. Soc., 18 (1967), 1142-1143.

[44] Cohn, R. M. Systems of ideals. Canadian J. Math., 21 (1969), 783-807.

[45] Cohn, R. M. A difference-differential basis theorem. Canadian J. Math.,22 (1970). no.6, 1224-1237.

[46] Cohn, R. M. Types of singularity of components of difference polynomial.Aequat. Math., 9, no. 2 (1973), 236-241.

[47] Cohn, R. M. The general solution of a first order differential polynomial.Proc. Amer. Math. Soc., 55 (1976), no. 1, 14-16.

498 BIBLIOGRAPHY

[48] Cohn, R. M. Solutions in the general solution. Contribution to algebra.Collection of Papers Dedicated to Ellis Kolchin. Academic Press, NewYork, 1977, 117-127.

[49] Cohn, R. M. Specializations of differential kernels and the Ritt problem.J. Algebra, 61 (1979), no. 1, 256-268.

[50] Cohn, R. M. The Greenspan bound for the order of differential systems.Proc. Amer. Math. Soc., 79 (1980), no. 4, 523-526.

[51] Cohn, R. M. Order and dimension. Proc. Amer. Math. Soc., 87 (1983),no. 1, 1-6.

[52] Cohn, R. M. Valuations and the Ritt problem. J. Algebra, 101 (1986), no.1, 1-15.

[53] Cohn, R. M. Solutions in the general solution of second order algebraicdifferential equations. Amer. J. Math, 108 (1986), no. 3, 505-523.

[54] Cox, D.; Little, J.; O’Shea, D. Ideals, Varieties and Algorithms. AnIntroduction to Computational Algebraic Geometry and CommutativeAlgebra. 2nd ed. Springer-Verlag, New York, 1997.

[55] Einstein, A. The Meaning of Relativity. Appendix II (Generalization ofgravitation theory), 4th ed. Princeton, 1953, 133-165.

[56] Eisen, M. Ideal theory and difference algebra. Math. Japon., 7 (1962),159-180.

[57] Eisenbud, D. Commutative Algebra with View toward Algebraic Geome-try. Springer-Verlag, New York, 1995.

[58] Elaydi, S. An Introduction to Difference Equations. 2nd ed. Springer-Verlag, New York, 1999.

[59] Etingof, P. I. Galois groups and connection matrices of q-difference equa-tions. Electron Res. Announc. amer. Math. Soc., 1 (1995), no. 1, 1-9 (elec-tronic).

[60] Evanovich, Peter. Algebraic extensions of difference fields. Trans. Amer.Math. Soc., 179 (1973), no. 1, 1-22.

[61] Evanovich, Peter. Finitely generated extensions of partial difference fields.Trans. Amer. Math. Soc., 281 (1984), no. 2, 795-811.

[62] Fliess, M. Esquisses pour une theorie des systemes non lineaires en tempsdiscret. Conference on linear and nonlinear mathematical control the-ory. Rend. Sem. Mat. Univ. Politec. Torino, 1987, Special Issue, 55-67.(French)

BIBLIOGRAPHY 499

[63] Fliess, M. Automatique en temps discret et algebre aux differences. ForumMath, 2 (1990), no. 3, 213-232. (French)

[64] Fliess, M. A fundamental result on the invertibility of discrete time dynam-ics. Analysis of controlled dynamic systems (Lyon, 1990). Progr. SystemsControl Theory. (Boston, MA), 8 (1991), 211-223.

[65] Fliess, M. Invertibility of causal discrete time dynamical system. J. PureAppl. Algebra, 86 (1993), no. 2, 173-179.

[66] Fliess, M.; Levine, J.; Martin, P.; Rouchon, O. Differential flatness anddefect: an overview. Geometry in nonlinear control and differential inclu-sions (Warsaw, 1993). Banach Center Publ., Warsaw, 1995, 209-225.

[67] Franke, C. Picard-Vessiot theory of linear homogeneous difference equa-tions. Trans. Amer. Math. Soc., 108 (1963), no. 3, 491-515.

[68] Franke, C. Solvability of linear homogeneous difference equations by ele-mentary operations. Proc. Amer. Math. Soc., 17, no. 1, 240-246.

[69] Franke, C. A note on the Galois theory of linear homogeneous differenceequations. Proc. Amer. Math. Soc., 18 (1967), 548-551.

[70] Franke, C. The Galois correspondence for linear homogeneous differenceequations. Proc. Amer. Math. Soc., 21 (1969), 397-401.

[71] Franke, C. Linearly reducible linear difference operators. Aequat. Math.,6 (1971), 188-194.

[72] Franke, C. Reducible linear difference operators. Aequat. Math., 9 (1973),136-144.

[73] Franke, C. A characterization of linear difference equations which are solv-able by elementary operations. Aequat. Math., 10 (1974), 97-104.

[74] Gelfand, S. I.; Manin, Yu. I. Methods of Homological Algebra. Springer-Verlag, New York, 1996.

[75] Greenspan, B. A bound for the orders of the components of a system ofalgebraic difference equations. Pacific J. Math., 9 (1959), 473-486.

[76] Grothendieck, A. Sur quelques points d’algebre homologique. TohokuMath. J., 9 (1957), 119-221.

[77] Hackbush, W. Elliptic Differential Equations. Theory and NumericalTreatment. Springer-Verlag, New York, 1987.

[78] Hendrics, P. A. An algorithm for determining thr difference Galois groupfor second order linear difference equations. Technical report , Rijksuniver-siteit, Groningen, 1996.

500 BIBLIOGRAPHY

[79] Hendrics, P. A. Algebraic aspects of linear differential and difference equa-tions. Ph. D. Thesis, Rijksuniversiteit, Groningen, 1996.

[80] Hendrics, P. A. An algorithm for computing a standard form for second-order linear q-difference equations. Algorithms for Algebra (Eindhoven,1996). J. Pure and Appl. Algebra, 117/118 (1997), 331-352.

[81] Hendrics, P. A.; Singer, M. F. Solving difference equations in finite terms.J. Symbolic Computation, 27 (1999), 239-259.

[82] Herzog, Fritz. Systems of algebraic mixed difference equations. Trans.Amer. Math. Soc., 37 (1935), 286-300.

[83] Hilton, P. J.; Stammbach, U. A Course in Homological Algebra. Springer-Verlag, New York, 1997.

[84] van Hoeij, M. Rational solutions of linear difference equations. Technicalreport, Dept. of Mathematics, Florida State University, 1998.

[85] Hrushovski, Ehud. The Manin-Mamford conjecture and the model theoryof difference fields. Ann. Pure Appl. Logic, 112 (2001), no. 1, 43-115.

[86] Infante, R. P. Strong normality and normality for difference fields. Aequat.Math., 20 (1980), 121-122.

[87] Infante, R. P. Strong normality and normality for difference fields. Aequat.Math., 20 (1980), 159-165.

[88] Infante, R. P. The structure of strongly normal difference extensions. Ae-quat. Math., 21 (1980), no. 1, 16-19.

[89] Infante, R. P. On the Galois theory of difference fields. Aequat. Math., 22(1981), 112-113.

[90] Infante, R. P. On the Galois theory of difference fields. Aequat. Math., 22(1981), 194-207.

[91] Infante, R. P. On the inverse problem in difference Galois theory. Alge-braists’ homage: papers in the ring theory and related topics. (New Haven,Conn., 1981). Contemp. Math., 13 (1982), 349-352.

[92] Iwasawa, K.; Tamagawa, T. On the group of automorphisms of a func-tional field. J. Math. Soc. Japan, 3 (1951), 137-147.

[93] Jacobi, C. G. J. Gesammelte Werke, vol. 5. Berlin, 1890, 191-216.

[94] Johnson, Joseph L. Differential dimension polynomials and a fundamentaltheorem on differential modules. Amer. J. Math, 91 (1969), no. 1, 239-248.

[95] Johnson, Joseph L. Kahler differentials and differential algebra. Ann. ofMath. (2), 89 (1969), 92-98.

BIBLIOGRAPHY 501

[96] Johnson, Joseph L. A notion on Krull dimension for differential rings.Comment. Math. Helv., 44 (1969), 207-216.

[97] Johnson, Joseph L. Kahler differentials and differential algebra in arbitrarycharacteristic. Trans. Amer. Math. Soc., 192 (1974), 201-208.

[98] Johnson, Joseph L.; Sit, W. On the differential transcendence polynomi-als of finitely generated differential field extensions. Amer. J. Math., 101(1979), 1249-1263.

[99] Kaplansky, I. An Introduction to Differential Algebra. Hermann, Paris,1957.

[100] Karr, M. Theory of summation in finite terms. J. Symbolic Comput., 1(1985), no. 3. 303-315.

[101] Kelley, W.; Peterson, A. Difference Equations. An Introduction with Ap-plications. 2nd ed. Acad. Press, San Diego, 2001.

[102] Kelly, J. General Topology. Springer-Verlag, New York, 1985.

[103] Kolchin E. R. The notion of dimension in the theory of algebraic differen-tial equations. Bull Amer. Math. Soc., 70 (1964), 570-573.

[104] Kolchin, E. R. Some problems in differential algebra. Proc. Int’l Congressof Mathematicians (Moscow - 1966), Moscow, 1968, 269-276.

[105] Kolchin, E. R. Differential Algebra and Algebraic Groups. AcademicPress, New York - London, 1973.

[106] Kondrateva, M. V.; Pankratev, E. V. A recursive algorithm for the compu-tation of Hilbert polynomial. Lect. Notes Comp. Sci., 378 (1990), 365-375.Proc. EUROCAL 87, Springer-Verlag.

[107] Kondrateva, M. V.; Pankratev, E. V. Algorithms of computation of char-acteristic Hilbert polynomials. Packets of Applied Programs. AnalyticTransformations, 129-146. Nauka, Moscow, 1988. (In Russian)

[108] Kondrateva, M. V.; Pankratev, E. V.; Serov, R. E. Computations in dif-ferential and difference modules. Proceedings of the Internat. Conf. onthe Analytic Computations and their Applications in Theoret. Physics,Dubna, 1985, 208-213. (In Russian)

[109] Kondrateva,M. V.; Levin, A. B.; Mikhalev, A. V.; Pankratev, E. V. Com-putation of dimension polynomials. Internat. J. of Algebra and Comput., 2(1992), no. 2, 117-137.

[110] Kondrateva, M. V.; Levin, A. B.; Mikhalev, A. V.; Pankratev, E. V. Dif-ferential and Difference Dimension Polynomials. Kluwer Academic Pub-lishers., Dordrecht, 1998.

502 BIBLIOGRAPHY

[111] Kowalski, P.; Pillay, A. A note on groups definable in difference fields.Proc. Amer. math. Soc., 130 (2002), no. 1, 205-212 (electronic).

[112] Kreimer, H. F. The foundations for extension of differential algebra. Trans.Amer. Math. Soc., 111 (1964), 482-492.

[113] Kreimer, H. F. An extension of differential Galois theory. Trans. Amer.Math. Soc., 118 (1965), 247-256.

[114] Kreimer, H. F. On an extension of the Picard-Vessiot theory. Pacific J.Math., 15 (1965), 191-205.

[115] Kunz, E. Introduction to Commutative Algebra and Algebraic Geometry.Birkhauser, Boston, 1985.

[116] Lando, B. Jacobi’s bound for the order of systems of first order differentialequations. Trans. Amer. Math. Soc., 152 (1970), no. 1, 119-135.

[117] Lando, B. Jacobi’s bound for the first order difference equations. Proc.Amer. Math. Soc., 32 (1972), no. 1, 8-12.

[118] Lando, B. Extensions of difference specializations. Proc. Amer. Math. Soc.,79, no. 2 (1980), 197-202.

[119] Lang, S. Introduction to Algebraic Geometry. Adison-Wesley, MA, 1972.

[120] Levin, A. B. Characteristic polynomials of filtered difference modules andof difference field extensions. Uspehi Mat. Nauk , 33 (1978), no. 3, 177-178 (Russian). English transl.: Russian Math. Surveys, 33 (1978), no. 3,165-166.

[121] Levin, A. B. Characteristic polynomials of inversive difference mod-ules and some properties of inversive difference dimension. Uspehi Mat.Nauk , 35 (1980), no. 1, 201-202 (Russian). English transl.: Russian Math.Surveys, 35 (1980), no. 1, 217-218.

[122] Levin, A. B. Characteristic polynomials of difference modules and someproperties of difference dimension. VINITI (Moscow, Russia), 1980, no.2175-80. (Russian)

[123] Levin, A. B. Type and dimension of inversive difference vector spacesand difference algebras. VINITI (Moscow, Russia), 1982, no. 1606-82.(Russian)

[124] Levin, A. B. Characteristic polynomials of ∆-modules and finitely gen-erated ∆-field extensions. VINITI (Moscow, Russia), 1985, no. 334-85.(Russian)

[125] Levin, A. B. Inversive difference modules and problems of solvability ofsystems of linear difference equations. VINITI (Moscow, Russia), 1985,no. 335-85. (Russian)

BIBLIOGRAPHY 503

[126] Levin, A. B. Computation of Hilbert polynomials in two variables. J.Symbolic Comput., 28 (1999), 681-709.

[127] Levin, A. B. Characteristic polynomials of finitely generated modules overWeyl algebras. Bull. Austral. Math. Soc., 61 (2000), 387-403.

[128] Levin, A. B. Reduced Grobner bases, free difference-differential modulesand difference-differential dimension polynomials. J. Symbolic Comput.,29 (2000), 1-26.

[129] Levin, A. B. On the set of Hilbert polynomials. Bull. Austral. Math. Soc.,64 (2001), 291-305.

[130] Levin, A. B. Multivariable dimension polynomials and new invariants ofdifferential field extensions. Internat. J. Math. and Math. Sci., 27, no. 4,201-213.

[131] Levin, A. B.; Mikhalev, A. V. Difference-differential dimension polynomi-als. VINITI (Moscow, Russia), 1988, no. 6848-B88. (Russian)

[132] Levin, A. B.; Mikhalev, A. V. Dimension polynomials of filteredG-modules and finitely generated G-field extensions. Algebra (collectionof papers). Moscow State University, Moscow, 1989, 74-94. (Russian)

[133] Levin, A. B.; Mikhalev, A. V. Type and dimension of finitely generatedvector G-spaces. Vestnik Mosk. Univ., Ser. I, Mat. Mekh., 1991, no. 4, 72-74 (Russian). English transl.: Moscow Univ. Math. Bull, 46, no. 4, 51-52.

[134] Levin, A. B.; Mikhalev, A. V. Dimension polynomials of difference-differential modules and of difference-differential field extensions. AbelianGroups and Modules, 10 (1991), 56-82. (Russian)

[135] Levin, A. B.; Mikhalev, A. V. Dimension polynomials of filtered differ-ential G-modules and extensions of differential G-fields. Contemp. Math.,131 (1992), Part 2, 469-489.

[136] Levin, A. B.; Mikhalev, A. V. Type and dimension of finitely generatedG-algebras. Contemp. Math., 184 (1995), 275-280.

[137] Macintyre, A. Generic automorphisms of fields. Ann. Pure Appl. Logic,88 (1997), no. 2-3, 165-180.

[138] Matsumura, H. Commutative ring theory. Cambridge Univ. Press, NewYork, 1986.

[139] Mikhalev, A. V.; Pankratev, E. V. Differential dimension polynomial ofa system of differential equations. Algebra (collection of papers). MoscowState Univ., Moscow, 1980, 57-67. (Russian)

504 BIBLIOGRAPHY

[140] Mikhalev, A. V.; Pankratev, E. V. Differential and Difference Algebra.Algebra, Topology, Geometry, 25, 67-139. Itogi Nauki i Tekhniki, Akad.Nauk SSSR, 1987 (Russian). English transl.: J. Soviet. Math., 45 (1989),no. 1, 912-955.

[141] Mikhalev, A. V.; Pankratev, E. V. Computer Algebra. Calculations inDifferential and Difference Algebra. Moscow State Univ,, Moscow, 1989.(Russian)

[142] Mishra, B. Algorithmic Algebra. Springer-Verlag , New York, 1993.

[143] Moosa, R. On difference fields with quantifier elimination. Bull. LondonMath. Soc., 33 (2001), no. 6, 641-646.

[144] Morandi, P. Field and Galois Theory. Springer-Verlag, New York, 1996.

[145] Nastasescu, C.; Van Oystaeyen, F. Graded and Filtered Rings and Mod-ules. Springer-Verlag, Berlin - New York, 1979.

[146] Nastasescu, C.; Van Oystaeyen, F. Methods of Graded Rings. Springer-Verlag, Berlin - New York, 2004.

[147] Nishioka, K. A note on differentially algebraic solutions of first order lineardifference equations. Aequat. Math., 27 (1984), 32-48.

[148] Nonvide, S. Corps aux differences finies. C. R. Acad. Sci. Paris Ser. IMath., 314 (1992), no. 6, 423-425. (French)

[149] Osborne, Scott M. Basic Homological Algebra. Springer-Verlag, NewYork, 2000.

[150] Ovsiannikov, L. V. Group Analysis of Differential Equations. Acad. Press,New York, 1982.

[151] Pankratev, E. V. The inverse Galois problem for the extensions of differ-ence fields. Algebra i Logika, 11 (1972), 87-118 (Russian). English transl.:Algebra and Logic, 11 (1972), 51-69.

[152] Pankratev, E. V. The inverse Galois problem for extensions of differencefields. Uspehi. Mat. Nauk , 27 (1972), no. 1, 249-250. (Russian)

[153] Pankratev, E. V. Fuchsian difference modules. Uspehi. Mat. Nauk , 28(1973), no. 3, 193-194. (Russian)

[154] Pankratev, E. V. Computations in differential and difference modules.Symmetries of partial differential equations, Part III. Acta Appl. Math.,16 (1989), no. 2, 167-189.

[155] Petkovsek, M. Finding closed form solutions of difference equations bysymbolic methods. Thesis, Dept. of Comp. Sci., Carnegie Melon Univer-sity, 1990.

BIBLIOGRAPHY 505

[156] Pillay, A. A note on existentially closed difference fields with algebraicallyclosed fixed field. J. Symbolic Logic, 66 (2001), no. 2, 719-721.

[157] Praagman, C. The formal classification of linear difference operators. Proc.Kon. Ned. Ac. Wet., Ser. A, 86 (1983), 249-261.

[158] Praagman, C. Meromorphic linear difference equations. Thesis, Universityof Groningen, 1985.

[159] van der Put. M., Singer, M. F. Galois theory of difference equations.Springer , Berlin, 1997.

[160] Richtmyer, R. D.; Morton, K. W. Difference Methods for Initial-ValueProblems. 2nd ed. Interscience Publ., New York, 1967.

[161] Riordan, J. Combinatorial Identities. John Wiley and Sons Inc., NewYork, 1968.

[162] Ritt, J. F. Algebraic difference equations. Bull. Amer. Math. Soc., 40(1934), 303-308.

[163] Ritt, J. F. Complete difference ideals. Amer. J. Math, 63 (1941), 681-690.

[164] Ritt, J. F. Differential Algebra. Amer. Math. Soc. Coll. Publ., Vol. 33.New York, 1950.

[165] Ritt, J. F.; Doob, J. L. Systems of algebraic difference equations. Amer.J. Math, 55 (1933), 505-514.

[166] Ritt, J. F., Raudenbush, H. W. Ideal theory and algebraic difference equa-tions. Trans. Amer. Math. Soc., 46 (1939), 445-453.

[167] Roman, S. Field Theory. Second Edition. Springer-Verlag, New York,2006.

[168] Sachkov, V. N. Combinatorial Methods in Discrete Mathematics. Ency-clopedia of Mathematics and its Applications, Vol. 55. Cambridge Univ.Press, New York, 1996.

[169] Scanlon, T.; Voloch, J. F. Difference algebraic subgroups of commutativealgebraic groups over finite fields. Manuscripta Math, 99 (1999), no. 3,329-339.

[170] Sit, W. Well-ordering of certain numerical polynomials. Trans. Amer.Math. Soc., 212 (1975), 37-45.

[171] Spindler, K. Abstract Algebra with Applications. Vol. I, II. Marcel Dekker,New York, 1994.

[172] Strodt, W. Systems of algebraic partial difference equations. Master essay ,Columbia Univ., 1937.

506 BIBLIOGRAPHY

[173] Strodt, W. Principal solutions of difference equations. Amer. J. Math., 69(1947), 717-757.

[174] Tikhonov, A. N.; Samarskii, A. A. Equations of Mathematical Physics.Dover, New York, 1990.

[175] Trendafilov, I. Radical closures in difference rings and modules. Applica-tions of mathematics in engineering and economics (Sozopol, 2001). HeronPress, Sofia, 2002, 212-217.

[176] Vermani, L. R. An Elementary Approach to Homological Algbera. Chap-man and Hall/CRC, New York, 2003.

[177] Weibel, C. An Introduction to Homological Algebra. Cambridge Univ.Press. New York, 1994.

[178] Zariski, O., Samuel, P. Commutative Algebra. Princeton: Van Nostran.Vol. I (1958), Vol. II (1960).

Index

AAbelian category, 10–12Additive category, 9

functor, 9Adhered function, 429, 431Admissible order, 96

transformation, 202–204, 206–209Affine K-algebra, 35

algebraic K-variety, 30, 31A-leader of a difference (σ-)

polynomial, 130Algebra of difference polynomials, 117

of inversive differencepolynomials, 118

of σ-polynomials, 116–119of σ∗-polynomials, 116–118

Algebraic element, 65field extension, 65–69closure, 66–69

Algebraically closed field, 67dependent element, 76

set, 76disjoint fields, 76, 79independent element, 76independent set, 76

Almost every specialization, 86, 87α-derivation, 94, 95α-invariant element, 114αi-periodic, 114Antisymmetric relation, 3Amonadic difference (σ-) field

extension, 354–356Artinian module, 22–23

ring, 22–23Aperiodic difference (σ-) ring, 114Ascending chain condition, 22Associated graded module, 44

object, 11ring, 44prime ideals, 25primes belonging to an ideal, 26

Automorphism, 12Autoreduced set, 130–132, 134, 135,

138, 270(<1, . . . , <p)-autoreduced set, 212,

213, 215, 216Axiom of Choice, 5

BBabbitt’s Decomposition Theorem,

336–338Basic rectangle, 456–458Basic set, 103–121Basis for a set, 6

for transformal transcendence,248

of a perfect difference ideal, 141,142

of a set in a difference ring, 141of a solution field, 475, 476

Benign decomposition, 338, 340extension, 334

Bijective mapping, 2Bimorphism, 8Binary relation, 2, 5Biproduct, 9Boundaries, 13Buchberger algorithm, 99–100

criterion, 99, 102

CCartesian product, 2, 3Casorati determinant, 472–475

507

508 INDEX

Category, 6–12with enough projective objects,

11, 13with enough injective objects, 11,

13Cauchy sequence, 89Chain, 4Chain complex, 13, 14Characteristic set, 128, 132, 134,

215–217, 270, 271(<1, . . . , <p)-characteristic set,

216–218Characteristic polynomial, 161, 162,

198, 200, 232, 234, 235Chinese Remainder Theorem, 19Clopen subset, 85Cokernel, 10, 12

object, 10Codomain of a morphism, 7Coherent autoreduced set, 136, 138Coheight of a prime ideal, 27, 34

coheight of an ideal, 27Cohomological spectral sequence, 12Compatible difference field extensions,

311Complete difference ideal, 123, 124

set of parameters, 33–35, 395–397system of difference (σ-) over

fields, 428Completely aperiodic difference field,

298Component of a filtration, 11, 12, 42Complete set of conjugates, 74Composition series, 24

of morphisms, 6Compositum, 65, 66Condition of minimality, 4Confluent relation, 99Constant, 106Contravariant functor, 8Coordinate extension, 359–364

field, 30Coprime ideals, 19

in pairs, 19Coproduct, 9

Core, 286, 290of a Galois group, 464, 465

Covariant functor, 7Criterion of Compatibility, 342–345Cycles, 13

DDecomposable ideal, 25, 26Defining difference (σ-) ideal, 117, 118

σ∗-ideal, 117, 118Degree of a difference kernel with

respect to a subindexing, 319of a difference (σ-) polynomial,

117with respect to a variable τyi,

117of a field extension, 65of inseparability, 73lexicographic order, 96, 97reverse lexicographic order, 96of a monomial, 15

with respect to a variable, 15of a polynomial, 15

with respect to a variable, 15of a σ∗-polynomial, 117

with respect to a variable γyi,117

of a skew polynomial, 95dependence relation, 5–6dependent element, 5

set, 6derivation, 89–95descending chain condition, 522defining difference (σ-) ideal, 117

σ-ideal of an s-tuple, 118σ∗-ideal of an s-tuple, 118

defining ideal of an n-tuple, 32diagonal sum of a matrix, 36Dickson’s Lemma, 97difference algebra, 300–309

automorphism, 108dimension, 166, 273

polynomial, 161, 258, 273, 274,442, 444

endomorphism, 108epimorphism, 156

INDEX 509

field, 104extension, 104subextension, 104

Galois group, 355, 356homomorphism, 108, 109, 156ideal, 104ideals separated in pairs, 123–126

strongly separated in pairs,123–126

indeterminates, 115isomorphism, 108, 156kernel, 319, 320, 372–374module, 155monomorphism, 156operator, 155, 156of order (l1,. . . ,lp), 48overfield, 104overring, 104place, 385polynomial, 115, 117R0-homomorphism, 108ring, 103–105

extension, 104specialization, 115

of a difference field, 386transcendence basis, 248transcendence degree, 249, 250vector space, 156subfield, 104subring, 104, 105

of invariant elements, 114of periodic elements, 114

type, 164, 257–258, 273valuation, 386

ring, 385–387variety, 149

differential of an element, 92dimension of a difference variety, 394

of a difference variety relative toyi1, . . . , yiq, 394

of an irreducible variety, 33of a module with respect to a

family of submodules, 240of a prime inversive difference

ideal, 394

of a prime inversive differenceideal relative to yi1, . . . , yiq,394

of a ring, 28dimension polynomial, 183, 184, 218,

219, 268of a set in Nm, 54of a σ∗-K-algebra, 301, 302

directed set, 4discrete filtration, 43discrete valuation, 36disjoint sets, 3domain of a relation, 2

of a morphism, 7domination, 386

Eeffective order of an ordinary

difference field extension,295–296

of a difference (σ-) polynomial,129

of a difference polynomial withrespect to yj , 129

of a prime inversive differenceideal, 394


of a difference variety, 398of a difference variety relative to

yi1, . . . , yiq, 394embedding, 12endomorphism, 12epimorphism, 8, 12ε-filtration, 239equivalence, 7, 8, 10

class, 3relation, 3

equivalent basic sets, 202difference (σ-) fields, 335, 336difference (σ-) field extensions,

336s-tuples, 86, 118kernels, 319

510 INDEX

realizations of a difference kernel,378

essential prime divisors, 144–147separated divisors, 147

components of a variety, 153strongly separated divisors, 146

essentially continuous function, 429ε-subvariety, 150, 154ε-variety, 150, 151, 153, 154exact chain complex, 13

pair of homomorphisms, 12sequence, 12

excellent filtration, 161, 195, 196,226–229, 234, 235, 241, 263

p-dimensional filtration, 167, 168,210

exhaustive filtration, 43extension of a specialization, 86

Ffactor object, 8, 10family of sets, 1–3, 5field of algebraic functions, 88, 89

of definition, 30fil-basis, 45, 46filtered object, 11

G-A-module, 226module, 43–46ring, 42–46

filtration, 11, 42, 43, 155, 156, 195,196, 226, 227, 264, 265

final object, 8finite field extension, 64–65

filtration, 226finitely generated difference (σ-) field

extension, 107difference (σ-) ideal, 107difference (σ-) module, 156difference (σ-) overfield, 107difference (σ-) overring, 107difference ring extension, 107field extension, 64–65G-A-module, 226G-ring extension, 224inversive difference (σ∗-) field

extension, 108

inversive difference (σ∗-)overfield, 108

inversive difference (σ∗-)overring, 108

inversive difference (σ∗-) ringextension, 108

σ-ε∗-module, 233, 236σ-K-algebra, 300σ∗-K-algebra, 300σ∗-ideal, 107

first difference, 48relative to a variable, 48

flat module, 13free fields over a common subfield, 79

filtered module, 45σ∗-R-module, 199

generators, 168join, 81σ-K-module, 168σ∗-K-module, 210vector σ-K-space, 168vector σ∗-K-space, 210

formal algebraic solution, 119–121fundamental matrix, 486, 487, 491

replicability theorem, 352system of solutions, 473theorem of infinite Galois theory,

85

GGalois closed subgroup, 479

field extension, 70group, 84

of finite type, 467G-A-module, 225, 226, 228G-A-homomorphism, 225G-dimension, 231

G-dimension polynomial, 229–231Generelized Picard-Vessiot extension

(GPVE), 481, 483Liouvillian extension (GLE), 483,

484Generators of a field extension, 15

of a ring extension, 14generic prolongation of a difference

(σ-) kernel, 320, 374

INDEX 511

solution field, 481, 483specialization, 85, 86zero of an ideal, 32

of a set of differencepolynomials, 118

of a set of σ∗-polynomials, 118of a variety, 33, 152of σ-polynomials, 118

G-epimorphism, 224, 226G-field, 224

extension, 224G-generators, 224G-homomorphism, 224, 225G-ideal, 224G-isomorphism, 224, 226G-module, 225G-monomorphism, 224, 226“Going down” Theorem, 29“Going up” Theorem, 29good filtration, 226G-operator, 225, 226G-overfield, 224gradation, 37, 38graded homomorphism, 38

homomorphism of degree r, 38graded ideal, 37module, 38ring, 37submodule, 38subring, 37

graded lexicographic order, 53G-ring, 224, 225

extension, 224of quotients, 224

ground difference (σ-) field, 149greatest element, 4greatest lower bound, 4Greenspan number, 433Grothendieck’s axiom A5, 39Grobner basis, 96–102

with respect to the orders<1, . . . , <p, 169, 170, 223

G-subfield, 224G-subring, 224G-torsor, 489, 490G-type, 231

HHausdorff Maximal Principle, 5high order condition, 432height of a prime ideal, 27

height of an ideal, 27Hilbert function, 41Hilbert polynomial, 41, 42Hilbert Basis Theorem, 23, 95, 97Hilbert’s Nullstellenzats, 32Hilbert Syzygy Theorem, 42homogeneous component of a graded

module, 38component of a graded ring, 37component of an element, 37, 38degree of a component, 37, 38

of an element, 37, 38element of degree n, 37, 38homomorphism, 37, 38ideal, 37submodule, 38subring, 37

homogenezation, 163homology module, 13homomorphism of filtered modules, 43

of filtered modules of degree p, 43of filtered difference (σ-)

modules, 156of filtered σ-R-modules, 157of G-A-modules, 225of graded modules, 38

modules of degree r, 38of graded rings, 37rings of degree r, 37

hypersurface, 31

II-adic filtration, 43ideal of a variety, 30identity morphism, 6, 7incompatible difference field

extensions, 311–318indecomposible difference ideal, 127independent element, 5

set, 6index set, 2, 3indexing, 3

512 INDEX

induced automorphism, 356, 358, 359induction condition, 4infimum, 4initial object, 8, 9

of a difference (σ-) polynomial,129

of a difference (σ-) polynomialwith respect to yi, 427

of a σ∗-polynomial, 139subset, 57, 58, 62, 63

injective mapping, 2module, 11object, 11, 13resolution, 14

integral closure, 28integral element, 28integral extension of a ring, 28, 29inseparable degree, 73intermediate field, 64, 65

difference field, 104σ-field, 104σ∗-field, 104

integrally closed integral domain, 29invariant difference (σ-) ring, 114

element, 114subset of a difference ring, 112

invariants of a dimension polynomial,183

inverse difference field, 284, 285limit degree, 284inverse reduced limit degree, 284

inversive closure, 109–112difference algebra, 300

dimension, 202, 273field, 104, 105field extension, 104field subextension, 104module, 188operator, 185, 186overfield, 104polynomial, 115type, 201, 202, 264, 273vector space, 186

difference ring, 103, 104, 106subfield, 104

irreducible component, 31–33ε-component, 150, 151ε-variety, 150, 151ideal, 26topological space, 31variety, 150

invertible morphism, 7irredundant intersection, 27, 143, 144

primary decomposition, 25representation of a perfect

difference ideal, 143, 144of an ε-variety, 150, 151, 154of a variety, 150, 154

isolated difference (σ-) field extension,364

prime ideal, 17set of prime ideals, 26

isomorphic kernels, 319isomorphism in a category, 7

of algebraic structures, 12of field extensions, 64

JJacobi number, 434

Jacobi number of a matrix, 36Jacobson radical, 18j-leader, 168j-leading coefficient, 168Jordan-Holder series, 24

Kkernel, 9, 10

object, 9K-homomorphism, 64K-isomorphism, 64K isomorphic fields, 64(<k, <i1, . . . , <il)-reduced element,

169, 212(<k, <i1, . . . , <il)-reduction, 169, 212<k-leader, 167k-leading coefficient, 168, 211K-linear derivation, 89Kolchin polynomial, 55, 56Krull dimension, 27

Theorem, 27topology, 85

INDEX 513

k-leader, 168, 211kth order, 168kth S-polynomial, 172Kuratowski Lemma, 5

Lleader of a difference (σ-) polynomial,

129, 130of a σ∗-polynomial, 133

leading coefficient, 97, 101, 269leading monomial, 97, 101term, 97, 101

least common multiple, 99, 100, 101,168, 269

least upper bound, 4left derived functor, 14

exact functor, 12limited graded module, 40Ore domain, 95

length of an element of a freeσ∗-module, 446

of a kernel, 319, 320, 371of a module, 24of a σ∗-operator, 448

lexicographic order, 53, 96limit degree, 274–277

of a difference variety, 394of a difference variety relative to

yi1, . . . , yiq, 394of a prime inversive difference

ideal, 394of a prime inversive difference

ideal relative to yi1, . . . , yiq,394

limit of a spectral sequence, 11σ-transcendence basis, 253, 254

linear order, 4difference (σ-) ideal, 135, 136σ∗-ideal, 135, 136

linearly disjoint field extensions, 34,75

linearly ordered set, 4, 5Liouvillian extension (LE), 482local difference (σ-) ring, 385

ring, 17

σ∗-K-algebra of finitely generatedtype, 305

localization, 20, 21low weight condition, 432, 433lower bound, 4Luroth’s Theorem, 78

MMacaulay’s Theorem, 28mapping, 2, 6Mapping Theorem, 363matrix order, 97

of an admissible transformation,202, 203

maximal ideal, 15–16difference ideal, 104

(σ-) place, 385specialization of a difference

field, 385σ-ideal, 104σ∗-ideal, 104

maximum spectrum of a ring, 18m-basis, 141, 142mild difference(σ-) field extension, 335minimal degree, 335

element, 4generator, 336, 337normal standard generator, 336polynomial, 66primary decomposition, 25, 26prime ideal, 17σ-dimension polynomial, 267σ∗-dimension polynomial, 267standard generator, 336τ -invariant reduced set, 488

mixed difference ideal, 126module of derivations, 89

module of differentials, 92module of fractions, 20, 21of Kahler differentials, 91

monadic difference (σ-) fieldextension, 354

monic morphism, 8, 10monomial, 96, 97, 101, 169, 211

ideal, 97order, 96, 97, 100, 101

514 INDEX

monomorphism, 8–12morphism, 6–11m-tuple, 3, 53, 54multiple, 168, 169

realization, 420, 421multiplicative set, 15–17multiplicatively closed set, 15multiplicity, 74

NNakayama Lemma, 18natural homomorphism, 20, 21

isomorphism, 8transformation, 7, 8

nilpotent, 18nilradical, 18Noether Normalization Theorem, 35Noetherian module, 22

ring, 22nondegenerate extension of a

specialization, 87normal closure, 69

element, 67, 332field extension, 67, 68standard generator, 333, 334

null object, 8numerical polynomial, 47–53N-Z-dimension polynomial, 64

Oobject, 6–11

with filtration, 11order, 128, 129, 133, 185, 225, 232,

269, 270, 296of a difference kernel, 322of a difference kernel with respect

to a subindexing, 322of a difference (σ-) operator, 155of a difference (σ-) polynomial,

129of a difference polynomial with

respect to yj , 129of a difference variety, 394of a difference variety relative to

yi1, . . . , yiq, 394of a node, 444

of a prime inversive differenceideal, 394


of a term, 128, 133of τ with respect to σi, 167

orderly ranking, 129, 133ordinary difference ring, 104–106ortant, 57, 132overfield, 15overring, 14

Pp-adic completion, 89pairwise coprime ideals, 19parameters, 33–35partial order, 4, 5

difference ring, 104σ-ring, 104

partially ordered set, 4, 5partition, 54–57pathological difference (σ-) field

extension, 354p-dimensional filtration, 167, 168perfect closure of a field, 34, 72

of a set in a difference ring, 121difference (σ-) ideal, 121difference ideal generated by a

set, 121field, 71

periodic element, 114difference (σ-) ring, 114with respect to αi, 114

permitted difference field, 428ring, 428function, 428, 429

Picard-Vessiot extension (PVE), 473,475

ring (PVR), 486, 487, 489place, 88, 89points, 30, 31positive filtration, 43, 46positively graded module, 40

graded ring, 37power series, 34, 35

INDEX 515

P -prime ideal, 24preadditive category, 9primary field extension, 83

ideal, 24decomposition, 24–26

prime ideal, 15, 16difference ideal, 104σ-ideal, 104σ∗-ideal, 104system of algebraic σ∗-equations,

444primitive element, 71principal component of a difference

variety, 396, 397of a difference variety with

respect to yk, 410prime element, 17realization of a difference kernel,

324, 325, 378product, 8, 9product order, 53projective module, 11, 13

object, 11resolution, 13, 14

prolongation of a difference (σ-)kernel, 319, 372

proper field subextension, 64ε-subvariety, 150, 151subvariety, 150

properly monadic difference (σ-) fieldextension, 354

property L∗, 373P, 372P∗, 372, 373, 375

purely inseparable closure, 73inseparable element, 72inseparable field extension, 72, 73inseparable part, 73σ-transcendental extension, 248,

250transcendental field extension, 78

Qq-chain, 484qLE σ∗-overfield, 484

quasi-linearly disjoint field extensions,34, 81–83

quotient field, 20G-field, 224,object, 8

σ-field, 114

Rradical ideal, 18, 19radical of an ideal, 18range of a relation, 2rank, 129–132, 134, 215–217, 269–270ranking of a family of

σ-indeterminates, 129of a family of σ∗-indeterminates,

133realization of a difference kernel, 324,

378reduced degree of a difference kernel

with respect to asubindexing, 322

of a polynomial, 70reduced element of a free module, 100

limit degree, 282σ∆-limit degree, 282polynomial, 98–99σ-polynomial, 131, 269–271σ∗-polynomial, 134

reducible ε-variety, 151variety, 151

reduction modulo an autoreduced set,131–132, 134

modulo an element, 98, 101a set, 98, 101–102

reflexive closure of a relation, 3of a difference (σ-) ideal, 107relation, 3σ-ideal, 104

regular automorphism, 358field extension, 82–83local integral domain, 36realization of a difference kernel,

324, 378remainder, 98, 101, 130, 134–135replicability, 352–354

516 INDEX

residue field, 17field of a place, 88

resolvent, 426ideal, 426

right derived functor, 14exact functor, 13limited graded module, 40Ore domain, 95

ring extension, 14of constants, 106-107of difference (σ-) operators, 156of σ-ε∗-operators, 233–235of fractions, 20of G-A-operators, 225of G-operators, 225of inversive difference (σ∗-)

operators, 185of a place, 88of skew polynomials, 94–95

Ritt difference ring, 141–149number, 434

Ssaturated multiplicatively closed set,

15saturation, 16separable closure, 73

degree, 73element, 69factor of degree, 73field extension, 70, 79part, 73polynomial, 69

separably algebraic field extension,70, 417

algebraically closed field, 71generated field extension, 79

separant, 427separated difference (σ-) ideals, 123

filtration, 43varieties, 153

separating transcendence basis, 78–79set of constants, 187

of coordinate extensions, 359–361of parameters, 395, 396

short exact sequence, 12

shuffling, 121σ-algebraic element, 115

field extension, 115s-tuple, 117

σ-algebraically dependent family, 115on a set element, 245σ-algebraically independent on a

set element, 245family, 115indexing, 379

σ-automorphism, 108σ∆-limit degree, 282σ-dimension, 165, 273σ-dimension polynomial, 161, 257, 273σ-endomorphism, 108σ-ε∗-operator, 232σ-ε∗-dimension, 237

polynomial, 235, 236σ-ε∗-epimorphism, 233σ-ε∗-homomorphism, 233

of filtered σ-ε∗-modules, 233σ-ε∗-isomorphism, 233σ-ε∗-linearly dependent elements, 238σ-ε∗-linearly independent, 238σ-ε∗-module, 233σ-ε∗-monomorphism, 233σ-ε∗-ring, 232σ-ε∗-type, 238σ-field, 104

extension, 104subextension, 104

σ-filtration, 239σ-Galois group, 356σ-generators, 107σ-homomorphism, 108σ-ideal, 104σ-indeterminates, 115σ-invariant subgroup, 356σ-isomorphism, 108σ-K-algebra, 300σ-kernel, 319, 372σ-maximal ideal, 127σ-normal extension, 478, 479σ-operator, 155σ-overfield, 104σ-overring, 104

INDEX 517

σ-place, 385σ-polynomial, 115σ-prime ideal, 128σ-R-module, 156σ-R0-homomorphism, 108σ-ring, 103

extension, 104of fractions, 113

σ-specialization, 115of a σ-field, 385

σ-stable difference field extension, 356subgroup, 356

σ-subfield, 104σ-subring, 104

of invariant elements, 114of periodic elements, 114

σ-subset, 112σ-transcendental basis, 248σ-transcendence degree, 248, 300σ-transcendental element, 115σ-type, 165, 258σ-valuation, 389

ring, 389(σ1; :::;σp)-dimension polynomial, 183(σ1; :::;σp)∗-dimension polynomial,

218, 219σ′-periodic element, 114σ∗-algebraically independent family,

116σ∗-dimension, 201, 274σ∗-dimension polynomial, 198, 262,

263, 265, 273σ∗-field, 104

extension, 104subextension, 104

σ∗-generators, 107, 108σ∗-ideal, 104σ∗-indeterminates, 116σ∗-K-algebra, 300σ∗-linearly independent elements, 201σ∗-operator, 185σ∗-order of a σ∗-polynomial, 456σ∗-overfield, 104σ∗-overring, 104σ∗-polynomial, 116

σ∗-ring, 103extension, 104of fractions, 113

σ∗-R-module, 187σ∗-subfield, 104σ∗-subring, 104σ∗-subset, 112σ∗-type, 201, 264

similar elements, 211terms, 211

simple difference ring, 107field extension, 71σ-ring, 107

singular component of a differencevariety, 427

realization, 420skew derivation, 94

polynomial, 94smallest element, 4solution field, 473

of a set of difference (σ-)polynomials, 118

of a system of algebraic difference(σ-) equations, 118

of a set of σ∗-polynomials, 118solvability by elementary operations,

484special set, 322specialization, 85–88, 115

of a domain, 87spectral sequence, 11

functor, 12spectrum of a ring, 18splitting field, 66S-polynomial, 99, 102standard basis, 97

filtration, 156, 187, 226, 234, 235form of a σ∗-operator, 441generator, 333gradation, 41p-dimensional filtration, 167position of a difference

polynomial, 412ranking, 129, 133representation of a polynomial, 97Z-dimension polynomial, 60

518 INDEX

stepwise compatibility condition, 317strength a system of equations in finite

differences, 443s-tuple from a family of difference

(σ-) overfields, 149over a difference field, 117

strongly normal extension, 485separated difference (σ-) ideals,

123σ-stable difference field

extension, 357subcategory, 7subfield, 104subindexing, 3subobject, 8subring, 104

integrally closed in a ring, 28substitution, 117subvariety, 150supremum, 4surjective mapping, 2symmetric closure of a relation, 3

relation, 3σ∗-operator, 441system of algebraic σ∗-equations,

118system of defining equations of a

variety, 30

Tterm, 97, 101, 127, 133, 168, 210terminal object, 8total degree of a monomial, 15

of a polynomial, 15Picard-Vessiot ring (TPVR), 491

total order, 4totally ordered set, 4transcendence basis, 77

degree, 27, 33, 77, 319of an integral domain, 27

transcendental element, 65transform of an element, 107, 133, 211

of a term, 129, 133, 211

transformally algebraic element, 115dependent family, 115dependent on a set element, 245difference (σ-) field extension, 115independent family, 115independent on a set element, 245transcendental element, 115

transitive closure of a relation, 3transitive relation, 3translation, 103trivial filtration, 43type, 240, 241, 301, 302typical difference (σ-) dimension,

164–165, 273difference (σ-) transcendence

degree, 258G-dimension, 231inversive difference (σ∗-)

dimension, 201, 273difference (σ∗-) transcendence

degree, 264σ-ε∗-dimension, 237σ∗-transcendence degree, 264

Uunderlying field, 104

ring, 103universal compatibility condition, 317

extension, 477field, 32K-linear derivation, 92system of difference (σ-)

overfields, 149system of σ∗-overfields, 153

universally compatible extension, 465,471

upper bound, 4

Vvaluation ring, 36vanishing ideal of a variety, 30variety, 30, 149vector σ-R-space, 156vector σ∗-R-space, 186

INDEX 519

σ-ε∗-space, 233V -ring, 88

Wweight function, 430

parameter, 430, 431well-ordered set, 5, 49Well-ordering Principle, 5

ZZariski topology, 31Z-dimension polynomial, 59Zermelo Postulate, 5zero morphism, 8

object, 8Zorn Lemma, 5

Difference Algebra (Algebra and Applications)

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