Subalgebra Bases in Local Rings and its Efficient Computation in Polynomial Rings

Name : JUNAID ALAM KHAN

Year of Admission : 2006

Registration No : 67-GCU-PHD-SMS-06

Abdus Salam School of Mathematical Sciences

GC University, Lahore, Pakistan

i

Subalgebra Bases in Local Rings and its Efficient Computation in Polynomial Rings

Submitted to

Abdus Salam School of Mathematical Sciences

GC University Lahore, Pakistan

in the partial fulfillment of the requirements for the award of degree of

Doctor of Philosophy in

Mathematics By

Name : JUNAID ALAM KHAN

Year of Admission : 2006

Registration No : 67-GCU-PHD-SMS-06

Abdus Salam School of Mathematical Sciences GC University, Lahore, Pakistan

ii

DECLARATION

I Junaid Alam Khan Registration No. 67-GCU-PHD-SMS-06 student at Abdus Salam

School of Mathematical Sciences GC University in the subject of Mathematics, Year

of Admission (2006), hereby declare that the matter printed in this thesis titled

“Subalgebra Bases in Local Rings and its Efficient Computation in Polynomial Rings”

is my own work and that

i) I am not registered for the similar degree elsewhere contemporaneously.

ii) No direct major work had already been done by me or anybody else on this

topic; I worked on, for the Ph. D degree.

iii) The work, I am submitting for the Ph. D. degree has not already been

submitted elsewhere and shall not in future be submitted by me for obtaining

similar degree from any other institution.

Dated: ____________________ ____________________

Signature

iii

RESEARCH COMPLETION CERTIFICATE

Certified that the research work contained in this thesis titled “Subalgebra Bases in Local

Rings and its Efficient Computation in Polynomial Rings” has been carried out and

completed by Junaid Alam Khan Registration No. 67-GCU-PHD-SMS-06 under my

supervision.

___________________ __________________ Dated Supervisor Gerhard Pfister

Submitted Through Prof. Dr. A. D. Raza Choudary __________________

Director General Controller of Examination

Abdus Salam School of Mathematical Sciences GC University, Lahore GC University, Lahore, Pakistan Pakistan

I dedicate all of my efforts and success to my ever-loving

parents. To whom never shall I return their love, nor shall

fulfill their sacrifices.

iv

Table of Contents

Table of Contents v

Abstract vii

Acknowledgements viii

Introduction 1

1 Preliminaries 5

1.1 Monomial Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Standard Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Sagbi Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4 Grobner Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.5 Grobner basis under Composition . . . . . . . . . . . . . . . . . . . . 16

2 Subalgebra Analogue to Standard basis for Ideal 20

2.1 Sasbi basis of F [[Γ]] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2 Sasbi Basis in the Localization of F [x1, . . . , xn] . . . . . . . . . . . . . 24

2.3 Implementation in SINGULAR . . . . . . . . . . . . . . . . . . . . . 34

3 Converting Subalgebra Bases with the Sagbi Walk 41

3.1 Sagbi cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2 Sagbi Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.2.1 Crossing Cones . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.2.2 Converting Sagbi Basis . . . . . . . . . . . . . . . . . . . . . . 49

3.2.3 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.3 Implemetation in SINGULAR . . . . . . . . . . . . . . . . . . . . . . 58

v

4 Sagbi basis under Composition 60

4.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3 Implementation in SINGULAR . . . . . . . . . . . . . . . . . . . . . 69

Bibliography 76

vi

Abstract

In Chapter 1, there are given some necessary definitions and results about monomial

orderings, Standard basis and Sagbi basis in polynomial ring over the field along with

a description on the Grobner walk algorithm and Grobner basis under composition.

In Chapter 2 we develop a theory of subalgebra basis analogous to Standard basis

for ideals in polynomial rings over a field. We call this basis Sasbi Basis, standing

for Subalgebra Analogue to Standard Basis for Ideals. Sasbi bases may be infinite. In

this chapter we consider subalgebras admitting a finite Sasbi basis and give algorithms

to compute them. Sasbi basis theory is given in my paper [22].

In Chapter 3, we present an algorithm which converts the Sagbi basis with respect

to one ordering to the Sagbi basis with respect to another ordering, under the as-

sumption that the subalgebra admits a finite Sagbi basis with respect to all monomial

orderings. We called it Sagbi walk algorithm. Sagbi Walk algorithm is given in my

paper [20].

Composition is an operation of replacing variables in a polynomial by other poly-

nomials. In Chapter 4, we study the behavior of Sagbi basis under composition. Some

related results are from my paper [21].

vii

Acknowledgements

To begin with the name of ALLAH the most benevolent and merciful, the creator of

the universe, who inculcated the consecration upon me to fulfill the requirements of

this dissertation. I invoke peace for Holy Prophet Hazrat MUHAMMAD (S.A.A.W)

who is forever a torch of guidance for humanity as whole.

I am pleased to acknowledge my deepest gratitude, heart-felt regards and obliga

tion to my learned supervisor Prof Dr. Gerhard Pfister whose guidance, creativity

and keen interest enabled me to complete this dissertation.

I would also like to acknowledge Dr. A. D. R. Choudary whose elegant personality,

devotion and invigorating encouragement will always remain a source of inspiration

for me.

I am indebted to all my honorable teachers whose teaching has brought me to this

stage of academic zenith, and for their special care and concern during my stay in

the school.

Words wane in expressing my veneration for my loving parents, I love the most,

who nurtured and guided me to become a good muslim, and inspired to learn. I owe

my heartiest gratitude for their assistance and never ending prayers for my success.

I would never have been able to stand today without their continuous support and

generous help.

I highly commend the co-operative behavior displayed by my brothers, sister, who

prayed and endeavored for my edification and betterment. Also many love for my

niece; Ayesha and Rumaisa.

I further categorically acknowledge my bosom friends Fahad, Hussain bhai, Saqib

and Karachi University group members Irfan, Irshad, Hasham, Arsalan and Afroze for

their constant encouragement, invaluable suggestions, cooperation and constructive

criticism that they extended to me during my studies.

I am obliged to thankful to all my class mates Abdul Mueed, Athar bhai, Bhatti,

viii

ix

Imran, Irshaad, Kamran bhai, Khurram, Mobeen, Malik, Peer jee, Sarwar, Shafique,

Shoaib, Shah jee, Waseem and my senior collogue Dr Saeed Akram for their concern,

continuous help and positive attitude. I also want to thanks the other members of

my research group Afshan, Nazeran and Ahsan for many interesting discussions.

I am also particularly grateful to Mr. Awais, Mr. Nauman and Mr. Shoukat for

their extremely thorough and professional work in smoothing out the completion of

this dissertation.

I am failing in my duties and it will be skimpy on my part not to acknowledge

the benevolence of the Higher Education Commission for providing me Indigenous

Fellowship and Government of Punjab for providing financial support, and providing

me an excellent opportunity to complete my Ph.D.

It is not possible for me to name all those who have contributed, directly or indi-

rectly, towards the completion of my work. I am grateful to all my well-wishers for

their sincere support. I apologize to anyone whose name I may have forgotten.

Lahore, Pakistan Junaid Alam Khan

August 2010

Introduction

In the ring F [x] of univariate polynomials over a field F , most symbolic algorithmic

methods are based on the Division algorithm for polynomials. An iterated application

of this algorithm leads to the “Euclidean algorithm”. Given two polynomial ξ, ∂ ∈F [x], this algorithm give the greatest common divisor (gcd) of ξ and ∂. In the language

of gcd of ideal theory, an iterated application of Euclidean algorithm to polynomials

ξ1, . . . , ξm, computes their gcd, d ∈ F [x]. It is well known that the ring of univariate

polynomials over a field F is a Principal Ideal Domain, and d is a generator of the

ideal J ⊂ F [x] where J is generated by ξ1, . . . , ξm. By means of this representation of

J as principal ideal, we may derive algorithms of basic operations on ideals of F [x].

Let F [x1 . . . , xn] be a polynomial ring in n variables over the field F . In whole

dissertation we denote F [x1 . . . , xn] by R. The polynomial ring R is no longer prin-

cipal. In this context, a Grobner Basis for an ideal J ⊆ R, generated by finite set,

is similar to the gcd in the univariate case. In [2], Buchberger gave an algorithm

(Buchberger’s algorithm) to compute Grobner bases of an ideal in R. Grobner was

the name of Buchberger’s PhD thesis advisor. This algorithm is a kind of generaliza-

tion of Euclidean Algorithm. To prove the correctness of his algorithm, Buchberger

used a criterion for detecting whether a set is a Grobner basis for the ideal it gener-

ates. For this criterion he defined the notion of s-polynomials (little ”s” stands for

syzygies). Buchberger’s method by solving algorithmically “The Ideal Membership

Problem”transformed Grobner bases theory into a fundamental tool in Computational

Ideal Theory.

After ideals, subalgebras are important objects in polynomial ring theory, and

they are also concerned with the ”Membership Problem”. Shannon and Sweedler in

[34], presented a method that reduces a given subalgebra membership problem to

an ideal membership problem, and give a solution using Grobner basis techniques.

Nevertheless, many examples revealed that this method is computationally imprac-

tical. This urged the development of a theory leading to the construction of special

1

2

subalgebra bases analogous to Grobner bases for ideals. In order to solve ” Subal-

gebra membership problem”, algorithmically and intrinsically, Kapur and Madlener

in 1986 discovered a direct approach using rewritten techniques to give answer to

this problem in R and proposed in [19] a completion procedure to compute what

they called ” Canonical bases”. Independently Robbiano and Sweedler ([33]), defined

the same objects, which they called Sagbi bases, standing for Subalgebra Analog to

Grobner Bases for Ideals. They showed that it is possible to convey many proper-

ties of Grobner basis into a vocabulary of Sagbi bases. In both works, a Sagbi basis

construction algorithm has theoretically the same architecture and requires the same

ingredients as Buchberger’s one. Like Buchberger’s algorithm , Sagbi basis construc-

tion algorithms have two main ingredients. First is a reduction process and second is

a criterion for a set to be a Sagbi basis for the subalgebra it generates. For the second

purpose we have the notion of S-polynomials (capital ”S” standing for Subalgebra).

In the first chapter we review some basic notions and results related with monomial

orderings, Standard basis, Sagbi basis, Grobner Walk algorithm and Grobner basis

under composition.

Let F [[x1, . . . , xn]], the formal power series ring in n variables over the field F

be denoted by R. The localization of the polynomial ring, F [x1, . . . , xn]⟨x1,...,xn⟩ is

denoted by Rm where m = ⟨x1, . . . , xn⟩ is the maximal ideal of R. Let Γ ⊂ M\0where M is the maximal ideal of R. In extending Buchberger’s theory of Grobner

bases of polynomial ideals, Grobner bases of ideals of type ⟨Γ⟩ in R have been dis-

cussed in [16],[24]. They are called Standard bases 1. Let Λ be a finite subset of Rand consider the ideal J = ⟨Λ⟩R. Suppose we want to compute a Standard basis of J .

There are at least three possibilities. Using Buchberger’s algorithm for well orderings

we can compute it up to a given degree. There exists a theory of Standard bases in

Rm which is induced by a local degree ordering ( [10]) and we can compute a Stan-

dard basis of ⟨Λ⟩Rm using Mora’s tangent cone algorithm ( [10], [28]). It can also be

computed via homogenization (Lazard’s algorithm [25]). In chapter 2 we present sub-

algebra bases analogous to Standard basis for ideals in R and Rm. We call these bases

“SASBI Bases” standing for “Subalgebra Analogue to Standard Bases for Ideals”.

In [17], analogously to the theory of Standard bases of ideals in formal power ring R,

the theory of Sasbi basis of subalgebras of type F [[Γ]] is developed, which we review in

first section of chapter 2. In section 2.2 in analogy to the Standard basis for ideals in

localization of polynomial ring Rm we develop the theory of Sasbi basis in a suitable

1Standard basis is the notion of Hironaka ([16]).

3

localization of F [Λ], which was developed by Khan ( [22]). They can be computed

up to a certain degree similar to the Standard basis case (Algorithm 2.1.4). They can

be computed via a Sagbi bases if the homogenized algebra has a finite Sagbi basis

(Theorem 2.2.2). Using homogenization we get a finiteness condition: if the Sagbi

basis of homogenized algebra is finite then the Sasbi basis of subalgebra is finite. If

finiteness condition holds then they can also be computed directly using a generaliza-

tion of the tangent cone algorithm. For this first we need a reduction process, which

leads us to the notion of a weak Sasbi normal form (Definition 2.2.2), which can be

computed by Algorithm 2.2.4. Secondly we need a criterion to decide whether a given

set is a Sasbi basis for the subalgebra it generates. Such a criterion is presented in

Theorem 2.2.6. On the basis of this criterion we develop Sasbi basis construction

algorithm(Algorithm 2.2.7). This algorithm computes a Sasbi basis for subalgebras

admitting a finite Sasbi bases. In section 2.3 we deal with the implementation of a

Sasbi basis algorithm in SINGULAR([12]).

In several cases for the fast computation of Grobner bases, the Grobner walk ([5])

algorithm is used, which converts the Grobner basis w.r.t one monomial ordering

to the Grobner basis w.r.t another monomial ordering along a path in the Grobner

Fan ([29]) of a given ideal. In chapter 3 we study a procedure which is the Subalge-

bra analogue of the Grobner walk which is developed by Khan ([20]). We called it

Sagbi walk algorithm. This algorithm converts the Sagbi basis w.r.t one monomial

ordering to the Sagbi basis w.r.t another monomial ordering under the assumption

that the subalgebra admits a finite Sagbi basis w.r.t all monomial orderings. The

Sagbi Walk algorithm follows a path in Sagbi fan (Definition 3.1.2) which is the ana-

logue of the Grobner fan. In section 3.1 we have established the theory of Sagbi fan.

More precisely a Sagbi fan is the collection of all Sagbi Cones (Definition 3.1.2) of

subalgebras with faces. This collection is finite due to Theorem 3.1.3 which tells that

the number of Sagbi bases of a subalgebra w.r.t to all monomial orderings is finite.

Section 3.2 is devoted for the construction of the Sagbi Walk algorithm (Algorithm

3.2.5). The correctness of this algorithm is based on Theorem 3.2.3, which we have

proved using Lemma 3.2.4. Also we have shown that this algorithm terminates. This

algorithm have been implemented by Khan ([23]) in SINGULAR. In section 3.3 we

give some examples in which the Sagbi Walk algorithm works more efficient than the

usual Sagbi basis construction algorithm.

The behavior of Grobner bases under composition of polynomials is discussed in

[17],[18]. In chapter 4 we discuss the behavior of Sagbi basis under composition . Let

4

Γ ⊂ R and >, >′be two different monomial ordering in R. Let Λ be a Grobner

basis ( w.r.t > ) of the ideal generated by Γ and Φ be a list of n polynomials. The

composition by Φ commutes with the Grobner basis computation if the composed set

Λ Φ is also a Grobner basis ( w.r.t >′) of Γ Φ. When composition of polynomials

commutes with the Grobner basis computation? If > and >′are the same monomial

orderings, the complete answer is discussed in [18]. There is a sequel of this paper

also by Hong (c.f. [17]), which is devoted to the case when Γ Φ is a Grobner basis

with respect to >′( possibly different from >). He shows that this happens if the list

of the leading monomials of Φ is a permuted powering (Definition 1.5.6). Now we ask

the same question for the Sagbi basis case. The case of the same monomial ordering

is treated in [30](Theorem 4.1.2). Let S be a Sagbi basis ( w.r.t > ) of the subalgebra

generated by Γ. In [20] Khan shows that S Φ is a Sagbi basis of Γ Φ w.r.t some

monomial ordering >′( possibly different from > ) if the list of the leading monomials

of Φ is a permuted powering (Theorem 4.1.4). In section 4.2 we give some examples

which show that if we want to compute a Sagbi basis of Γ Φ when the list of the

leading monomials of Φ is a permuted powering then it would be more efficient first

to compute the Sagbi basis of Γ then compose it with Φ to obtain the Sagbi basis.

Chapter 1

Preliminaries

In this chapter we recall basic notions and some results that will be used later. Let

F [x1, . . . , xn] be a polynomial ring over the field F in n variables. Throughout the

dissertation we denote F [x1, . . . , xn] by R.

1.1 Monomial Ordering

Definition 1.1.1. Let Monn=xa= xa11 . . . xan

n | a = (a1, . . . , an) ∈ Nn be the set of

all monomials in R. Let > be a relation on Monn, > is called monomial ordering if

it is total ordering relation and its satisfies

xa > xb =⇒ xcxa > xcxb

for all a, b, c ∈ Nn. We call > a monomial ordering on R if > is a monomial ordering

on Monn.

Definition 1.1.2. Let > be a fixed monomial ordering. For ξ ∈ R, ξ = 0, we can

write as

ξ = λaxa + λbx

b + . . .+ λcxc, xa > xb > . . . > xc,

and λa, λb, . . . , λc ∈ F\0.

5

1. The leading monomial of ξ w.r.t > is xa, we denote it by LM>(ξ).

2. The leading exponent of ξ w.r.t > is a, we denote it by LE>(ξ).

3. The leading term of ξ w.r.t > is λaxa, we denote it by LT>(ξ)

4. The leading coefficient of ξ w.r.t > is λa, we denote it by LC>(ξ).

5. tail(ξ) := ξ − LT>(ξ),

6. support(ξ) := xa, xb, . . . , xc, the set of all monomials of ξ with non-zero coef-

ficients.

7. ecart(ξ) = deg(ξ)− deg(LM>(ξ))

Definition 1.1.3. Let > be a monomial ordering on Monn.

1. If ∀ a = (0, . . . , 0), xa > 1 then > is called global ordering.

2. If ∀ a = (0, . . . , 0), xa < 1 then > is called local ordering.

3. If > is neither global nor local then > is called mixed ordering.

Definition 1.1.4. The monomial order > is well ordering if every non-empty subset

of M of Monn has smallest element w.r.t >.

Lemma 1.1.1. ([10], Lemma 1.25) Let > be a monomial ordering, then > is global

if and only if > is well ordering.

Definition 1.1.5. For any monomial ordering > on Monn,

S> := u ∈ R\0 |LM>(u) = 1

is a multiplicatively closed set. The the ring associated to R and > is define as

R> := S−1> R = ξ

∂| ξ, ∂ ∈ R and ∂ ∈ S>

6

This ring is the localization of R w.r.t S>. The ring R> is Noetherian because it is

the localization of a Noetherian ring R.

Lemma 1.1.2. ([10], lemma 1.5.2) Let > be a monomial ordering on R. Then

1. R ⊂ R> ⊂ Rm.

2. R> = R if and only if > is a global ordering.

3. R> = Rm if and only if > is a local ordering.

Definition 1.1.6. Let > be a monomial ordering on R.

Let ξ be an element of R>. Choose u ∈ R such that LM>(u) = 1 and uξ ∈ R. Then

LM>(ξ) := LM>(uξ),

LC>(ξ) := LC>(uξ),

LT>(ξ) := LT>(uξ),

LE>(ξ) := LE>(uξ).

Definition 1.1.7. Let v = (v1, . . . , vn) ∈ Rn and xa ∈ Monn. The v-degree of xa is

define by

degv(xa) = v.a = v1a1 + · · ·+ vnan

Let ξ ∈ R such that ξ =∑

a λaxa. The v-degree of ξ is define as

degv(ξ) = max degv(xa) |λa = 0 .

Let degv(ξ) = d. The initial form of ξ w.r.t v denoted by Inv(ξ) is defined as

Inv(ξ) =∑

degv(xa)=d

λaxa

If ξ = Inv(ξ) then ξ is called v-homogenous.

7

Definition 1.1.8. Let A ∈ GL(n,R) and >lex denote the lexicographical ordering

on Rn. The matrix A defines a monomial ordering >A by setting

xa >A xb :⇐⇒ Aa >lex Ab,

Lemma 1.1.3. (cf. [10], page 18 ) Let > be a monomial ordering. There exist

A ∈ GL(n,R) such that > and >A are the same monomial ordering.

Now we discuss characterization of monomial orderings which was first discussed

in [32].

Definition 1.1.9. Let v is a non-zero integer vector. The monomial ordering > is

called v-weighted degree ordering if for all monomials p, q ∈ Monn

degv(p) > degv(q) ⇒ p > q

Lemma 1.1.4. (c.f. [10], page 15 )Let > be a monomial ordering on R and M is

finite subset Monn. Then ∃ v ∈ Nn such that p > q if and only if degv(p) > degv(q)

∀ p, q ∈ M .

1.2 Standard Bases

Let Γ ⊂ R and > be any monomial ordering. The LM>(Γ), the set of leading

monomials of Γ is defined : LM>(Γ) := LM>(ξ) | ξ ∈ Γ\0 .

Definition 1.2.1. For each Γ ⊂ R>

L>(Γ) = ⟨LM>(Γ)⟩R

is called the leading ideal of Γ.

8

Definition 1.2.2. Let J be an ideal in R>. A set Γ ⊂ R> is called a Standard basis

of J w.r.t > if Γ ⊂ J , and L>(J) = L>(Γ).

If > is global monomial ordering then Standard basis is also called a Grobner basis.

Definition 1.2.3. Let Γ ⊂ R>, Γ is called reduced if

1) LC>(∂) = 1 ∀∂ ∈ Γ and 0 /∈ Γ.

2) LM(∂) - LM(ξ) ∀∂, ξ ∈ Γ.

3) ∀∂, no monomial in the power series expansion of tail(∂) is contained in L>(Γ).

Remark 1.2.1. Reduced Grobner bases can always be computed and they are unique

(c.f. [10]). In general reduced Standard bases do not exist.

Definition 1.2.4. Let Γ be a finite subset of R>and ∂ ∈ R>. We say that a

polynomial ζ is a normal form of ∂ w.r.t Γ, and we write ζ = NF (∂ |Γ), if

1. ζ = NF (0 | Γ) = 0.

2. ζ = 0 ⇒ LM>(ζ) /∈ L>(Γ).

3. If Γ = ∂1, . . . , ∂s, then ∃ u ∈ S> such that g := u∂ −NF (∂ |Γ)

has a Standard representation w.r.t Γ, i.e

g =s∑

i=1

fi · ∂i

for fi ∈ R> and LM>(g) = maxsi=1LM>(fi)LM>(∂i).

Definition 1.2.5. Given non zero polynomials ξ, ∂ ∈ R.

lcm(LM>(ξ), LM>(∂)) stands for the least common multiple of LM>(ξ) and LM>(∂).

The polynomial

spoly(ξ, ∂) = lcm(LM>(ξ),LM>(∂))LC>(ξ)LM>(ξ)

ξ − lcm(LM>(ξ),LM>(∂))LC>(∂)LM>(∂)

∂

is called the s-polynomial of ξ and ∂.

Now we give an algorithm to compute normal form. This algorithm is called

Mora’s Normal form algorithm.

9

Algorithm 1.2.2. ([10]).

Let > be a monomial ordering.

Input: ξ ∈ R and Γ = ∂1, . . . , ∂s ⊂ R with ∂i = 0 ∀ i = 1, . . . , s.

Output: ζ ∈ R a normal form of ξ w.r.t Γ.

• ζ := ξ;

• ∆ := Γ;

• while(ζ = 0 and ∆ζ := ∂ ∈ ∆ | LM>(∂) | LM>(ζ) = ∅)

choose ∂ ∈ ∆ζ such that ecart(∂) is minimal;

if (ecart(∂) > ecart(ζ))

∆ := ∆ ∪ ζ;

ζ := spoly(ζ, ∂);

• return ζ;

In algorithm 1.2.2, if > is a global monomial ordering then ecart condition is

always not satisfied. In that case we avoid ecart condition in algorithm 1.2.2.

Now we give a criterion for a system of generators of the ideal J in R> to be a

Standard basis. This criterion is called Buchberger’s criterion.

Theorem 1.2.3. ([10], Theorem 1.7.3) Let Γ = ∂1, . . . , ∂s ⊂ R> and J be the

ideal generated by Γ in R>. Then Γ is a Standard basis of J if and only if ∀i = j,

NF (spoly(∂i, ∂j),Γ) = 0.

1.3 Sagbi Basis

In this section we present ”subalgebras basis” theory for global ordering case . These

bases are called Sagbi bases where SAGBI stands for Subalgebra Analogue to Grobner

Basis for Ideal. For this section we consider only global monomial orderings i.e well

orderings. If Γ is a subset of R, then the subalgebra generated by Γ is usually

10

denoted by F [Γ]. This notion is natural since the elements of F [Γ] are precisely the

polynomials in the set of formal variables Γ, viewed as elements of F [Γ]. If Γ is finite

then F [Γ] is finitely generated algebra.

Definition 1.3.1. A Γ-monomial is a finite power product of the form Γa = ∂a11 . . . ∂am

m

where ∂i ∈ Γ for i = 1, . . . ,m, and a = (a1, . . . , am) ∈ Nm. The set MonΓ = Γa | a ∈

Nm, m ∈ N is the set of all Γ-monomials.

Let > be a monomial orderings in R.

Definition 1.3.2. We define in>(F [Γ]) to be the subalgebra generated by LM>(Γ).

in>(F [Γ]) := F [LM>(Γ)]

Definition 1.3.3. Let F [Γ] be a subalgebra of R. A subset S ⊂ F [Γ] is called a

Sagbi basis of F [Γ] w.r.t > if

in>(F [Γ]) = F [LM>(S)]

Sagbi basis S generates F [Γ], i.e. F [Γ] = F [S]. If S is a Sagbi basis it means that

S is a Sagbi basis of F [S].

Definition 1.3.4.

(1) A subset Γ ofR is s-interreduced if 0 /∈ Γ and for any polynomial ξ ∈ Γ, LM>(ξ) /∈

F [LM>(Γ\ξ)].

(2) ξ ∈ R is called s-reduced w.r.t Γ if no monomial of ξ is contained in F [LM>(Γ)].

(3) A subset Γ of R is s-reduced if LC>(ξ) = 1 ∀ξ ∈ Γ and Γ is s-interreduced, and

for all ξ ∈ Γ, tail(ξ) is s-reduced w.r.t Γ.

Theorem 1.3.1. A s-reduced Sagbi basis exists and is uniquely determined.

Proof of theorem 1.3.1 is similar to the case of Grobner basis.

At this point we define the analogue of normal form in subalgebras theory.

11

Definition 1.3.5. Let ∂ be a polynomial and Γ be a finite subset in R. A a polyno-

mial ζ is a Sagbi normal form of ∂ w.r.t Γ, and we write ζ = SNF (∂ |Γ), if

0. ζ = SNF (0 | Γ) = 0

1. ζ = 0 ⇒ LM>(h) /∈ F [LM>(Γ)]

2. ∂ − ζ has a representation w.r.t Γ, that is either ∂ − ζ = 0 or ∂ − ζ =∑v

i=1 ciΓai

where ci ∈ F and LM>(∂) = maxvi=1LM>(ciΓai). This representation is called a

Sagbi representation.

We give an algorithm to compute Sagbi normal form.

Algorithm 1.3.2. ([33])

Input: Let Γ = ∂1, . . . , ∂s and ξ are subset and polynomial in R.

Output: ζ ∈ R a polynomial Sagbi normal form of ξ with respect to Γ.

• ζ := ξ ;

• while (ζ = 0 and Γζ = cΓa, Γ-monomial, c ∈ F |LT>(cΓa) = LT>(ζ) = ∅)

choose cΓa ∈ Γζ;

ζ = ζ − cΓa;

• return ζ;

We have seen that the s-polynomial is a key ingredient of Buchberger’s criterion.

We need a counter part to s-polynomial in Sagbi basis theory. For convenience we

call them S-polynomials. Once we define the S-polynomial then we formulate Sagbi

basis criterion.

Definition 1.3.6. Let Γ = ∂1, . . . , ∂m ⊂ R. Let

AR(Γ) := h ∈ F [y1, . . . , ym] |h(LM>(∂1), . . . , LM>(∂m)) = 0 ⊂ F [y1, . . . , ym].

We called AR(Γ) the ideal of algebraic relations between leading monomials of ∂′is. If

S is a finite generating set of AR(Γ), then each polynomial P (y1, . . . , ym) ∈ S gives

12

rise by substituting the y′is by the ∂′is, to a polynomial P (Γ) = P (∂1, . . . , ∂m). P (Γ)

is called S-polynomial.

The next theorem gives a criterion for a set to be a Sagbi basis of F [Γ].

Theorem 1.3.3. (c.f. [33]) Given Γ ⊂ R, Γ is a Sagbi basis of F [Γ] if and only if

every S-polynomial has a vanishing Sagbi normal form.

On the basis of Theorem 1.3.3 we build an algorithm that computes a Sagbi basis

for finitely generated subalgebras. This algorithm is known as Sagbi basis construc-

tion algorithm. As Sagbi basis may be infinite, this algorithm doesn’t necessarily

terminate. This algorithm terminates if and only if the considered subalgebra admits

a finite Sagbi basis.

Algorithm 1.3.4. ([33]) Let > be a monomial ordering.

Input: A finite subset Γ ⊂ F [x1, ...., xn].

Output: A Sagbi basis S for F [Γ].

• S = Γ;

• oldS = ∅;

• while (S = oldS)

Compute a generating set S for AR(S);

P = S(S);

Red = SNF (p | S) | p ∈ P \ 0;

oldS = S;

S = S ∪Red;

• return S;

Example 1.3.5. Let > be the degree reverse lexicographical ordering on Q[x, y], and

Γ = x2, y2, xy + x be a subset of Q[x, y]. In start we have

13

S = x2, y2, xy + x

oldS = ∅.

In the first turn the while loop gives

P = y4 + 2xy2

Red = xy2

oldS = x2, y2, xy + x.

In the next turn

P = 2xy2 + y2, 2xy4 + y4, 4x3y4 + 6x2y4 + x4y4 + y

Red = ∅

oldS = x2, y2, xy + x, xy2

S = x2, y2, xy + x, xy2.

Now S = oldS, the algorithm stop and S = x2, y2, xy + x, xy2

is a Sagbi basis.

1.4 Grobner Walk

The Grobner walk ([5, 35]) algorithm is used to convert a Grobner basis w.r.t one

monomial ordering to Grobner basis w.r.t another monomial ordering along a path in

the Grobner Fan ([29]). In this section we consider only global monomial orderings.

We know that Grobner bases of the same ideal for different monomial orderings

may be different. Is the collection of all Grobner basis for a fixed ideal finite? The

following theorem gives the answer.

Theorem 1.4.1. ([7], chapter 8) For an ideal J in polynomial ring R, the set

Lead(J) = L>(J) | > a monomial ordering is finite.

Definition 1.4.1. Let Γ ⊂ R and v ∈ Rn be a fixed vector.

14

We denote Inv(Γ) as

Inv(Γ) := Inv(ξ) | ξ ∈ Γ

We define Lv(Γ) to be the ideal generated by Inv(Γ).

Lv(Γ) := ⟨Inv(Γ)⟩

Lv(Γ) not need to be a monomial ideal. For generic v, Lv(Γ) is a monomial ideal.

Theorem 1.4.2. ([15], Theorem 3.12) For Γ ⊂ R and monomial ordering > there

exist a weight vector v such that

L>(Γ) = Lv(Γ)

Definition 1.4.2. Let Γ = ∂1, . . . , ∂s be a Grobner basis of ideal J w.r.t a mono-

mial ordering > such that LM>(∂i) = xa(i) and we write

∂i = xa(i) +∑b

ci,bxb ci,b ∈ F,

where xa(i) > xb whenever ci,b = 0. We define the ΩΓ,> as

ΩΓ,> = v ∈ (Rn)+ : a(i).v ≥ b.v whenever ci,b = 0

= v ∈ (Rn)+ : (a(i)− b).v ≥ 0 whenever ci,b = 0

ΩΓ,> is called the Grobner Cone 1. It is a closed convex polyhedral cone2.

Definition 1.4.3. The collection of all the Grobner cones ΩΓ,> and faces 3 as Γ ranges

over all Grobner bases with respect to > of J is called Grobner Fan. This collection

is finite according to Theorem 1.4.2.

1It is also defined as the topological closure in Qn of v ∈ (Qn)+ | ⟨LT>(J)⟩ = Lv(J) ([5]).2a convex polyhedral cone with vertex at the origin is the intersection of finitely many half-spaces

containing the origin.3Let T = 0 is a non-trivial linear equation. A face of a cone η is η ∩ T = 0, such that T ≥ 0

on η.

15

For more details of Grobner fan see [7], chapter 8, section 4.

Grobner walk algorithm : Here we overview the idea of the Grobner walk algo-

rithm. Let Γ1 and Γ2 be the reduced Grobner basis of J under the monomial orderings

>1 and >2 respectively. The Grobner walk algorithm computes Γ2, given >1 and >2

and Γ1. Suppose >1 and >2 are obtain by matrices M1 and M2. The first row of

M1 and M2 is denoted by w and v respectively. In the Grobner walk algorithm we

traverse on the line segment between w and v, (1−u)w+uv for u ∈ [0, 1]. When we

cross from Grobner cone Ω to new Grobner cone Ω′we convert the reduced Grobner

basis corresponding to Ω into the reduced Grobner basis corrresponding to Ω′. As

the number of Grobner cones is finite, after a finite number of steps we reach the

Ω>2,Γ2(J) and obtain the reduced Grobner basis with respect to >2.

1.5 Grobner basis under Composition

First we overview several results about the behavior of Grobner basis under com-

position. In this section we consider only global monomial orderings. We fix some

notations which also will be used in chapter 4.

• ξ, ∂ two non-zero polynomials and s, t two monomials in R.

• Φ = (Φ1, . . . ,Φn) where Φi are non-zero polynomials in R.

• LM>(Φ) = (LM>(Φ1), . . . , LM>(Φn)).

• Mat(LM>(Φ)), the exponent matrix of LM>(Φ), that is, the n by n matrix

whose (i, j)-th entry is given by degxi(LM>(Φj)).

Now define the process of the composition of polynomials.

16

Definition 1.5.1. Let ξ ∈ R. We define the composition of ξ by Φ, written ξ Φ,

as the polynomial obtained from ξ by replacing each xi by Φi. We also define, for a

subset Γ ⊂ R, Γ Φ = ∂ Φ | ∂ ∈ Γ.

Proposition 1.5.1. (c.f. [17])

a) (ξ + ∂) Φ=ξ Φ + ∂ Φ.

b) (ξ∂) Φ=(ξ Φ)(∂ Φ).

c) LM>(s Φ)=s LM>(Φ), ∀ monomials s.

Let > and >′are monomial orderings.

Definition 1.5.2. Let Λ be a Grobner basis w.r.t > of the ideal ⟨Γ⟩. We say that

composition by Φ commutes with the Grobner basis computation if the composed set

Λ Φ is also a Grobner basis of ⟨Γ Φ⟩ w.r.t >′.

The question is when the composition commutes with Grobner basis computation?

When > and >′are same monomial ordering the complete answer is given in [17].

Now we introduce some notations

Definition 1.5.3. Composition by Φ is compatible with > if all monomials s and t

satisfy the following condition

s > t ⇒ s LM>(Φ) > t LM>(Φ)

Definition 1.5.4. Composition by Φ is compatible with nondivisibility if all mono-

mials s and t satisfy the following condition

s - t ⇒ s LM>(Φ) - t LM>(Φ)

The following theorem gives answer to our question.

Theorem 1.5.2. ([18]) Composition by Φ commutes with Grobner basis computation

if and only if composition by Φ is compatible with nondivisibility and compatible with

>.

17

Now we investigate the answer of our question when >′is possibly different from

>. It is discussed in [17]. First we introduce some notations.

Definition 1.5.5. The composition of > by Φ,( denoted by > Φ ) is the binary

relation over the Monn if ∀ monomials s, t

s > Φ t ⇐⇒ s LM>(Φ) > t LM>(Φ)

The relation > Φ is not necessary a monomial ordering. For a counter example

see [17]. However under some condition on Φ it becomes a monomial ordering.

Definition 1.5.6. The list LM>(Φ) is called a permuted powering if

LM>(Φ) = (xλ1

π(1), . . . , xλn

π(n))

for some permutation of π of (1, . . . , n) and some λ1, . . . , λn > 0.

Lemma 1.5.3. ( Lemma 7, [17]) Let LM>(Φ) be a permuted powering. Then

(i) The binary relation > Φ is a monomial ordering. In this case we denote it

by >Φ.

(ii) LM>(ξ Φ) = LM>Φ(ξ) LM>(Φ).

Next theorem describes the condition on composition when it commutes with

Grobner basis computation possibly under different monomial orderings.

Theorem 1.5.4. (Theorem 2, [17]) If the list LM>(Φ) is a permuted powering and

Λ is a Grobner basis of ⟨Γ⟩ with respect to >Φ then ΛΦ is a Grobner basis of ⟨ΓΦ⟩

w.r.t >.

Example 1.5.5. Consider the ring Q[x, y, z] and > is degree reverse lexicographical

ordering. Let

18

Γ = xyz − 1, xy + xz + yz, x+ y + z

Φ = ((x2 + y + z)3, (x+ y2 + z)3, (x+ y + z2)3)

We obviously see that LM>(Φ) is a permuted powering, thus the Theorem 1.5.4 ap-

plies. Note that Mat(LM>(Φ)) is a diagonal matrix and all diagonal entries are

same so > and >Φ induce the same monomial ordering. Therefore first we compute

a Grobner basis Λ of Γ with respect to > and obtain that

Λ = yz + y2 + z2, x+ y + z, z3 − 1

We compose Λ by Φ obtaining Grobner basis of Γ Φ w.r.t >.

19

Chapter 2

Subalgebra Analogue to Standardbasis for Ideal

Let F [[x1, . . . , xn]] be the formal power series ring over the field F which we denote

by R. Let Γ ⊂ M\0 where M is the maximal ideal of R. We define

F [[Γ]] = Q(∂1, . . . , ∂s) |Q ∈ F [[y1, . . . , ys]] and ∂1, . . . , ∂s ∈ Γ for some s.

2.1 Sasbi basis of F [[Γ]]

We fix a local degree ordering > and use the notation of Definition 1.1.2 which make

sense in R too. In this section we recall some result of [17] and give an algorithm

which computes Sasbi basis of F [[Γ]] up to a certain certain degree.

Definition 2.1.1. Given two elements ∂, ζ ∈ R, we will say that ∂ reduces to ζ with

respect to Γ if ∃ Γ-monomial Γa and γ ∈ F such that

ζ = ∂ − γΓa, with ζ = 0 or LM>(ζ) < LM>(∂)

In this case we will write

∂Γ→ ζ,

20

and we have that ∂ − ζ ∈ F [[Γ]].

Consider a chain (possibly infinite) of reductions

∂Γ→ ζ1

Γ→ ζ2Γ→ . . .

Γ→ ζmΓ→ . . .

This implies there exist Γ-monomials Γa(i) and λa(i) ∈ F\0 such that

ζm = ∂ −m∑i=1

λa(i)Γa(i) ,

and because of the definition of the reduction

LM>(λa(1)Γa(1)) > LM>(λa(2)Γ

a(2)) > . . .

If the chain is infinite, then we have following sequence in R :

sm =m∑i=1

λa(i)Γa(i) , m ≥ 1.

This sequence happens to be convergent in R w.r.t the M-adic topology. Let s

denote the limit of the sequence (sm)m≥1. Since all the terms are in the complete

subalgebra F [[Γ]] so we have that s ∈ F [[Γ]].

Definition 2.1.2. ζ ∈ R is called a Sasbi normal form of ∂ up to degree d with respect

to Γ if there exist ∂1, . . . , ∂s ∈ Γ and H ∈ F [Y1, . . . , Ys] such that ∂ ≡ H(∂1, . . . , ∂s)+

ζ modMd+1 and ∀xβ ∈ support(ζ) we have xβ /∈ F [LM>(∂1), . . . , LM>(∂s)]. We

denote it by S-NF (∂ |Γ, d).

For computational reason we give an algorithm which computes a normal form up

to the degree d.

Algorithm 2.1.1. Let > be any local degree ordering in R.

Input: Γ ⊂ M\0 , ∂ ∈ R, d ∈ Z.

21

Output: ζ = S-NF (∂ |Γ, d) (a normal form of ∂ w.r.t Γ up to degree d)1.

• ζ := ∂;

• while(ζ = 0 and ord(ζ) ≤ d)

∆ζ = Γa |LM>(Γa) = LM>(ζ) = ∅;

if (∆ζ = ∅)

choose Γa ∈ ∆ζ;

ζ = ζ − LC>(ζ)LC>(Γa)

Γa ;

else

return (LT>(ζ) + S-NF (ζ − LT>(ζ) |Γ, d);

• end(while)

• return ζ;

Definition 2.1.3. A set Γ ⊂ M\0 is Sasbi basis2 of F [[Γ]] if

F [LM>(F [Γ])] = F [LM>(Γ)]

where LM>(F [Γ]) = LM>(∂) | ∂ ∈ F [[Γ]]\0 , i.e Γ is a Sasbi basis if ∀ ξ ∈

F [[Γ]]\0,

LM>(ξ) = LM>(Γa)

for some Γ-monomial Γa.

Example 2.1.2. The set Γ = x2,∑∞

i=3 xi ⊂ R is a Sasbi basis for F [[Γ]]. Indeed,

if ∂ ∈ F [[Γ]]\0, then LM>(∂) = 1 or LM>(∂) = xa, for some a ≥ 2. Hence

LM>(∂) ∈ F [x2, x3] = F [LM>(Γ)].

1For theoretical reasons we allow Γ to be infinite and d = ∞. We have seen that for d → ∞ thenormal form S-NF (∂ |Γ, d) converges in the M-adic topology. We call this limit S-NF (∂ |Γ), thenormal form of ∂ w.r.t Γ induced by S-NF (∂ |Γ, d).

2In [17] this is called a Standard basis of subalgebras. We use this notation to be similar to Sagbibases introduced in [33].

22

Now for the characterization of the Sasbi bases in R similar to those in R we need

to define an analogue of the S-polynomial.

Definition 2.1.4. Let Γ = ∂1, . . . , ∂s ⊂ M\0. An S-polynomial3 is an element

of the form

λ1 Γa − λ2 Γ

b

where λ1, λ2 ∈ F\0 and Γa and Γb are Γ-monomials, such that LT>(λ1 Γa) =

LT>(λ2 Γb).

Next theorem gives criteria for a set to be a Sasbi basis of F [[Γ]].

Theorem 2.1.3. (cf. [17], page 50) Given Γ = ∂1, . . . , ∂s ⊂ M\0, Γ is a Sasbi

basis of F [[Γ]] if and only if every S-polynomial of Γ has a vanishing Sasbi normal

form w.r.t Γ.

Now we give an algorithm to compute Sasbi basis in R.

Algorithm 2.1.4.

Input: A finite subset Γ of M\0.

Output: A Sasbi basis S for F [[Γ]].

• S = Γ;

• oldS = ∅ ;

• while (S = oldS)

S := s | s is an S-polynomial of S;

R := r | r = S-NF (s ∈ S |S) and r = 0;3In case of standard bases the S-polynomials of ∂1, . . . , ∂s come form a special set of generators of

the syzygies of LM>(∂1), . . . , LM>(∂s). In case of algebras one needs instead of the syzygies the alge-braic relations. A special set of generator for the algebraic relations between LM>(∂1), . . . , LM>(∂s)are of type λ1Y

a − λ2Yb such that LT>(λ1 Γ

a) = LT>(λ2 Γb). Therefore we also call the induced

polynomials λ1Γa − λ2Γ

b S-polynomials.

23

oldS = S;

S = S ∪R;

• return S;

2.2 Sasbi Basis in the Localization of F [x1, . . . , xn]

Now we will introduce Sasbi bases in F [Γ]>. Let Γ = ∂1, . . . , ∂s be a subset R and

> be a local ordering. We define

F [Γ]> = (S> ∩ F [∂1, . . . , ∂s])−1F [∂1, . . . , ∂s].

Definition 2.2.1. A subset S ⊂ F [Γ]> is called SASBI4 Basis of F [Γ]> if

F [LM>(F [Γ]>)] = F [LM>(S)]

i.e for all ∂ ∈ F [Γ]>\0

LM>(∂) = LM>(Sa)

for some S-monomial Sa.

If > is a global monomial ordering then Sasbi basis is also called a Sagbi basis.

If S is a Sasbi basis then it means that S is a Sasbi basis of F [S]>.

Next theorem shows that Sasbi basis of F [Γ]> is also Sasbi basis of F [[Γ]].

Theorem 2.2.1. Let R> ⊂ R be equipped with local degree ordering >. Let Γ =

∂1, . . . , ∂s be a subset of R. If Γ is a Sasbi basis of F [Γ]> then Γ is a Sasbi basis of

F [[Γ]].

4SASBI stands for “Subalgebra Analogue to Standard Basis For Ideal”.

24

Proof. For ∂ ∈ F [[Γ]] we have to prove that there exist Γ-monomial Γa such that

LM>(∂)=LM>(Γa). If ∂ ∈ F [[Γ]] then ∃ Θ ∈ F [[y1, . . . , ys]] such that we have

∂ = Θ(∂1, . . . , ∂s). There exist a decomposition of Θ = Θ(0)+Θ(1), Θ(0) ∈ F [y1, . . . , ys]

and Θ(1) ∈ F [[y1, . . . , ys]] such that

LM>(Θ(0)(∂1, . . . , ∂s)) = LM>(Θ(∂1, . . . , ∂s)) = LM>(∂).

Since Γ is Sasbi bases of F [Γ]> there exist a Γ-monomial Γa, such that LM>(Γa) =

LM>(Θ(0)(∂1, . . . , ∂s)). We get LM>(∂) = LM>(Γ

a) which shows that Γ is a Sasbi

basis for F [[Γ]].

Now we want to show how to reduce the Sasbi bases computation for local order-

ings using homogenization with respect to a variable “t” to the computation of Sagbi

bases for global orderings.

Theorem 2.2.2. Let Γ = ∂1, . . . , ∂s ⊂ R and F [Γ] = F [∂1, . . . , ∂s]>. Here >

is a local monomial ordering given by a matrix M . Consider F [t, x1, . . . , xn] with

monomial ordering >h defined by the matrix

1 1 · · · 1

0

... M

0

.

>h is a global ordering. We define ξi to be ξi := ∂hi ∈ F [t, x1, . . . , xn]. Let Ψ =

ξ1, . . . , ξs. Assume S = S1, . . . , Sk ⊂ F [t, x1, . . . , xn] is a Sagbi basis of F [Ψ]

with respect to >h. Let sj := Sj(t = 1) , 1 ≤ j ≤ k, then S = s1, . . . , sk is a Sasbi

basis of F [Γ]>.

Proof. We want to show S is a Sasbi basis for F [Γ]>. For this we have to show that

1. S ⊂ F [Γ]>.

25

2. For ∂ ∈ F [Γ]> there exist a = (a1, a2, . . . , ak) ∈ Nk such that LM>(∂) = LM>(Sa).

1) We know that S = S1, . . . , Sk is a Sagbi basis of F [Ψ] so Si =∑

γi,jΨai,j with

γi,j ∈ F . Putting t = 1 we get si =∑

γi,jΓai,j this implies si ∈ F [Γ]>.

2) For ∂ ∈ F [Γ]> ∃ u ∈ S>∩F [Γ] such that u∂ =∑

γjΓaj , then there exists ρ ∈ Z such

that tρuh∂h =∑

γi(Γaj)h =

∑γjΨ

aj . We have that S is Sagbi basis of F [Ψ] therefore

we can find a = (a1, a2, . . . , ak) ∈ Nk such that LM>(tρuh∂h) = LM>(S

a). Since

LM>(Ψ)|t=1 = LM>(Ψ|t=1), we have LM>(tρuh∂h)|t=1 = LM>(∂), since LM>(u) = 1

as u ∈ S> and LM>(Sa)t=1 = LM>(S

a), we obtain LM>(∂) = LM>(Sa).

Theorem 2.2.2 shows that Sasbi bases are computable in many cases but similar

to the theory of Standard bases w.r.t to local orderings for ideals this approach is not

very efficient. Therefore one would like to have an efficient way for computing Sasbi

bases. The basis for this is the concept of the Sasbi normal form.

Definition 2.2.2. Let ∂ and Γ be a polynomial and a finite subset in R respectively,

such that F [Γ]> admits a finite Sasbi bases and ∂ ∈ F [Γ].

A polynomial ζ is a Weak Sasbi normal form of ∂ w.r.t Γ, and we write polynomial

ζ = WS-NF (∂|Γ), if

0. ζ = WS-NF (0| Γ) = 0

1. ζ = 0 ⇒ LM>(ζ) /∈ F [LM>(Γ)]

2. There exist unit u ∈ S> ∩ F [Γ] such that u∂ − ζ has a representation w.r.t Γ,

that is either u∂ − ζ = 0 or u∂ − ζ =∑v

i=1 γiΓai where γi ∈ F and LM>(∂) =

maxvi=1LM>(γiΓai). This representation is called Sasbi representation.

Algorithm 2.2.3.

Input: ξ, Γ, > a local monomial ordering. We assume that Γ = ∂1, . . . , ∂s and ξ

are subset and polynomial in R such that ξ ∈ F [Γ]. We also assume there exist a

26

finite Sagbi basis of F [Ψ] where Ψ = Γh the homogenization of Γ w.r.t “t”, a new

variable.

Output: ζ ∈ R a polynomial weak Sasbi normal form of ξ w.r.t Γ.

• ζ := ξ;

• ∆ := Γ;

• while(ζ = 0 and ∆ζ = ∆a, ∆-monomial |LM>(∆a) = LM>(ζ) = ∅

choose ∆a ∈ ∆ζ such that ecart(∆a) is minimal;

if ecart(∆a) > ecart(ζ)

∆ := ∆ ∪ ζ;

ζ = ζ − LC>(ζ)LC>(∆a)

∆a

• return ζ;

Theorem 2.2.4. The algorithm terminates and computes a weak Sasbi normal form.

Proof. We can prove termination by homogenization: start with ζ := ξh and Ψ :=

Γh = ∂h| ∂ ∈ Γ.

At this step we have while loop as follows

• while(ζ = 0 and ∆ζ = Ψa, Ψ-monomial |LM>(Ψa) = tbLM>(ζ) for some b)

choose ∂ ∈ ∆ζ in a way with b ≥ 0 is minimal;

if b > 0

∆ = ∆ ∪ ζ;

ζ := ζ − LC>(ζ)LC>(Ψa)

Ψa;

ζ := (ζ|t=1)h;

Let ∆v denotes the set ∆ after the v-th turn of the while loop. By our assumption F [Ψ]

has a finite Sagbi bases therefore ∃ integer N > 0 such that F [LM>(∆v)] becomes

27

stable for v ≥ N . The next ζ satisfies LM>(ζ) ∈ F [LM>(∆N)] = F [LM>(Ψ)], when

LM>(ζ) = LM>(Ψa) for some Ψa ∈ F [Ψ] and b = 0. Therefore for v ≥ N the set

∆v itself becomes stable. With this fixed ∆, the algorithm continues and after finite

number of turns its terminates, since > is a well ordering on F [t, x].

Now we prove correctness. At the i-th turn in the while loop, we create a set

∆i = ∂1, ∂2, . . . , ∂s, ζ0, ζ1, . . . , ζi−2 such that ζi = ζi−1 − γi∆a(i) and LM>(∆a(i)) =

LM>(ζi−1) > LM>(ζi) where ∆a(i) is ∆i-monomial.

Suppose, by induction, that in the first i− 1 steps we have constructed Sasbi repre-

sentations

ujξ =v(j)∑l=1

γ(j)l Γa

(j)l + ζj where γj

l ∈ F and LM>(ξ) = maxv(j)

l=1LM>(γ(j)l Γa

(j)l ).

where uj ∈ S> ∩ F [Γ] and 1 ≤ j ≤ i− 1

We have to prove ∃ui ∈ S> ∩ F [Γ] and uiξ =v(i)∑l=1

γ(i)l Γa

(i)l + ζi and LM>(ξ) =

maxv(i)

l=1LM>(γ(i)l Γa

(i)l ).

We have two possibilities

1) ∆a(i) = Γa(i) is Γ-monomial.

2) ∆a(i) = ∆a(i)

i is ∆i-monomial.

Induction step: Consider the Sasbi representation for j = i− 1.

ui−1ξ =v(i−1)∑l=1

γ(i−1)l Γa

(i−1)l + ζi−1 and LM>(ξ) = maxv

(i−1)

l=1 LM>(γ(i−1)l Γa

(i−1)l ).

For the first case in induction step, replace ζi−1 by γiΓa(i) + ζi, and obtain

ui−1ξ =v(i−1)∑l=1

γ(i−1)l Γa

(i−1)l + γiΓa(i) + ζi.

28

Put ui = ui−1 and γ(i−1)l Γa

(i−1)l = γ

(i)l Γa

(i)l , 1 ≤ l ≤ v(i−1), γiΓa(i) = γ

(i)vi Γ

a(i)vi we get the

required representation

uiξ =v(i)∑l=1

γ(i)l Γa

(i)l + ζi.

As LM>(γ(i)vi Γ

a(i)vi ) < LM>(γ

(i)l Γa

(i)l ), 1 ≤ l ≤ vi − 1, from this condition we get

LM>(ξ) = maxv(i)

l=1LM>(γ(i)l Γa

(i)l ) which shows representation is Sasbi.

For the second case in induction step replace the ζi−1 by γi∆a(i)

i + ζi , it becomes

ui−1ξ =v(i−1)∑l=1

γ(i−1)l Γa

(i−1)l + γi∆a(i)

i + ζi.

We can write ∆a(i)

i = Γb(i)Φγ(i)where Φ = ζ0, ζ1, . . . , ζi−2. Since we are in the

second case, not all the components of γ(i) are zero. Since LM>(ζi−1) < LM>(ζj) for

j ≤ i − 2 and LM>(∆a(i)) = LM>(ζi−1) < LM>(ξ) it follows that LM>(Γ

b(i)) < 1.

Since ζj = ujξ −v(j)∑l=1

γ(j)l Γa

(j)l ,1 ≤ j ≤ i− 2, we can replace ζj’s by this expression.

Therefore

∆ia(i) = Γb(i)R(u0, u1, . . . , ui−2, ξ, ∂1, . . . , ∂s)ξ + Γb(i)Lγ(i).

For a suitable polynomial R and L = v(0)∑l=1

γ(0)l Γa

(0)l , . . . ,

v(i−2)∑l=1

γ(i−2)l Γa

(i−2)l . From

u0, u1, . . . , ui−2, ξ ∈ F [Γ] and LM>(Γb(i)) < 1 it follows that the

ui = ui−1 − γiΓb(i)R ∈ S> ∩ F [Γ].

Since LM>(γ(j)l Γa

(j)l ) ≤ LM>(ξ) it follows that the leading monomial of any Γ-

monomial occuring in Γb(i)Lγ(i) is smaller than leading monomial of ξ. This implies

that

uiξ =v(i−1)∑l=1

γ(i−1)l Γa

(i−1)l + γiΓa(i)Lγ(i) + ζi

is a Sasbi representation since LM>(ξ) = maxv(i−1)

i=1 LM>(γ(i−1)l Γa

(i−1)l ).

29

Example 2.2.5. In the localization of the univariate polynomial ring F [x]> where >

is the local ordering take ∂ = x3 + x4 and Γ = ∂1 = x3 + x6, ∂2 = x − x2 we want

to compute the weak Sasbi normal form of ∂ with respect to Γ. Put ζ0 = ∂.

In the first reduction we select the Γ-monomial ∂1 = x3 + x6 with minimal ecart

such that LM>(ζ0) = LM>(∂1) = x3, we have ecart(ζ0) = 1, ecart(∂1) = 3, so

ecart(∂1) >ecart(ζ0) therefore we have to enlarge Γ = ∂1 = x3+x6, ∂2 = x−x2, ∂3 =

x3 + x4 and

ζ1 = ζ0 − ∂1

x4 − x6 = x3 + x4 − (x3 + x6)

In the second reduction we select the Γ-monomial ∂2∂3 = (x − x2)(x3 + x4) =

x4 − x6 with minimal ecart such that LM>(ζ1) = LM>(∂2∂3) = x4. Now we have

ecart(∂2∂3) = 2, ecart(ζ1) = 2 so Γ remains the same and

ζ2 = ζ1 − ∂2∂3,

0 = x4 − x6 − (x4 − x6),

we get ζ2 = 0. Now we summarize and obtain

ζ2 = ζ1 − ∂2∂3,

0 = x4 − x6 − (x− x2)(x3 + x4).

As ζ1 = ∂ − ∂1

ζ2 = ∂ − ∂1 − ∂2∂3,

0 = x3 + x4 − (x3 − x6)− (x− x2)(x3 + x4),

30

∂3 = ∂ we get

∂ − ∂∂2 = ∂1 + ζ2,

x3 + x4 − (x− x2)(x3 + x4) = x3 − x6,

(1− ∂2)∂ = ∂1 + ζ2

(1− x− x2)(x3 + x4) = x3 − x6.

we have 1− ∂2 = 1− x− x2 ∈ S> ∩ F [Γ] and ζ2 = 0 is the weak Sasbi normal form.

Definition 2.2.3. Let Γ = ∂1, . . . , ∂m ⊂ R. Let

AR(Γ) := h ∈ F [y1, . . . , ym] |h(LM>(∂1), . . . , LM>(∂m)) = 0 ⊂ F [y1, . . . , ym].

Definition 2.2.4. Let Γ ⊆ R and ∂ =∑ν

i=1 γiΓai ∈ F [Γ]. The height of the element

∂ w.r.t the presentation ∂ =∑ν

i=1 γiΓai is maxν

i=1LM>(Γai). If the presentation is

fixed then we will write ht(∑ν

i=1 γiΓai) = maxν

i=1LM>(Γai).

Theorem 2.2.6. (SASBI basis criterion) Let Γ=∂1, ∂2, . . . , ∂m be a subset of

R. Assume that F [Ψ] admits a finite Sagbi basis where Ψ = ∂1h, . . . , ∂mh, the

homogenization of Γ. Let S := P1, ..., Pk be a generating set5 of AR(Γ). Then Γ is

a Sasbi basis for F [Γ]> if and only if for each 1 ≤ j ≤ k, WS-NF (Pj(Γ) |Γ) = 0.

Proof. (⇒) Suppose that WS-NF (Pj(Γ) |Γ) = 0. This implies that LM>(WS-

NF (Pj(Γ) |Γ)) /∈ F [LM>(Γ)] by the property of the weak Sasbi normal form. We

have Pj(Γ) ∈ F [Γ], therefore WS-NF (Pj(Γ) |Γ) ∈ F [Γ]. Since Γ is a Sasbi basis of

F [Γ]> we have LM>(WS-NF (Pj(Γ) |Γ)) ∈ F [LM>(Γ)]. This is contradiction to the

assumption that LM>(WS-NF (Pj(Γ) |Γ)) /∈ F [LM>(Γ)].

5Note that the set of S-polynomials defined in Definition 2.2.3 is a generating set of AR(Γ).

31

(⇐) We have to show that Γ is Sasbi basis. For this purpose we prove that ∂ ∈ F [Γ]>

has a Sasbi representation w.r.t Γ, that is there exist u ∈ S> ∩ F [Γ] such that

u∂ =ν∑

i=1

γiΓai with LM>(∂) = ht(

ν∑i=1

γiΓai)

Let ∂ ∈ F [Γ]>, choose u ∈ S> ∩ F [Γ] such that u∂ =∑ν

i=1 γiΓai , furthermore, we

assume that this representation has the smallest possible height of all possible rep-

resentations of u∂ in F [Γ]. We denote this height by X := maxνi=1LM>(Γ

ai).

It is clear that LM>(∂) ≤ X. Suppose that LM>(∂) X. Without loss of

generality, let the first µ summands in the above representation of ∂ be the ones

for which X = LM>(Γai). Then cancelation of their leading terms must occur,

that is,∑µ

i=1 γiLM>(Γai) = 0, and hence we obtain a polynomial in F [y1, ..., ym],

P (y) =∑µ

i=1 γiyai ∈ AR(Γ). Since S = P1, ..., Pk is a generating set of AR(Γ) we

can write

P (y) =k∑

j=1

fjPj(y) (2.2.1)

for suitable fj ∈ F [y1, . . . , ym]. Furthermore, note that

ht(P (Γ)) = maxkj=1ht(fj(Γ))ht(Pj(Γ)) = X

hand:where, fj(Γ) andPj(Γ) are considered as expressions in the ∂i′s.

We have assumption that for all 1 ≤ j ≤ k, WS-NF (Pj(Γ) | Γ) = 0, which means

that wjPj(Γ) has a Sasbi representation, wjPj(Γ) =∑νj

l=1 γljΓalj , for suitable wj ∈

S> ∩ F [Γ] and LM>(Pj(Γ)) = maxνjl=1LM>(Γ

alj ) ht(Pj(Γ)). The inequality is

strict since Pj ∈ AR(Γ), so we may assume that w = wj, where 1 ≤ j ≤ k . For each

j, we have

wfj(Γ)Pj(Γ) =

νj∑l=1

γlj∂j(Γ)Γalj. (2.2.2)

32

Let Xj is the height of the right hand side in eq 2.2.2, then obtain

Xj maxkj=1ht(fj(Γ))ht(Pj(Γ)) = X.

Finally, the equations 2.2.1 and 2.2.2 imply that:

u∂ = P (Γ) +ν∑

i=µ+1

γiΓai

=k∑

j=1

νj∑l=1

γjjfj(Γ)Γalj

︸ ︷︷ ︸sum1

+ν∑

i=µ+1

γiΓai

︸ ︷︷ ︸sum2

.

If we examine the expressions of the above equation, we see that Xj < X ; for all 1 ≤

j ≤ k therefore ht(sum1) = maxkj=1Xj < X. By the choice of µ, ht(sum2 < X).

But this is not possible because we have chosen a representation of ∂ with smallest

possible height, therefore we get contradiction. Thus, Γ is a Sasbi basis of F [Γ]>.

This theorem is the basis of the following algorithm:

Algorithm 2.2.7. Let > be a local monomial ordering on R.

Input: A finite subset Γ ⊂ R. Assume F [Γ]> admits a finite Sasbi basis and F [Ψ]

admits a finite Sagbi basis where Ψ = Γh is the homogenization with respect to new

variable “t”.

Output: A Sasbi basis S for F [Γ]>.

• S = Γ;

• oldS = ∅;

• while (S = oldS)

Compute a generating set S for AR(S);

P = S(S);

33

Red = WS-NF (p | S) | p ∈ P \ 0;

oldS = S;

S = S ∪Red;

• return S;

Example 2.2.8. Let Γ = ∂1 = x4, ∂2 = x4 + x5 + x6, ∂3 = y2, ∂4 = x7, ∂5 = y3 + x8

is a subset F [x, y] and > the degree lexicographical local monomial ordering. We

consider F [Γ]> = F [x4, x4+x5+x6, y2, x7, y3+x8]>. Then we have an ideal AR(Γ) =

(y52 − y3

3, y2 − y1) therefore AR(Γ)(Γ) = (s1 = x8y3 + 12x16, s2 = x5 + x6). We can

take the reduction of s1 by ∂12∂5 (with minimal ecart) and obtain

h = s1 − ∂12∂5

x8y3 +1

2x16 − (x4)

2(y3 + x8) = 0

so WS-NF (s1 |Γ) = 0. There is no Γ-monomial Γa such that LM>(Γa) = LM>(s2),

so WS-NF (s2 |Γ) = x5 + x6. We have now Γ = Γ ∪ ∂6 = x5 + x6. Then we

have AR(Γ)(Γ) = (s1 = x8y3 + 12x16, s1 = x5 + x6). By the above argument WS-

NF (s1 |Γ) = 0. On the other hand s2 = ∂6 and therefore WS-NF (s2 |Γ) = 0. This

shows that Γ = ∂1, ∂2, ∂3, ∂4, ∂5, ∂6 is a Sasbi basis of F [x4, x4 + x5 + x6, y2, x7, y3 +

x8]>.

2.3 Implementation in SINGULAR

In this section we will give an overview of the main procedures which we have imple-

mented in SINGULAR. In this overview we will present these procedures and give by

concrete SINGULAR examples to explain their usage. We have implemented three

34

types of procedures:

1) Weak Sasbi Normal form procedure

” WSNF procedure”: It is an implementation of Algorithm 2.2.4 (ecart driven

normal form) to obtain weak Sasbi normal form of a polynomial.

SINGULAR Procedure:

LIB"algebra.lib" ; // we need this library for "algebra containment"

// procedure

proc WSNF(poly f,ideal I)

ideal G=I ;

poly h=f ;

poly h1,j ;

list L ;

map psi ;

while(h!=0 && h1!=h)

L= algebra containment(lead(h),lead(G),1) ;

if (L[1]==1)

def s= L[2] ;

psi= s,maxideal(1),G ;

35

j= psi(check) ;

if (ecart(h)<ecart(j))

G[size(G)+1]=h ;

h1=h ;

h=h-j ;

kill s ;

return (h) ;

SINGULAR Example

ring r=0,(x,y),Ds ;

ideal i=x2,x4+x5+x6,x7,y2,y3+x8 ;

poly f=x4y3+y5 ;

WSNF(f, i) ;

=> x5y3-x6y3-x8y2-x12-x13-x14

ring r=0,x,ls ; // example 2.2.5

ideal i=x3+x4 ;

poly g=x3+x6, x-x2;

WSNF(g, i) ;

36

=> 0

2) Procedure to compute S-polynomials

”sasbiSpoly procedure”: This procedure computes the generators of AR(G) (def-

ined in Definition 2.2.3) which are S-polynomials.

SINGULAR Procedure:

LIB"elim.lib" ; // we need this library for "nselect" procedure

proc sasbiSpoly(ideal id)

def bsr= basering ;

ideal vars = maxideal(1) ;

int n=nvars(bsr) ;

int m=ncols(id) ;

int z ;

ideal p ;

if(id==0)

return(p) ;

else

execute("ring R1=("+charstr(bsr)+"),(@y(1..m),"+varstr(bsr)+"),

37

(ds(m),ds(n));");

ideal id =imap(bsr,id) ;

ideal A ;

for (z=1; z<=m; z++)

A[z]=lead(id[z])-@y(z) ;

A=std(A) ;

ideal kern=nselect(A,m+1,m+n) ;

export kern,A ;

setring bsr ;

map phi= R1,id ;

p=simplify(phi(kern),1) ;

return (p) ;

SINGULAR Example

ring r=0,(x,y),Ds ;

ideal i=x2,x4+x5+x6,x7,y2,y3+x8 ;

sasbiSpoly(i);

[1]=x5+x6

[2]=x8y3+1/2x16

38

3) SASBI BASIS construction algorithm

”Sasbi procedure”: It is an iterative consequence of previous procedures to com-

pute Sasbi basis.

SINGULAR Procedure:

proc Sasbi(ideal id)

ideal S,oldS,Red ;

list L ;

int z,n ;

S=id ;

while( size(S)!=size(oldS))

L=sasbiSpoly(S) ;

n=size(L) ;

for (z=1; z<=n; z++)

Red=L[1][z] ;

Red=WSNF(Red[1],S) ;

oldS=S ;

S=S+Red ;

39

return(S) ;

SINGULAR Example

ring r=0,(x,y),Ds ;

ideal i=x2,x4+x5+x6,x7,y2,y3+x8 ;

Sasbi(i);

[1]=x2

[2]=x4+x5+x6

[3]=x7

[4]=y2

[5]=y3+x8

[6]=x5+x6

40

Chapter 3

Converting Subalgebra Bases withthe Sagbi Walk

In this chapter we will consider only global monomial orderings, i.e well orderings.

Consider the subalgebra A ⊂ R = F [x1, . . . , xn] which admits a finite Sagbi basis

w.r.t all monomial orderings.

3.1 Sagbi cone

In this section we study Sagbi fan and its related concept.

Lemma 3.1.1. Let Γ = ∂1, . . . , ∂s ⊂ A and >1 and >2 are two monomial orderings

such that LT>1(∂i) = LT>2(∂i) for all i. If Γ is a Sagbi basis of A w.r.t >1 then Γ is

a Sagbi basis of A w.r.t >2.

Proof. let ∂ ∈ A. Reducing ∂ by Γ using >2, we obtain

∂ = γ1Γa1 + · · ·+ γsΓ

as + ζ

where ζ is the Sagbi normal form of ∂ w.r.t Γ. Using >2, either ζ is zero or ∀

xb ∈ support(ζ), xb = LM>2(Γa) ∀ Γ-monomials Γa. However, we have LT>2(∂i) =

41

LT>1(∂i) ∀ i. We assumed that Γ is a Sagbi basis for A w.r.t >1, also we have

ζ = ∂ −∑s

i=1 aiΓai ∈ A, therefore ζ = 0. Above we used the Sagbi normal form

algorithm, therefore LT>2(∂) = LT>2(aiΓai) for some i, this implies that Γ is also

Sagbi basis for A w.r.t >2.

Definition 3.1.1. Let Γ ⊂ R and v be a fixed vector.

We denote Inv(Γ) as

Inv(Γ) := Inv(ξ) | ξ ∈ Γ

We define inv(Γ) to be the subalgebra generated by Inv(Γ).

inv(Γ) = F [Inv(Γ)]

inv(Γ) need not to be a monomial subalgebra. For generic v, inv(I) is a monomial

subalgebra.

Theorem 3.1.2. Given a subalgebra A ⊂ R and monomial ordering >. There exist

a weight v such that in>(A) = inv(A).

Proof. Let Γ = ∂1, ∂2, . . . , ∂s be a Sagbi basis with respect to >. By Lemma 1.1.4 ∃

a weight v such that for any two monomials u > v occurring in Γ, degv(u) > degv(v).

Now we raise two question.

• The collection of all possible Sagbi bases of a fixed subalgebra A is finite or not?

• Under which condition, s-reduced Sagbi basis for A w.r.t different monomial order-

ing are same ?

42

Given a subalgebra A ⊂ R, we define the set

IN(A) = in>(A) | > a monomial ordering

Now we answer the first question

Theorem 3.1.3. For a subalgebra A ⊂ R the set IN(A) is finite.

Proof. We proof it by contradiction. Assume IN(A) is an infinite set. If we take

N ∈ IN(A) then there exist monomial ordering >N such that N = in>N(A). Let

Υ = >N |N ∈ IN(A) . Our assumption implies that Υ is an infinite set.

As A is a finitely generated subalgebra we have A = F [∂1, . . . , ∂s] for polynomials

∂i ∈ R. Each ∂i is a finite sum of terms. By pigeonhole principle ∃ Υ1 ⊂ Υ which is

infinite and leading terms LT>(∂i) are the same ∀ >∈ Υ1 and ∀i, 1 ≤ i ≤ s. For any

monomial ordering >∈ Υ we write N1 = F [LT>(∂1), . . . , LT>(∂s)] .

Let Γ = ∂1, . . . , ∂s. For any >1∈ Υ1, if Γ is a Sagbi basis for A w.r.t >1, then

by Lemma 3.1.1 ∀ >∈ Υ1, Γ would be a Sagbi basis for A w.r.t >.

However, this cannot be the case since Υ1 is a subset of Υ which is selected in

such a way that the in>(A) were all distinct for > in Υ . If we take a monomial

ordering >1∈ Υ then LT>1(∂s+1) /∈ N1 for some ∂s+1 ∈ A. Now we compute Sagbi

normal form of ∂s+1 reducing it by ∂1, . . . , ∂s. Now replace ∂s+1 by its Sagbi normal

form. We may assume that ∀ xb ∈ support(∂s+1), xb = LM>1(Γ

a) for all Γ-monomial

Γa where Γ = ∂1, . . . , ∂s .

Again we can find infinite subset Υ2 ⊂ Υ1 by pigeonhole principle such that

∀ >∈ Υ2 the leading terms of ∂1, . . . , ∂s+1 are the same. Now we suppose that

N2 = F [LT>1(∂1), . . . , LT>1(∂s+1)] for all > in Υ2. Here we have N1 ⊂ N2. By

43

the argument above we can show that ∂1, . . . , ∂s+1 cannot be a Sagbi basis w.r.t any

monomial ordering in Υ1. Now we fix any monomial ordering >2∈ Υ2. There exist

∂s+2 ∈ A such that for all xb ∈ support(∂s+2), xb = LM>2(Γ

a) for all Γ-monomial Γa,

Γ = ∂1, . . . , ∂s+1 .

If we continue this way, we produce a descending chain of infinite subsets Υ ⊃

Υ1 ⊃ Υ2 ⊃ Υ3 . . . , and also produce a chain N1 ⊂ N2 ⊂ N3 ⊂ . . . , which is an infinite

strictly ascending chain. This contradicts the condition that A admits a finite Sagbi

basis.

Let S = ξ1, . . . , ξt be one of the Sagbi bases of A w.r.t a monomial ordering

> such that LT>(ξi) = xa(i), and the corresponding element of In(A) w.r.t > is

N = F [xa(1), . . . , xa(t)]. Our next aim is to answer the second question. This will

answer the second question. We write

ξi = xa(i) +∑b

γi,bxb, γi,b ∈ F

where xa(i) > xb whenever γi,b = 0. Any ordering > can be represent by non-singular

matrix A (c.f. Lemma 1.1.3). In order to select the leading term in ξi where 1 ≤ i ≤ t

we first compare monomials in support(ξi) with the first row v of A.

The weight vector v chooses the right leading term in ξi if a(i).v > b.v for all b

with γi,b = 0. If we have tie in the first computation then we use the other rows of A

to select the leading term. This leads to the following definition

Definition 3.1.2. Let S be a Sagbi basis of A w.r.t a monomial ordering > as above

. We define the ΩS,> as

ΩS,> = v ∈ (Rn)+ : a(i).v ≥ b.v whenever γi,b = 0

44

= v ∈ (Rn)+ : (a(i)− b).v ≥ 0 whenever γi,b = 0

ΩS,> is called the Sagbi cone1. It is polyhedral cone which is closed and convex2.

By Lemma 1.1.4, its interior3 is non-empty. For a matrix ordering >A, let v be the

first row of A. If v ∈ int(ΩS,>) then S is also a Sagbi basis with respect to >A. This

gives an answer to the second question.

Definition 3.1.3. The collection of all the cones ΩS,> and faces as S ranges over all

Sagbi bases with respect to > of A is called Sagbi Fan. This collection is finite due

to Theorem 3.1.3.

Example 3.1.4. Consider the subalgebra A = F [xy + y2, x2y2 + y3]. The s-reduced

Sagbi basis w.r.t the degree lexicographical ordering >1 with x >1 y is

S(1) = xy + y2, xy3 +1

2y4 − 1

2y3

Then v = (a, b) ∈ ΩS(1),>1if and only if following inequalities hold :

(1, 1).(a, b) ≥ (0, 2).(a, b) or a ≥ b ,

(1, 3).(a, b) ≥ (0, 4).(a, b) or a ≥ b ,

(1, 3).(a, b) ≥ (0, 3).(a, b) or a ≥ 0.

From these inequalities, ΩS(1),>1is the shaded area ( dark gray) in Fig 3.1.

1If S is a Grobner basis of some ideal I with respect to >, then ΩS,> is called Grobner Cone(Definition 1.4.2). It is also defined as the closure in Qn of v ∈ (Qn)+ | ⟨LT>(I)⟩ = ⟨Inv(I)⟩ ([5]). Similarly for a subalgebra A we can define the Sagbi cone as the closure in Qn of v ∈(Qn)+ | in>(A) = inv(A) . By Theorem 3.1.2 its interior is nonempty so it is well defined.

2Sagbi cones have the same geometrical structure as Grobner cones. For Grobner cones propertiessee [7].

3Its interior is defined as int(ΩS,>) = v ∈ (Rn)+

: a(i).v > b.v whenever γi,b = 0 .

45

Figure 3.1: Sagbi Fan of A

Now we take the weighted lexicographical ordering >2 with weight v = (1, 2) and

x >2 y. For this ordering the s-reduced Sagbi basis for A is

S(2) = y2 + xy, x2y2 + y3

Proceeding as above for v = (a, b) ∈ ΩS(2),>2, we get the inequalities

a ≤ b, 2a ≥ b.

From these inequalities, ΩS(2),>2is the shaded area (light gray) in Fig 3.1. Note that

ΩS(1),>1∩ ΩS(2),>2

is a common face of both cones.

Now we take the lexicographical ordering >3 with y >3 x, the s-reduced Sagbi basis

of A with respect to >3 is

S(3) = y2 + xy, y3 + x2y2.

For v = (a, b) ∈ ΩS(3),>3, we get inequalities

a ≤ b, 2a ≤ b.

46

From these inequalities, ΩS(2),>2is shaded area (white) in Fig 3.1.

We have SF (A) = ΩS(1),>1,ΩS(2),>2

,ΩS(3),>3 is a Sagbi Fan 4 of A.

3.2 Sagbi Walk

Let >s is a monomial ordering represent by matrix As with first row vs. Assume

that S is a Sagbi basis of A w.r.t >s. The starting monomial ordering for the Sagbi

walk is denote by >s. In the Sagbi fan of A, S corresponds to a cone ΩS,>s . Now we

want to compute a Sagbi basis of A w.r.t monomial ordering >t. We call >t target

ordering. Let the matix At represent the monomial ordering >t with first row vt.

As we have both >s and >t are global orderings so vs,vt ∈ (Rn)+ which is convex

therefore they can be joined by a straight line between vs and vt, (1− θ)vs + θvt for

θ ∈ [0, 1]. The Sagbi walk method basically consist of two basic steps. The first step

is to cross from one cone to next cone. In the next step we compute the Sagbi basis

of A corresponding to the new cone.

3.2.1 Crossing Cones

In this section we discuss the procedure to cross from one cone to another cone of the

Sagbi Fan, which is already given for the Grobner fan in [7], Chapter 8. We can use

it for the Sagbi fan because geometrically they are same. We review this procedure

for Sagbi cones to use it in Sagbi walk algorithm. For details see [7], chapter 8, page

437.

Let >s is a monomial ordering represented by matrix Aold with first row vold. Assume

4The union of all cones is positive orthant (R)+.

47

that Sold = ξ1, . . . , ξt is the Sagbi basis of A w.r.t >old and Ωold,>oldis the corre-

sponding cone of Sold. Now we move along the path from vold. Let vnew be the last

point on the path such that vnew ∈ Ωold,>old.

Assume ξi = xa(i) +∑

i,b γi,bxb : 1 ≤ i ≤ t with LT>Aold

(ξi) = xa(i) . Let

σ1, . . . , σm denote the vectors a(i)− b where 1 ≤ i ≤ t and γi,b = 0. The new weight

vector vnew is given by vnew = (1− θlast)vold + θlastvt. There is an algorithm given in

[7], page 437 from which we can compute θlast. For the readers convenience we give

this algorithm :

Algorithm 3.2.1.

Input : vold,vt, σ1, . . . , σm

Output : θlast.

• θlast = 1;

• for j = 1, . . . ,m

if (vt.σj < 0);

θj =vold.σj

vold.σj−vt.σj;

if (θj < θlast)

θlast = θj;

• return θlast;

After computing vnew, we have to go to the next cone in the Sagbi fan. Now we

define the monomial ordering >new which is a vnew-weighted degree ordering, if the

vnew-degree of two monomials is equal then >new uses the target ordering >t. We

know that the matrix At represents >t , therefore >new is represented by (vnew, At).

The monomial ordering >new gives the next cone Ωnew,>new . In the above process

whenever vold = vt, the following lemma shows that the after each step vnew is closer

48

to vt.

Lemma 3.2.2. (c.f. [7], chapter 8, page 438 ) Assume that the matrix (vold, At)

represents the monomial ordering >old and let θlast be as in algorithm 3.2.1. Then

θlast > 0.

3.2.2 Converting Sagbi Basis

Assume Sold and Snew are the Sagbi bases of A w.r.t >old and >new respectively where

>new is represented by the matrix (vnew, At). After crossing from the old cone Ωold,>old

into new cone the Ωnew,>new we have to convert Sold into Snew.

Here we note that some of the inequalities defining the old cone Ωold,>oldbecome

equalities as vnew lies on the boundary of the old cone Ωold,>old. Therefore for some

ξ ∈ Sold, the vnew-degree of the leading term of ξ and some other term in ξ are the

same. As vnew ∈ Ωold,>oldthe LT>old

(ξ) is in Inw(ξ). Suppose that Θ is Sagbi basis

of inwnew(Sold) w.r.t >new. Through Θ we can convert Sold into Snew.

Let Sold = ξ1, . . . , ξt be a Sagbi basis for the subalgebra A w.r.t >old. Let

vnew ∈ Ωold and >new be a monomial ordering represented by (vnew, At), let the

monic Sagbi basis of subalgebra invnew(Sold) w.r.t >new be Θ = ζ1, . . . , ζs. For

1 ≤ j ≤ s, ζj ∈ invnew(Sold) and therefore we can express it as

ζj = Pj(Invnew(ξ1), . . . , Invnew(ξt)), Pj ∈ F [y1, . . . , yt], ξi ∈ Sold

Now we replace each Invnew(ξi) by the ξi , the above polynomials become

ζj = Pj(ξ1, . . . , ξt), 1 ≤ j ≤ s

Theorem 3.2.3. Θ = ζ1, . . . , ζs form a Sagbi basis of A w.r.t >new.

49

Let > be a monomial ordering and v a weight vector. If the leading term of ξ

w.r.t > is in Inv(ξ) ∀ nonzero polynomials ξ then we say that v is compatible with

>. Before proving Theorem 3.2.3 we will prove the following lemma.

Lemma 3.2.4. Fix v ∈ (Rn)+\0 and let S be the Sagbi basis of the subalgebra A

w.r.t >.

a. If v is compatible with >, then LT>(A) = LT>(Inv(A)) = LT>(inv(A)).

b. If v lies in the cone ΩS,>, then Inv(S) form a Sagbi basis of inv(A) w.r.t >. In

particular,

inv(A) = inv(S)

Proof. (a): For any polynomial ∂ in R, LT>(∂) appears in Inv(∂) therefore the

equality LT>(A) = LT>(Inv(A)) is obvious. Now we prove second equality. First

we show the inclusion LT>(inv(A)) ⊂ LT>(Inv(A)). For this purpose we prove that

LT>(∂) ∈ LT>(Inv(A)) ∀ ∂ ∈ inv(A) . If ∂ ∈ inv(A) then we write it as

∂ = q(Inv(∂1), . . . , Inv(∂t)), q ∈ F [y1, . . . , yt], ∂i ∈ A

Each side in the above equation is a sum of v-homogenous terms. As we know that

the initial form Inv(∂i) is v-homogenous therefore any Inv(∂i)-monomial is also v-

homogenous. This implies that

Inv(∂) = q(Inv(∂1), . . . , Inv(∂t)), q ∈ F [y1, . . . , yt],

and we may assume that all the terms in q(Inv(∂1), . . . , Inv(∂t)) have the same v-

degree. It follows that Inv(∂) = Inv(q(∂1, . . . , ∂t)) ∈ Inv(A). Then compatibility of

v with > implies that LT>(∂) = LT>(Inv(∂)) ∈ LT>(Inv(A)). The other inclusion

hold obviously .

50

(b): We have S is a Sagbi basis w.r.t > therefore in>(A) = in>(S). Also we have

assume that v is compatible with > so we have equality in>(S) = in>(Inv(S)). It

follows that

in>(A) = in>(Inv(S)) (3.2.1)

From part a, we have equality 5

in>(A) = in>(Inv(S)) (3.2.2)

Combining 3.2.1 and 3.2.2 we get in>(inv(A)) = in>(Inv(S)), which show that

Inv(S) is a Sagbi basis of inv(A) w.r.t >.

Now we analyze the situation when v lies in the cone ΩS,>. Consider the monomial

ordering >v, which is v-weighted degree ordering. If v-degree of two monomials are

equal the it use >. The weight vector v is compatible with the monomial ordering

>v. Since v lies in the cone ΩS,>, therefore for each ξ ∈ S the leading terms of ξ

w.r.t >v and > are the same. Therefore it follows that S is Sagbi basis w.r.t >v.

From the initial part of the argument we have Inv(S) is a Sagbi basis of inv(A) w.r.t

>v because v is compatible with >v. Note that the leading term of Inv(ξ) w.r.t >

and >v are same for each ξ ∈ S. Again we conclude that Inv(S) is a Sagbi basis of

inv(A) w.r.t >.

Now we give the proof of Theorem 3.2.3.

Proof. The weight vector vnew is compatible with >new ( represented by (vnew, At) ).

By Lemma 3.2.4 (part a), we have

LT>new(A) = LT>new(invnew(A))

5We have F [LT>(A)] = F [LT>(inv(A))] = in>(inv(A)).

51

This implies

in>new(A) = in>new(invnew(A)) (3.2.3)

Note that the weight vnew lies in the cone Ωold,>old. Therefore from Lemma 3.2.4

(part b) we have

invnew(A) = invnew(Sold)

This implies

in>new(invnew(A)) = in>new(invnew(Sold)) (3.2.4)

Now we show that

in>new(invnew(Sold)) = in>new(Θ) = in>new(Θ). (3.2.5)

where Θ = ζ1, . . . , ζs is the Sagbi basis of invnew(Sold) w.r.t >new and Θ =

ζ1, . . . , ζs as described in the statement of Theorem 3.2.3.

In equation 3.2.5 the first equality is obvious because Θ is the Sagbi basis of

invnew(Sold) w.r.t>new. For the second equality it is sufficient to prove that LT>new(ζj) =

LT>new(ζj) where 1 ≤ j ≤ s.

ζj = Pj(ξ1, . . . , ξt) =∑a∈Jj

γj,a ξa11 . . . ξt

at where γj,a ∈ F, Jj ⊂ Nt a finite subset.

The vnew-degree of all the terms in the initial form of ξ1a1 . . . ξt

at is greater than

the the vnew-degree of all the terms in the ξ1a1 . . . ξt

at − (Inv(ξ1))a1 . . . (Inv(ξt))

at .

This shows that ζj is obtained by adding terms of smaller vnew-degree. The added

terms are smaller w.r.t >new because >new is compatible with vnew so the leading

term of ζj and ζj w.r.t >new is same.

Combining 3.2.3, 3.2.4 and 3.2.5 we obtain

in>new(A) = in>new(Θ)

52

Since ∀j, ζj ∈ A so we conclude that Θ is a Sagbi basis forA w.r.t>new as claimed.

3.2.3 The Algorithm

Now we present the Sagbi walk algorithm. This algorithm follows the straight line

segment from the start weight vector vs to the target weight vector vt. Now we define

three procedures which we will be used in the Sagbi walk algorithm.

NextCone : This procedure computes θlast from Algorithm 3.2.1.

Lift : This procedure lifts the Sagbi basis of invnew(Sold) w.r.t >new to the Sagbi basis

Snew following Theorem 3.2.3, and

s-interreduce : For a given set of polynomials Γ and a monomial ordering >, this

procedure s-interreduces Γ w.r.t >.

Algorithm 3.2.5.

Input: As with first row vs representing the starting monomial ordering and At with

first row vt representing the target monomial ordering, Ss = Sagbi basis w.r.t As.

Output: Snew = Sagbi basis w.r.t At.

• Aold := As ;

• Sold := Ss ;

• vnew := vs ;

• Anew := (vnew, At) ;

• done := false ;

• while(done = false)

In := Invnew(Sold) ;

InS :=Sagbibasis(In,>Anew) ;

Snew :=Lift(InS, Sold, In, Anew, Aold) ;

53

Snew :=s-interreduce(Snew, Anew) ;

θ :=NextCone(Snew,vnew,vt) ;

if (vnew = vt) then

done = true ;

else

Aold := Anew ;

Sold := Snew ;

vnew := (1− θ)vnew + θvt ;

Anew := (vnew, At) ;

• return (Snew) ;

Theorem 3.2.6. The algorithm 2 terminates and computes a Sagbi basis of A w.r.t

>t.

Proof. We first prove termination. We move along the line segment from vs to vt.

By Definition 3.1.2 each Sagbi cone has finitely many bounding hyperplanes. Also

the Sagbi fan of A = F [Ss] has only finitely many cones (Theorem 3.1.3). The line

segment from vs to vt is contained in some hyperplanes. We discard such type of

hyperplanes, the remaining hyperplanes determine a finite set of distinguished points

on the line segment from vs to vt.

In the Sagbi walk algorithm we have θlast=Nextcone(Snew,vnew,vt). It can be

computed by replacing vold with the current value of vnew in the Algorithm 3.2.1.

In our algorithm, a monomial ordering is always represented by matrix of the form

(vs, At). Therefore, monomial ordering always satisfies the hypothesis of Lemma

3.2.2. Note that the next value of vnew is vt when θlast = 1. If this happens the

algorithm terminates after one more turn through the while loop. On the other hand

54

if θlast = 1 than θlast = θj < 1 for some j. In this case the next value of vnew lies on

the hyperplane v.σj = 0. From Algorithm 3.2.1 we have vt.σj < 0, and vnew.σj ≥ 0,

therefore the line segment and hyperplane meet in a single point. Therefore the new

value of vnew is one of our distinguished points. Note that θlast > 0 (Lemma 3.2.2),

therefore if the current vnew and vt are different, then we must move to another

distinguished point further along the line segment. Our set of distinguished points

is finite so after a finite number of steps we reach vt, and the Sagbi walk algorithm

terminates.

Now we prove the correctness. Note that in each turn through the while loop, the

hypotheses of Theorem 3.2.3 are satisfied. Therefore when vnew reaches vt, the next

turn through the while loop computes the Sagbi basis of A w.r.t (vt, At) ( the matrix

representation of the monomial ordering >t). It follows that the final value of Snew

is a Sagbi basis for >t.

Example 3.2.7. Consider the subalgebra A = F [xy + z2, x2y2 + y3]. We have

Ss = z2 + xy, y3 + x2y2

is a Sagbi basis of A w.r.t lexicographical ordering induced by z > y > x. We want

to determine the Sagbi basis w.r.t lexicographical ordering induced by x > y > z. Let

As =

0 0 1

0 1 0

1 0 0

so vs = (0, 0, 1). Similarly we have

55

At =

1 0 0

0 1 0

0 0 1

and vt = (1, 0, 0). In all of the following computations the monomial orderings are

always represented by square matrices. For this purpose we delete appropriate linearly

independent rows in the matrices. Now we consider the monomial orderings defined

by

Anew =

0 0 1

1 0 0

0 1 0

(using the vnew = (0, 0, 1) first, then refining it by the target ordering). We have

In = z2, x2y2. The Sagbi basis of F [In] w.r.t Anew remains z2, x2y2. Therefore

the Sagbi basis of A w.r.t Anew does not change. We have

Snew = z2 + xy, x2y2 + y3

We then call the next cone procedure (Algorithm 3.2.1) with vnew in place of vold.

The cone of the monomial ordering >Anew is defined by the two inequalities which we

get by comparing z2 vs. xy and x2y2 vs y3. By Algorithm 3.2.1 θlast is the largest θ

such that (1− θ)(0, 0, 1) + θ(1, 0, 0) is in the cone is computed as follows:

z2 vs. xy : σ1 = (−1,−1, 2), vt.σ1 = −1 < 0 ⇒ θ1 =vnew.σ1

vnew.σ1−(−1)= 2

3

x2y2 vs. y3 : σ2 = (2,−1, 0), vt.σ2 = 2 ≥ 0 ⇒ θ2 = 1

Therefore the new weight vector is vnew = (1 − 23)(0, 0, 1) + 2

3(1, 0, 0) = (2

3, 0, 1

3),

and

56

Anew =

23

0 13

1 0 0

0 1 0

are updated for the next pass through main loop.

In second pass the leading term of z2 + xy w.r.t the monomial ordering >new is

xy. Therfore we have In = z2 + xy, x2y2. The Sagbi basis for F [In] w.r.t >new is

Θ = ζ1 = xy + z2, ζ2 = x2y2, ζ3 = xyz2 +1

2z4

We can express ζ1, ζ2, ζ3 in terms of generators of the subalgebra F [In], we have

xy + z2 = P1(z2 + xy, x2y2), P1 ∈ F [t1, t2] and P1(t1, t2) = t1

x2y2 = P2(z2 + xy, x2y2), P2 ∈ F [t1, t2] and P2(t1, t2) = t2

xyz2 +1

2z4 = P3(z

2 + xy, x2y2), P3 ∈ F [t1, t2] and P3(t1, t2) = t12 − t2

So by Theorem 3.2.3 in order to obtain the next Sagbi basis we lift

P1(z2 + xy, x2y2 + y3) = xy + z2

P2(z2 + xy, x2y2 + y3) = x2y2 + y3

P3(z2 + xy, x2y2 + y3) = xyz2 +

1

2z4 − 1

2y3.

s-reducing w.r.t >new, we obtain the Sagbi basis Snew given by

xy + z2, xyz2 +1

2z4 − 1

2y3

The call to NextCone returns θlast = 1, since for all ξ ∈ Snew, there is no pair of terms

in ξ having same weight for any point on the line segment (1− θ)(23, 0, 1

3) + θ(1, 0, 0).

57

Thus vnew = vt, after one more turn through the while loop, Snew does not change

therefore our algorithm terminates. Our final output is

St = xy + z2, xyz2 +1

2z4 − 1

2y3

which is Sagbi basis of A w.r.t the target ordering.

3.3 Implemetation in SINGULAR

In this section we provide examples with timing Sagbi walk algorithms and also

current implementations of Sagbi basis construction algorithm in SINGULAR. We

have implemented the Sagbi Walk algorithm using the programming language of the

computer algebra system SINGULAR [20]. swalk6 is the command which computes

Sagbi basis through Sagbi Walk algorithm, sagbi is the usual Sagbi basis implemented

in SINGULAR. Timings were conducted on a MacBookPro with 2.4 GHz Intel Core

2 Duo processor running Mac OS X.

We consider the ring Q[x, y, z] with the lexicographical ordering lp.

Example 3.3.1.

A = Q[x2y4, y4z2, xy4z + y2z, xy6z2 + y18z9].

Example 3.3.2.

ξ1 = x2 + 2xy + y2,

ξ2 = y2 + 2yz + z2,

6For a given ideal I, list L we have command swalk(I,L). Let v,w are integers vectors. Iflist L = v,w (resp ϕ) then the command compute the Sagbi basis of the subalgebra defined by thegenerators of ideal, calculated via the Sagbi walk algorithm from the ordering (a(v),lp) (resp dp)to the ordering (a(w),lp) ( resp lp), where (a(v),lp), dp, lp denote v-weighted lexicographical,degree reverse lexicographical and lexicographical orderings respectively.

58

ξ3 = xy + xz + y2 + yz + y + z,

ξ4 = 2xy2+4xyz+2xz2+y5+5y4z+10y3z2+2y3+10y2z3+4y2z+5yz4+2yz2+ z5.

A = Q[ξ1, ξ2, ξ3, ξ4].

Example 3.3.3.

A = Q[x2z2, y2z2, xyz2 + yz, 2xy2z3 + y7z7].

Example 3.3.4.

A = Q[x2, y2, xy + y, 2xy2 + y13].

Example 3.3.5.

A = Q[x2y4, y4z2, xy4z + y2z, xy6z2 + y14z7].

The following table compares the Sagbi Walk algorithm, (with times in seconds)

with the current implementations of the Sagbi basis algorithm in SINGULAR.

Example swalk sagbi3.3.1 66 80593.3.2 26 2103.3.3 17 1813.3.4 63 36003.3.5 36 276

Table 3.1:

59

Chapter 4

Sagbi basis under Composition

In this chapter we review some results about the behavior of Sagbi basis under com-

position. Throughout the chapter we consider only global monomial orderings and

we will use notations from section 1.5.

4.1 Main Results

Let Φ = (Φ1, . . . ,Φn), Φi ∈ R be a list of n nonzero polynomials and > be a monomial

ordering. For ξ ∈ R, we denote the composition of ξ with Φ by ξ Φ (see Definition

1.5.1).

Definition 4.1.1. Let S ⊂ R. Two S-monomials Sa, Sb form a critical pair (Sa, Sb)

of S if LM>(Sa) = LM>(S

b). If c ∈ F such that Sa and cSb have the same leading

coefficient, then we define the T -polynomial of (Sa, Sb) as T (Sa, Sb) = Sa − cSb.

The set of all T -polynomials of S form a generating set of AR(S)(S) (Definition

1.3.6). Therefore the criterion of Sagbi basis (Theorem 1.3.3) can be restated as

Theorem 4.1.1. (c.f. [33]) A subset S of R is a Sagbi basis w.r.t > if and only if

the T -polynomial of every critical pair (Sa, Sb) of S is either equal to zero, or can be

60

written as

T (Sa, Sb) =t∑

i=1

ciSai , LM>(S

a) = LM>(Sb) > LM>(S

ai) ∀i.

Definition 4.1.2. Let Γ ⊂ R and S be a Sagbi basis ( w.r.t the monomial ordering

>) of the subalgebra F [Γ]. We say that composition by Φ commutes with the Sagbi

basis computation if the composed set S Φ is also a Sagbi basis (w.r.t the monomial

ordering >′) of the subalgebra F [Γ Φ].

The question is when does the composition commute with Sagbi basis computa-

tion? if > and >′are the same monomial orderings the complete answer is given in

[30].

Theorem 4.1.2. ([30], Theorem 2) The Composition by Φ commutes with Sagbi basis

computation if and only if the composition by Φ is compatible with >.

Now we investigate the answer of our question in case that >′is possibly different

from >. Recall the notation of LM>(Φ) from section 1.5 :

LM>(Φ) = (LM>(Φ1), . . . , LM>(Φn)).

Definition 4.1.3. We say that composition by Φ is compatible with non-equality if

∀ monomials s, t in R satisfy following condition

s = t =⇒ s LM>(Φ) = t LM>(Φ)

Lemma 4.1.3. If the LM>(Φ) is a permuted powering (see Definition 1.5.6) then

the composition by Φ is compatible with non-equality.

61

Proof. Let s = xa1 . . . xan and t = xb1 . . . xbn . As LM>(Φ) is a permuted powering,

therefore we have s LM>(Φ) = xa1λ1

π(1) . . . xanλn

π(n) and t LM>(Φ) = xb1λ1

π(1) . . . xbnλn

π(n) . If

s = t then ai = bi for some i, this implies aiλi = biλi. This shows that s LM>(Φ) =

t LM>(Φ).

From Lemma 1.5.3 we know that if LM>(Φ) is a permuted powering than the

binary relation > Φ (see Definition 1.5.5) is a monomial ordering. In this case we

denote it by >Φ. The next theorem gives to answer of our question.

Theorem 4.1.4. If LM>(Φ) is a permuted powering and S is a Sagbi basis of F [Γ]

w.r.t >Φ then S Φ is a Sagbi basis of F [Γ Φ] w.r.t >.

On the base of following lemma we will give proof of Theorem 4.1.4

Lemma 4.1.5. If LM>(Φ) is a permuted powering and S is a Sagbi basis w.r.t >Φ

then S Φ is a Sagbi basis w.r.t >.

Before proving Lemma 4.1.5 we will give Lemma 4.1.6.

Lemma 4.1.6. Assume that LM>(Φ) is a permuted powering. For some S-monomials

Sa, Sb, if ((S Φ)a, (S Φ)b) is a critical pair of S Φ with respect to > then (Sa, Sb)

is a critical pair of S with respect to >Φ.

Proof. Assume that ((SΦ)a, (SΦ)b) is a critical pair of SΦ with respect to > then

LM>((SΦ)a) = LM>((SΦ)b). Obliviously we have (SΦ)a = SaΦ and (S Φ)b =

Sb Φ. By Lemma 1.5.3 we get LM>Φ(Sa) LM>(Φ) = LM>Φ

(Sb) LM>(Φ). Since

our composition is compatible with non-equality with respect to > (Lemma 4.1.3),

we get LM>Φ(Sa) = LM>Φ

(Sb) i.e (Sa, Sb) is a critical pair of S with respect to >Φ.

62

Now we give the proof of Lemma 4.1.5

Proof. Assume that LM>(Φ) is a permuted powering. Let S be a Sagbi basis with

respect to >Φ. We have to prove that S Φ is a Sagbi basis with respect to >. We

will use theorem 4.1.1. Let ((S Φ)a, (S Φ)b) be an arbitrary critical pair of S Φ

with respect to >. From Lemma 4.1.6, we know that (Sa, Sb) is a critical pair of S

with respect to >Φ. Since S is a Sagbi basis with respect to >Φ, by Theorem 4.1.1

we can write

Sa − cSb =∑i

ciSai (or zero) where c, ci ∈ F and (4.1.1)

LM>Φ(Sa) = LM>Φ

(Sb) >Φ LM>Φ(Sai) ∀i (4.1.2)

Composing the equation 4.1.1 with Φ and using Proposition 1.5.1 we get

(S Φ)a − c(S Φ)b =∑i

ci(S Φ)ai (or zero) (4.1.3)

Using the definition of relation >Φ the inequality in equation 4.1.2 becomes

LM>Φ(Sa) LM>(Φ) = LM>Φ

(Sb) LM>(Φ) > LM>Φ(Sai) LM>(Φ) ∀i

Using Lemma 1.5.3, this becomes

LM>((S Φ)a) = LM>((S Φ)b) > LM>((S Φ)ai) ∀i (4.1.4)

The leading terms of the left-hand side of 4.1.3 cancel. Thus 4.1.3 and 4.1.4

together give a representation as in Theorem 4.1.1, and since the critical pair ((S

Φ)a, (S Φ)b) of S Φ w.r.t > was arbitrary therefore S Φ is a Sagbi basis with

respect to >.

63

Lemma 4.1.7. F [S] = F [Γ] ⇒ F [S Φ] = F [Γ Φ].

Proof. Assume that F [S] = F [Γ]. First we will show that F [S Φ] ⊆ F [Γ Φ].

Let ζ ∈ F [S Φ] then

ζ =∑i

ci(S Φ)ai , ci ∈ F

After rewriting it becomes

ζ =∑i

ciSai Φ (4.1.5)

We have Sai ∈ F [S] = F [Γ] therefore

Sai =∑j

djΓai,j , dj ∈ F (4.1.6)

Putting 4.1.5 and 4.1.6 together, we obtain

ζ =∑i

ci(∑j

djΓai,j) Φ

By rewriting, we obtain

ζ =∑i

∑j

cidj(Γ Φ)ai,j

Therefore we get ζ ∈ F [Γ Φ]. Thus we have F [S Φ] ⊆ F [Γ Φ].

In the same way, we can prove that F [Γ Φ] ⊆ F [S Φ] by exchanging the roles

of S and Γ.

Lemma 4.1.8. Consider the following statements

(A) S is a Sagbi basis of F [Γ] w.r.t >Φ then S Φ is Sagbi basis of F [Γ Φ] w.r.t >.

(B) S is Sagbi basis w.r.t >Φ then S Φ is Sagbi basis w.r.t >.

Then (B) ⇒(A).

64

Proof. Let S be a Sagbi basis of F [Γ] w.r.t >Φ then we trivially have S is Sagbi

basis w.r.t >Φ . Then from (B), we have, S Φ is Sagbi basis w.r.t >. Since S is

Sagbi basis of F [Γ] w.r.t > therefore F [S] = F [Γ]. Then by Lemma 4.1.7, we have

F [S Φ] = F [Γ Φ]. Therefore, we have S Φ is Sagbi basis of F [Γ Φ] w.r.t >.

From Lemma 4.1.5 and Lemma 4.1.8 we can easily proof Theorem 4.1.4.

4.2 Examples

In this section we illustrate the use of the Theorem 4.1.4 by several examples. First

we give some examples of compositions Φ such that LM>(Φ) is a permuted powering.

Example 4.2.1. (Compositions)

Scaling

Φi = aixi, ci = 0.

Translation

Φi = xi − ci.

Permutation

Φi = xπ(i), where π is permutation of (1, . . . , n).

Powering

Φi = xaii , ai > 0.

Univariate

Φi ∈ F [xi] of degree ai > 0.

General

Φi ∈ R such that LM>(Φi) = xπ(i)ai where π is permutation

65

of (1, . . . , n) and ai > 0.

Theorem 4.1.4 can be apply to the case of finitely generated monomial subalgebras

and symmetric subalgebra.

Example 4.2.2. Let Γ be some set of monomials. We take composition Φ such

that LM>(Φ) is a permuted powering. The set Γ is a Sagbi basis of F [Γ] w.r.t every

monomial ordering, in particular with respect to >Φ . Then from Theorem 4.1.4, ΓΦ

is a Sagbi basis w.r.t to >.

For example, let Γ = x3y, xy2 and Φ = (x2 − 2xy + 7, 2y5 − y2 + 1). Let > be

the lexicographical ordering on Q[x, y] with x > y. We have LM>(Φ) = (x2, y5) is a

permuted powering. Therefore the composed set

Γ Φ = (x2 − 2xy + 7)3(2y5 − y2 + 1), (x2 − 2xy + 7)(2y5 − y2 + 1)

2

is a Sagbi basis w.r.t >.

The case of a symmetric algebra is similar because the set elementary symmetric

polynomials Γ in R is a Sagbi basis of symmetric algebra w.r.t all monomial order-

ings. Therefore for any Φ such that LM>(Φ) is a permuted powering, ΓΦ is a Sagbi

basis with respect >.

Now we give some examples which shows the use of Theorem 4.1.4.

Example 4.2.3. Consider ring Q[x, y, z]. Let

Γ = x2z, y2, xy + y, 2xy2 + y3

Φ = ((x3 + y + z)3, (x+ y2 + z)3, (x+ y + z)3)

66

We want to compute the Sagbi basis of Γ Φ w.r.t to v-degree reverse lexicograph-

ical monomial ordering with v = (2, 3, 6), where x > y > z. The monomial ordering

corresponds to the matrix A

2 3 6

0 0 −1

0 −1 0

First we check that LM>(Φ) = (x9, y6, z3) is a permuted powering. Thus the The-

orem 4.1.4 applies. We can find matrix of monomial ordering >Φ by multiplying the

matrix A with Mat(LM>(Φ))

9 0 0

0 6 0

0 0 3

obtaining a matrix for >Φ.

18 18 18

0 0 −3

0 −6 0

This corresponds to the degree reverse lexicographical ordering. We compute the

Sagbi basis S of Γ w.rt the degree reverse lexicographical ordering, obtaining

S = x2z, y2, xy + y, y3 + 2xy2, 2xy2 + y2

From this we obtain a Sagbi basis of Γ Φ w.r.t the v-degree reverse lexicographical

ordering, simply by composing of S with Φ.

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Next examples shows that the above illustration of the Theorem 4.1.4 can be used

to compute Sagbi basis efficiently. We have done some experiments1 in the computer

algebra system SINGULAR ([12]). Timings were conducted on a Pentium M 2.13

MHz system with 1 GB memory under Windows XP.

For following examples we consider ring Q[x, y, z] and > is the degree reverse

lexicographical ordering.

Example 4.2.4.

Γ = x2, y2, xy + y, 2xy2 + y9

Φ = ((x2 + y), (y2 + z), (x+ z2))

Example 4.2.5.

Γ = x2, x4 + x5 + x6, x7, y2, y3 + x8

Φ = ((x2 + yz)2, (y2 + xz)2, (x+ z2)2)

Example 4.2.6.

Γ = x2z2, y2z2, xyz2 + yz, 2xy2z3 + y7z7

Φ = ((x+ z3)3, (x2 + y)2, (y2 + z))

In examples 4.2.4 and 4.2.5 we obviously see that LM>(Φ) is a permuted powering

so we apply Theorem 4.1.4. Note that Mat(LM>(Φ)) is a diagonal matrix and all

diagonal entries are same so > and >Φ induce same monomial ordering. Therefore

first we compute a Sagbi basis S of Γ w.r.t >Φ and compose it by Φ obtaining Sagbi

basis of S Φ w.r.t >.

1See section 4.3 for the code of the procedures.

68

In example 4.2.6 we have LM>(Φ) = (z9, x4, y2), therefore the LM>(Φ) is a per-

muted powering. By the same process as in example 4.2.2 we obtain the matrix of >Φ

4 2 9

0 −2 0

−4 0 0

The following table compares the the time (in seconds) for the computation of

the Sagbi basis of Γ Φ using Theorem 4.1.4 and the current implementations of the

Sagbi basis algorithm in SINGULAR. In example 4.2.6 the system runs out of memory

Example comp sagbi4.2.4 56 5424.2.5 22 9164.2.6 149 -

Table 4.1:

OpenProblem. When does the composition commutes with s-reduced Sagbi basis

computation? The case of reduced Grobner basis is discuss in [13].

4.3 Implementation in SINGULAR

In this section we will give main procedures which we have implemented in SINGU-

LAR. In this overview we will present these procedures and give concrete SINGULAR

examples to explain their usage. We have implemented three types of procedures:

1) Procedure to check Iterated Powering

For a given list of compositions L, the procedure IP(L) return 1 (resp 0) if the

69

list of leading monomials of L is iterated powering ( resp not).

SINGULAR Procedure:

proc IP(List L)

int n,i,j,p,k,z ;

int n=nvars(basering) ;

ideal m=maxideal(1) ;

ideal I ;

for(z=1; z<=size(L); z++)

I[z]=L[z] ;

ideal J=lead(I) ;

p=1 ;

poly f ;

if (size(std(J))< n | | size(std(J))> n )

p=0 ;

for(i=1; i<=size(I); i++)

f=J[i] ;

for(k=1; k<=n; k++) ;

70

if ( leadmonom(f)/leadmonom(m[k])!=0)

for(j=k+1; j<=n; j++)

if(leadmonom(f)/leadmonom(m[j])!=0)

p=0 ;

break ;

return(p) ;

SINGULAR Example

ring r=0,(x,y,z),dp ;

list L=x+z6,x3+y,y2+z ;

IP(L) ;

1

list L=xz+y2,x3+z3,xy2+z ;

IP(L) ;

71

0

2) Exponent Matrix of the list of the leading monomials

For a given list of compositions L, the procedure IP(L) return the exponent mat-

-rix of the list of the leading monomials of L.

SINGULAR Procedure:

proc mat(list L)

int n,z,i,p ;

ideal J ;

n=nvars(basering) ;

for(z=1; z<=size(L); z++)

J[z]=L[z] ;

intvec v,w,u ;

p=n*n ;

intmat m[n][n] ;

w=leadexp(J[1]) ;

for(i=2; i<=n; i++)

v=leadexp(J[i]) ;

72

w=w,v ;

v=0 ;

m=w[1..p] ;

m=transpose(m) ;

return(m) ;

SINGULAR Example

ring r=0,(x,y,z),dp ;

list L =x+z6,x3+y,y2+z ;

mat(L) ;

0 3 0,

0 0 2,

6 0 0

3) Procedure to Compute Sagbi basis through Composition

For given ideal I ( the generators of ideal define the subalgebra ), list of

compositions L, matrix of monomial ordering M the procedure comp(I,L,M). It

computes the Sagbi basis IL with respect to M using Theorem

4.1.4.

SINGULAR Procedure:

LIB"sagbi.lib" ; // we need this library for "sagbi" procedure

73

proc comp(ideal I, list L, intmat M)

int n=nvars(basering) ;

def bsr=basering ;

ideal J ;

int p,z ;

for(z=1; z<=size(L); z++)

J[z]=L[z] ;

if (size(J)< n)

ERROR("number of polynomials in composition list is not equal

to number of variables in base ring") ;

p=IP(L) ;

if (p==0)

ERROR("composition is not iterated powering") ;

intmat W[n][n]= mat(L) ;

intmat O[n][n]= M*W ;

execute("ring R1=("+charstr(bsr)+"),("+varstr(bsr)+"),(M(O)) ;") ;

ideal H=imap(bsr,I) ;

74

H=sagbi(H,0) ;

setring bsr ;

map phi= R1,J ;

ideal GC=phi(H) ;

return(GC) ;

SINGULAR Example

ring r=0,(x,y,z),dp ;

ideal I=x2z,y2,xy+y,2xy2+y3 ;

list L=x2+y,y2+x,x+z2 ;

intmat M[3][3]=1,1,1,0,0,-1,0,-1,0 ;

comp(I,L,M) ;

[1]=x4z2+x5+2x2yz2+2x3y+y2z2+xy2

[2]=y4+2xy2+x2

[3]=x2y2+x3+y3+xy+y2+x

[4]=x2y4+1/2y6+2x3y2+3/2xy4+y5+x4+3/2x2y2+2xy3+1/2x3+x2y

75

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