Gröbner basis - What, Why and How?

Grobner basis - What, Whyand How?

Tushant Mittal

Agenda

1 Motivational Problems

2 Monomial Ordering

3 Division Algorithm

4 Grobner Basis

5 Buchberger’s Algorithm

6 Complexity

7 Applications

2/18 08/04/2017 Tushant Mittal Indian Institute of Technology

Motivational Problems

Ideal Membership ProblemGiven f ∈ k[x1, x2, · · · xn] and an ideal I =< f1, f2, · · · , fn >, determine iff ∈ I .

Solving Polynomial EquationsFind all solution in kn of a system of polynomial equationsfi (x1, x2, · · · , xn) = 0. In other words, given an ideal I , compute V (I ).

Implicitization ProblemGiven a parametric solution of xi ’s in terms of variables ti i.e.xi = gi (t1, t2, · · · , ti ), find a set of polynomials fi such thatxi ∈ V (< f1, f2, · · · , fn >). It can be easily observed that this isessentially the inverse of the above question i.e given V (I ) compute I .

But an immediate question arises.How do we even store these ideals which are possibly of infinite size ?

Noetherian Ring

A Noetherian ring is a ring that satisfies the ascending chain condition onideals; that is, given any chain of ideals:

I1 ⊆ · · · ⊆ Ik−1 ⊆ Ik ⊆ Ik+1 ⊆ · · ·

there exists an n such that: In = In+1 = · · · In+k ∀k ≥ 0

Equivalently, every ideal I in R is finitely generated, i.e. there existelements a1, ..., an in I such that I =< a1, a2, · · · , an >

Theorem (Hilbert Basis Theorem)

R is Noetherian ⇒ R[x ] is Noetherian

Noetherian Ring

I1 ⊆ · · · ⊆ Ik−1 ⊆ Ik ⊆ Ik+1 ⊆ · · ·

Noetherian Ring

I1 ⊆ · · · ⊆ Ik−1 ⊆ Ik ⊆ Ik+1 ⊆ · · ·

Special Cases

R = k[x ] i.e. n = 1.We know that k[x ] is a PID. Moreover, it is a Euclidean domain andhence, a polynomial g ∈ < f > iff f |g .

Linear Algebra techniques can be used efficiently when the degree of thepolynomials is restricted to 1 irrespective of n.

We will generalize both the idea of division and a basis to solve theproblem for the general case.

Special Cases

Monomial Ordering

We will use the notation xα to represent∏n

i xαii where α = (α1, α2, · · · , αn).

Definition (admissible ordering of monomials)

A total ordering on all monomials is an ordering for which holds:

xα < xβ ⇒ ∀δ: xαxδ < xβxδ.

∀α: 1 < xα.

A few popular orderings are:

1. Lexicographical ordering: In which we compare xα and xβ thus: if thefirst k − 1 indices agree, αi = βi , i ≤ k − 1 and the kth differ, we decidebased on that index αk ≤ βk ⇒ α ≤ β, and the reverse.

2. Graded lexicographical order: in which the order is by the degree of themonomials and ties are broken using lexicographical ordering.

Monomial Ordering

∀α: 1 < xα.

Monomial Ordering

∀α: 1 < xα.

Preliminary Definitions

Let f =∑

i aixαi be a polynomial. Associated with it are the following

definitions

Definition (Multidegree)

multideg(f ) = maxiαi

Definition (Leading Coefficient)

LC(f ) = amultideg(f )

Definition (Leading Monomial)

LM(f ) = xmultideg(f )

Definition (Leading Term)

LT (f ) = LC(f )LT (f )

Let f =∑

definitions

LT (f ) = LC(f )LT (f )

Let f =∑

definitions

LT (f ) = LC(f )LT (f )

Let f =∑

definitions

LT (f ) = LC(f )LT (f )

Let f =∑

definitions

LT (f ) = LC(f )LT (f )

Example

Let f = 7x3y 2z + 2x2yz4 + 9xy 4 + 3yz7 + 2.

Using the lex ordering,

multideg(f ) = (3, 2, 1)

LC(f ) = 7

LM(f ) = x3y 2z

LT (f ) = 7x3y 2z

Whereas using the grlex ordering we would get,

multideg(f ) = (0, 0, 7)

LC(f ) = 3

LM(f ) = yz7

LT (f ) = 3yz7

Example

Let f = 7x3y 2z + 2x2yz4 + 9xy 4 + 3yz7 + 2.

Using the lex ordering,

multideg(f ) = (3, 2, 1)

LC(f ) = 7

LM(f ) = x3y 2z

LT (f ) = 7x3y 2z

Whereas using the grlex ordering we would get,

multideg(f ) = (0, 0, 7)

LC(f ) = 3

LM(f ) = yz7

LT (f ) = 3yz7

Division AlgorithmThe division algorithm is essentially the same as the one in the univariate casebut there is a small change which has to be made. To see this, let us look atan example,

a1 : x + ya2 : 1 r

xy + 1) x2y + xy 2 + y 2

y 2 + 1x2y − x

xy 2 + x + y 2

xy 2 − y

x + y 2 + y → x

y 2 + y

y 2 − 1

1 → x + y

0 → x + y + 1

a1 : x + ya2 : 1 r

xy + 1) x2y + xy 2 + y 2

y 2 + 1

x2y − x

xy 2 + x + y 2

xy 2 − y

x + y 2 + y → x

y 2 + y

y 2 − 1

1 → x + y

0 → x + y + 1

a1 : x + ya2 : 1 r

xy + 1) x2y + xy 2 + y 2

y 2 + 1x2y − x

xy 2 + x + y 2

xy 2 − y

x + y 2 + y → x

y 2 + y

y 2 − 1

1 → x + y

0 → x + y + 1

a1 : x + ya2 : 1 r

xy + 1) x2y + xy 2 + y 2

y 2 + 1x2y − x

xy 2 + x + y 2

xy 2 − y

x + y 2 + y → x

y 2 + y

y 2 − 1

1 → x + y

0 → x + y + 1

a1 : x + ya2 : 1 r

xy + 1) x2y + xy 2 + y 2

y 2 + 1x2y − x

xy 2 + x + y 2

xy 2 − y

x + y 2 + y → x

y 2 + y

y 2 − 1

1 → x + y

0 → x + y + 1

a1 : x + ya2 : 1 r

xy + 1) x2y + xy 2 + y 2

y 2 + 1x2y − x

xy 2 + x + y 2

xy 2 − y

x + y 2 + y

y 2 + y

y 2 − 1

1 → x + y

0 → x + y + 1

a1 : x + ya2 : 1 r

xy + 1) x2y + xy 2 + y 2

y 2 + 1x2y − x

xy 2 + x + y 2

xy 2 − y

x + y 2 + y → x

y 2 + y

y 2 − 1

1 → x + y

0 → x + y + 1

a1 : x + ya2 : 1 r

xy + 1) x2y + xy 2 + y 2

y 2 + 1x2y − x

xy 2 + x + y 2

xy 2 − y

x + y 2 + y → x

y 2 + y

y 2 − 1

1 → x + y

0 → x + y + 1

a1 : x + ya2 : 1 r

xy + 1) x2y + xy 2 + y 2

y 2 + 1x2y − x

xy 2 + x + y 2

xy 2 − y

x + y 2 + y → x

y 2 + y

y 2 − 1

1 → x + y

0 → x + y + 1

a1 : x + ya2 : 1 r

xy + 1) x2y + xy 2 + y 2

y 2 + 1x2y − x

xy 2 + x + y 2

xy 2 − y

x + y 2 + y → x

y 2 + y

y 2 − 1

1 → x + y

0 → x + y + 1

a1 : x + ya2 : 1 r

xy + 1) x2y + xy 2 + y 2

y 2 + 1x2y − x

xy 2 + x + y 2

xy 2 − y

x + y 2 + y → x

y 2 + y

y 2 − 1

1 → x + y

0 → x + y + 1

Division Algorithm

Algorithm 1: Multi Divide(f , f1, f2, · · · fn)

1 a1 := 0; a2 := 0; · · · an := 0; r = 02 p := f3 while p 6= 0 do4 i := 15 divisionoccured := false6 while i ≤ s AND divisionoccured := false do7 if LT (fi )|p then8 ai := ai + LT (p)/LT (fi )9 p := p − (LT (p)/LT (fi ))fi

10 divisionoccured := true

11 else12 i := i + 1

13 if divisionoccured := false then14 r := r + LT (p)15 p := p − LT (p)

16 return a1, a2, · · · , an, r ;

Are we done? NO!!

The natural algorithm to check if f belongs to the ideal generated by fi swould be to check if remainder of f = 0 on division with the basiselements.

Although this gives us a sufficient condition, it is not a necessary one. Tosee this, observe that the output of the algorithm depends on the order ofinput and the ordering used. For example,

Multi Divide(xy 2 − x , xy + 1, y 2 − 1) = (y , 0,−(x + y))

Multi Divide(xy 2 − x , y 2 − 1, xy + 1) = (y 2 − 1, 0, 0)

We want to find a ”good” basis for a given ideal which preserves theproperty that nonzero remainder implies non-membership also called theremainder property

Does such a basis exist ? Is it computable ?

Are we done? NO!!

Although this gives us a sufficient condition, it is not a necessary one. Tosee this, observe that the output of the algorithm depends on the order ofinput and the ordering used.

For example,

Are we done? NO!!

Grobner basis

Definition

Fix a monomial order. A finite subset G = {g1, g2, · · · , gn} of an ideal I issaid to be a Grobner basis (or standard basis) if

< LT (g1), LT (g2) · · · , LT (gn) > = < LT (I ) >

Theorem

Let G be a Grobner basis for an ideal I and let f ∈ k[x1, · · · , xn]. Then there isa unique remainder r on division by G with the following two properties:

1. No term of r is divisible by any of LT (g1), · · · LT (gn).

2. There is g ∈ I such that f = g + r .

Grobner basis

Definition

Fix a monomial order. A finite subset G = {g1, g2, · · · , gn} of an ideal I issaid to be a Grobner basis (or standard basis) if

< LT (g1), LT (g2) · · · , LT (gn) > = < LT (I ) >

Theorem

Let G be a Grobner basis for an ideal I and let f ∈ k[x1, · · · , xn]. Then there isa unique remainder r on division by G with the following two properties:

1. No term of r is divisible by any of LT (g1), · · · LT (gn).

2. There is g ∈ I such that f = g + r .

Syzygy Polynomials

Definition

For two monomials xα, xβ , LCM(xα, xβ) = xγ where γi = max(αi , βi )

Definition

If LCM(LM(f ), LM(G)) = xγ , S-polynomial is defined as,

S(f , g) =xγ

LT (f )f − xγ

LT (g)g

Suppose we have a sum∑n

i=1 ci fi , where ci ∈ k and multideg(fi ) = α. Ifmultideg(

∑ni=1 ci fi ) < α , then

n∑i=1

ci fi =n∑

c ′ijS(fi , fj)

Syzygy Polynomials

Definition

S(f , g) =xγ

LT (f )f − xγ

LT (g)g

n∑i=1

ci fi =n∑

c ′ijS(fi , fj)

Syzygy Polynomials

Definition

S(f , g) =xγ

LT (f )f − xγ

LT (g)g

n∑i=1

ci fi =n∑

c ′ijS(fi , fj)

Buchberger’s Criterion

Theorem (Buchberger ’65)

Let I be a polynomial ideal. Then a basis G = g1, · · · gn for I is a Groebnerbasis for I if and only if for all pairs i 6= j , the remainder on division ofS(gi , gj) by G is zero.

Algorithm 2: Buchberger(F)

1 Start with G:= F2 do3 G ′ := G4 for pair of polynomials f1, f2 ∈ G ′ do5 h := remainder [G , S(f1, f2)]6 if h 6= 0 then7 G = G ∪ {h}

8 while G 6= G ′;9 output G

Buchberger’s Criterion

Theorem (Buchberger ’65)

Let I be a polynomial ideal. Then a basis G = g1, · · · gn for I is a Groebnerbasis for I if and only if for all pairs i 6= j , the remainder on division ofS(gi , gj) by G is zero.

Algorithm 3: Buchberger(F)

1 Start with G:= F2 do3 G ′ := G4 for pair of polynomials f1, f2 ∈ G ′ do5 h := remainder [G , S(f1, f2)]6 if h 6= 0 then7 G = G ∪ {h}

8 while G 6= G ′;9 output G

Using Groner Basis

System of polynomials

- It can be shown that computing Grobner basisusing the lex ordering gives a basis where the variables are eliminatedsuccessively. Also, the order of elimination seems to correspond to theordering of the variables.Example, the Grobner basis corresponding to

I = (x2 + y 2 + z2 − 1, x2 + Z 2 − y , x − z)

G = (x − z ,−y + 2z2, z4 +1

2z2 − 1

The Implicitization Problem Similarly, we can eliminate the t variablesand the rest of the equations define the ideal we require. Example,

I = (t4 − x , t3 − y , t2 − z)

G = {t2 + z , ty − z2, tz − y , x − z2, y 2 − z3}

Thus, (x − z2, y 2 − z3) is the required ideal.

Using Groner Basis

System of polynomials - It can be shown that computing Grobner basisusing the lex ordering gives a basis where the variables are eliminatedsuccessively. Also, the order of elimination seems to correspond to theordering of the variables.

Example, the Grobner basis corresponding to

I = (x2 + y 2 + z2 − 1, x2 + Z 2 − y , x − z)

G = (x − z ,−y + 2z2, z4 +1

2z2 − 1

I = (t4 − x , t3 − y , t2 − z)

G = {t2 + z , ty − z2, tz − y , x − z2, y 2 − z3}

Using Groner Basis

System of polynomials - It can be shown that computing Grobner basisusing the lex ordering gives a basis where the variables are eliminatedsuccessively. Also, the order of elimination seems to correspond to theordering of the variables.Example, the Grobner basis corresponding to

I = (x2 + y 2 + z2 − 1, x2 + Z 2 − y , x − z)

G = (x − z ,−y + 2z2, z4 +1

2z2 − 1

I = (t4 − x , t3 − y , t2 − z)

G = {t2 + z , ty − z2, tz − y , x − z2, y 2 − z3}

Using Groner Basis

I = (x2 + y 2 + z2 − 1, x2 + Z 2 − y , x − z)

G = (x − z ,−y + 2z2, z4 +1

2z2 − 1

The Implicitization Problem

Similarly, we can eliminate the t variablesand the rest of the equations define the ideal we require. Example,

I = (t4 − x , t3 − y , t2 − z)

G = {t2 + z , ty − z2, tz − y , x − z2, y 2 − z3}

Using Groner Basis

I = (x2 + y 2 + z2 − 1, x2 + Z 2 − y , x − z)

G = (x − z ,−y + 2z2, z4 +1

2z2 − 1

The Implicitization Problem Similarly, we can eliminate the t variablesand the rest of the equations define the ideal we require.

Example,

I = (t4 − x , t3 − y , t2 − z)

G = {t2 + z , ty − z2, tz − y , x − z2, y 2 − z3}

Using Groner Basis

I = (x2 + y 2 + z2 − 1, x2 + Z 2 − y , x − z)

G = (x − z ,−y + 2z2, z4 +1

2z2 − 1

I = (t4 − x , t3 − y , t2 − z)

G = {t2 + z , ty − z2, tz − y , x − z2, y 2 − z3}

Complexity

The worst case time complexity of Buchberger’salgorithm is O(22n ) time which restricts its usage.

Ideal membership problem is EXPSPACE-complete[Mayr-Meyer’82]

Polynomial System solving is in PSPACE . [Koll´ar’88,Fitchas-Galligo’90]

However, better algorithms can be constructed for specificpurposes. For example, computing a Grobner basis forthe radical of a zero dimensional Ideal takes randomizedO(d), deterministic O(dn) time. [Lakshman ’90]

Linear Algebra can also be used to compute GrobnerBasis by using Macaulay Matrices [Macaulay 1902].

Faster Algorithms by Jean-Charles Faugere (F4,F5) for acertain (broad) class of systems called regular sequencesin singly exponential time. Quite fast in the general caseas well, used in computer algebra systems.

Complexity

Applications

Effective computation with (holonomic) special functions

Solving Diophantine equations (Pell)

Automated geometry theorem proving.

Coding theory

Signal and image processing

Robotics

Graph coloring problems e.g. Sudoku puzzles

Extrapolating ”missing links” in palaeontology, and phylogenetic treeconstruction

References

Ali Ayad. “A Survey on the Complexity of Solving Algebraic Systems”.In: International Mathematical Forum 5.7 (2010), pp. 333–353.

Donal O’ Shea David Cox John Little. Ideals, Varieties and Algorithms.Springer, 2007.

William Fulton. Algebraic Curves, An Introduction to AlgebraicGeometry. 2008.

Madhu Sudan. “Algebra and Computation”. In: (1998).

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