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Gr obner basis - What, Why and How? Tushant Mittal Agenda - - PowerPoint PPT Presentation

Nov 09, 2022 162 likes •782 views

Gr obner basis - What, Why and How? Tushant Mittal Agenda Motivational Problems 1 2 Monomial Ordering 3 Division Algorithm 4 Gr obner Basis 5 Buchbergers Algorithm 6 Complexity 7 Applications 2/18 08/04/2017 Tushant Mittal Indian

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Gr¨

  • bner basis - What, Why

and How?

Tushant Mittal

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Agenda

1

Motivational Problems

2 Monomial Ordering 3 Division Algorithm 4 Gr¨

  • bner Basis

5 Buchberger’s Algorithm 6 Complexity 7 Applications

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

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Motivational Problems

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

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Motivational Problems

Ideal Membership Problem Given f ∈ k[x1, x2, · · · xn] and an ideal I =< f1, f2, · · · , fn >, determine if f ∈ I. Solving Polynomial Equations Find all solution in kn of a system of polynomial equations fi(x1, x2, · · · , xn) = 0. In other words, given an ideal I, compute V (I).

3/18 08/04/2017 Tushant Mittal Indian Institute of Technology

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Motivational Problems

Ideal Membership Problem Given f ∈ k[x1, x2, · · · xn] and an ideal I =< f1, f2, · · · , fn >, determine if f ∈ I. Solving Polynomial Equations Find all solution in kn of a system of polynomial equations fi(x1, x2, · · · , xn) = 0. In other words, given an ideal I, compute V (I). Implicitization Problem Given 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 that xi ∈ V (< f1, f2, · · · , fn >). It can be easily observed that this is essentially the inverse of the above question i.e given V (I) compute I.

3/18 08/04/2017 Tushant Mittal Indian Institute of Technology

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Motivational Problems

Ideal Membership Problem Given f ∈ k[x1, x2, · · · xn] and an ideal I =< f1, f2, · · · , fn >, determine if f ∈ I. Solving Polynomial Equations Find all solution in kn of a system of polynomial equations fi(x1, x2, · · · , xn) = 0. In other words, given an ideal I, compute V (I). Implicitization Problem Given 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 that xi ∈ V (< f1, f2, · · · , fn >). It can be easily observed that this is essentially 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 ?

3/18 08/04/2017 Tushant Mittal Indian Institute of Technology

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Noetherian Ring

A Noetherian ring is a ring that satisfies the ascending chain condition on ideals; 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

4/18 08/04/2017 Tushant Mittal Indian Institute of Technology

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Noetherian Ring

A Noetherian ring is a ring that satisfies the ascending chain condition on ideals; 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 exist elements a1, ..., an in I such that I =< a1, a2, · · · , an >

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Noetherian Ring

A Noetherian ring is a ring that satisfies the ascending chain condition on ideals; 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 exist elements a1, ..., an in I such that I =< a1, a2, · · · , an > Theorem (Hilbert Basis Theorem) R is Noetherian ⇒ R[x] is Noetherian

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Special Cases

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

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Special Cases

R = k[x] i.e. n = 1. We know that k[x] is a PID. Moreover, it is a Euclidean domain and hence, a polynomial g ∈ < f > iff f |g. Linear Algebra techniques can be used efficiently when the degree of the polynomials is restricted to 1 irrespective of n.

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Special Cases

R = k[x] i.e. n = 1. We know that k[x] is a PID. Moreover, it is a Euclidean domain and hence, a polynomial g ∈ < f > iff f |g. Linear Algebra techniques can be used efficiently when the degree of the polynomials is restricted to 1 irrespective of n. We will generalize both the idea of division and a basis to solve the problem for the general case.

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Monomial Ordering

We will use the notation xα to represent n

i xαi i

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.

6/18 08/04/2017 Tushant Mittal Indian Institute of Technology

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Monomial Ordering

We will use the notation xα to represent n

i xαi i

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 the

first k − 1 indices agree, αi = βi, i ≤ k − 1 and the kth differ, we decide based on that index αk ≤ βk ⇒ α ≤ β, and the reverse.

6/18 08/04/2017 Tushant Mittal Indian Institute of Technology

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Monomial Ordering

We will use the notation xα to represent n

i xαi i

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 the

first k − 1 indices agree, αi = βi, i ≤ k − 1 and the kth differ, we decide based on that index αk ≤ βk ⇒ α ≤ β, and the reverse.

  • 2. Graded lexicographical order: in which the order is by the degree of the

monomials and ties are broken using lexicographical ordering.

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Preliminary Definitions

Let f =

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

definitions

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Preliminary Definitions

Let f =

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

definitions Definition (Multidegree) multideg(f ) = maxiαi

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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 )

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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 )

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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 )

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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

8/18 08/04/2017 Tushant Mittal Indian Institute of Technology

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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

8/18 08/04/2017 Tushant Mittal Indian Institute of Technology

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Division Algorithm

The division algorithm is essentially the same as the one in the univariate case but there is a small change which has to be made. To see this, let us look at an example,

9/18 08/04/2017 Tushant Mittal Indian Institute of Technology

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Division Algorithm

The division algorithm is essentially the same as the one in the univariate case but there is a small change which has to be made. To see this, let us look at an example, a1 : x + y a2 : 1 r xy + 1 ) x2y + xy 2 + y 2 y 2 + 1

9/18 08/04/2017 Tushant Mittal Indian Institute of Technology

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Division Algorithm

The division algorithm is essentially the same as the one in the univariate case but there is a small change which has to be made. To see this, let us look at an example, a1 : x + y a2 : 1 r xy + 1 ) x2y + xy 2 + y 2 y 2 + 1 x2y − x

9/18 08/04/2017 Tushant Mittal Indian Institute of Technology

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Division Algorithm

The division algorithm is essentially the same as the one in the univariate case but there is a small change which has to be made. To see this, let us look at an example, a1 : x + y a2 : 1 r xy + 1 ) x2y + xy 2 + y 2 y 2 + 1 x2y − x xy 2 + x + y 2

9/18 08/04/2017 Tushant Mittal Indian Institute of Technology

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Division Algorithm

The division algorithm is essentially the same as the one in the univariate case but there is a small change which has to be made. To see this, let us look at an example, a1 : x + y a2 : 1 r xy + 1 ) x2y + xy 2 + y 2 y 2 + 1 x2y − x xy 2 + x + y 2 xy 2 − y

9/18 08/04/2017 Tushant Mittal Indian Institute of Technology

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Division Algorithm

The division algorithm is essentially the same as the one in the univariate case but there is a small change which has to be made. To see this, let us look at an example, a1 : x + y a2 : 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

9/18 08/04/2017 Tushant Mittal Indian Institute of Technology

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Division Algorithm

The division algorithm is essentially the same as the one in the univariate case but there is a small change which has to be made. To see this, let us look at an example, a1 : x + y a2 : 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

9/18 08/04/2017 Tushant Mittal Indian Institute of Technology

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Division Algorithm

The division algorithm is essentially the same as the one in the univariate case but there is a small change which has to be made. To see this, let us look at an example, a1 : x + y a2 : 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

9/18 08/04/2017 Tushant Mittal Indian Institute of Technology

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Division Algorithm

The division algorithm is essentially the same as the one in the univariate case but there is a small change which has to be made. To see this, let us look at an example, a1 : x + y a2 : 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 y + 1

9/18 08/04/2017 Tushant Mittal Indian Institute of Technology

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Division Algorithm

The division algorithm is essentially the same as the one in the univariate case but there is a small change which has to be made. To see this, let us look at an example, a1 : x + y a2 : 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 y + 1 1 → x + y

9/18 08/04/2017 Tushant Mittal Indian Institute of Technology

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Division Algorithm

The division algorithm is essentially the same as the one in the univariate case but there is a small change which has to be made. To see this, let us look at an example, a1 : x + y a2 : 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 y + 1 1 → x + y → x + y + 1

9/18 08/04/2017 Tushant Mittal Indian Institute of Technology

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Division Algorithm

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

1 a1 := 0; a2 := 0; · · · an := 0; r = 0 2 p := f 3 while p = 0 do 4

i := 1

5

divisionoccured := false

6

while i ≤ s AND divisionoccured := false do

7

if LT(fi)|p then

8

ai := ai + LT(p)/LT(fi)

9

p := p − (LT(p)/LT(fi))fi

10

divisionoccured := true

11

else

12

i := i + 1

13

if divisionoccured := false then

14

r := r + LT(p)

15

p := p − LT(p)

16 return a1, a2, · · · , an, r;

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Are we done? NO!!

The natural algorithm to check if f belongs to the ideal generated by fis would be to check if remainder of f = 0 on division with the basis elements.

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Are we done? NO!!

The natural algorithm to check if f belongs to the ideal generated by fis would be to check if remainder of f = 0 on division with the basis elements. Although this gives us a sufficient condition, it is not a necessary one. To see this, observe that the output of the algorithm depends on the order of input and the ordering used.

11/18 08/04/2017 Tushant Mittal Indian Institute of Technology

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Are we done? NO!!

The natural algorithm to check if f belongs to the ideal generated by fis would be to check if remainder of f = 0 on division with the basis elements. Although this gives us a sufficient condition, it is not a necessary one. To see this, observe that the output of the algorithm depends on the order of input 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)

11/18 08/04/2017 Tushant Mittal Indian Institute of Technology

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Are we done? NO!!

The natural algorithm to check if f belongs to the ideal generated by fis would be to check if remainder of f = 0 on division with the basis elements. Although this gives us a sufficient condition, it is not a necessary one. To see this, observe that the output of the algorithm depends on the order of input 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 the property that nonzero remainder implies non-membership also called the remainder property

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Are we done? NO!!

The natural algorithm to check if f belongs to the ideal generated by fis would be to check if remainder of f = 0 on division with the basis elements. Although this gives us a sufficient condition, it is not a necessary one. To see this, observe that the output of the algorithm depends on the order of input 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 the property that nonzero remainder implies non-membership also called the remainder property Does such a basis exist ? Is it computable ?

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Gr¨

  • bner basis

Definition Fix a monomial order. A finite subset G = {g1, g2, · · · , gn} of an ideal I is said to be a Gr¨

  • bner basis (or standard basis) if

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

12/18 08/04/2017 Tushant Mittal Indian Institute of Technology

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Gr¨

  • bner basis

Definition Fix a monomial order. A finite subset G = {g1, g2, · · · , gn} of an ideal I is said to be a Gr¨

  • bner basis (or standard basis) if

< LT(g1), LT(g2) · · · , LT(gn) > = < LT(I) > Theorem Let G be a Gr¨

  • bner basis for an ideal I and let f ∈ k[x1, · · · , xn]. Then there is

a 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.

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Syzygy Polynomials

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

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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

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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 Lemma Suppose we have a sum n

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

multideg(n

i=1 cifi) < α , then n

  • i=1

cifi =

n

  • i=1

c′

ijS(fi, fj)

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Buchberger’s Criterion

Theorem (Buchberger ’65) Let I be a polynomial ideal. Then a basis G = g1, · · · gn for I is a Gr¨

  • ebner

basis for I if and only if for all pairs i = j , the remainder on division of S(gi, gj) by G is zero.

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Buchberger’s Criterion

Theorem (Buchberger ’65) Let I be a polynomial ideal. Then a basis G = g1, · · · gn for I is a Gr¨

  • ebner

basis for I if and only if for all pairs i = j , the remainder on division of S(gi, gj) by G is zero. Algorithm 3: Buchberger(F)

1 Start with G:= F 2 do 3

G ′ := G

4

for pair of polynomials f1, f2 ∈ G ′ do

5

h := remainder[G, S(f1, f2)]

6

if h = 0 then

7

G = G ∪ {h}

8 while G = G ′; 9 output G

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Using Gr¨

  • ner Basis

System of polynomials

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Using Gr¨

  • ner Basis

System of polynomials - It can be shown that computing Gr¨

  • bner basis

using the lex ordering gives a basis where the variables are eliminated

  • successively. Also, the order of elimination seems to correspond to the
  • rdering of the variables.

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Using Gr¨

  • ner Basis

System of polynomials - It can be shown that computing Gr¨

  • bner basis

using the lex ordering gives a basis where the variables are eliminated

  • successively. Also, the order of elimination seems to correspond to the
  • rdering of the variables.Example, the Gr¨
  • bner basis corresponding to

I = (x2 + y 2 + z2 − 1, x2 + Z 2 − y, x − z) G = (x − z, −y + 2z2, z4 + 1 2z2 − 1 4)

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Using Gr¨

  • ner Basis

System of polynomials - It can be shown that computing Gr¨

  • bner basis

using the lex ordering gives a basis where the variables are eliminated

  • successively. Also, the order of elimination seems to correspond to the
  • rdering of the variables.Example, the Gr¨
  • bner basis corresponding to

I = (x2 + y 2 + z2 − 1, x2 + Z 2 − y, x − z) G = (x − z, −y + 2z2, z4 + 1 2z2 − 1 4) The Implicitization Problem

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Using Gr¨

  • ner Basis

System of polynomials - It can be shown that computing Gr¨

  • bner basis

using the lex ordering gives a basis where the variables are eliminated

  • successively. Also, the order of elimination seems to correspond to the
  • rdering of the variables.Example, the Gr¨
  • bner basis corresponding to

I = (x2 + y 2 + z2 − 1, x2 + Z 2 − y, x − z) G = (x − z, −y + 2z2, z4 + 1 2z2 − 1 4) The Implicitization Problem Similarly, we can eliminate the t variables and the rest of the equations define the ideal we require.

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SLIDE 52

Using Gr¨

  • ner Basis

System of polynomials - It can be shown that computing Gr¨

  • bner basis

using the lex ordering gives a basis where the variables are eliminated

  • successively. Also, the order of elimination seems to correspond to the
  • rdering of the variables.Example, the Gr¨
  • bner basis corresponding to

I = (x2 + y 2 + z2 − 1, x2 + Z 2 − y, x − z) G = (x − z, −y + 2z2, z4 + 1 2z2 − 1 4) The Implicitization Problem Similarly, we can eliminate the t variables and 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.

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Complexity

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

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Complexity

The worst case time complexity of Buchberger’s algorithm is O(22n) time which restricts its usage. Ideal membership problem is EXPSPACE-complete [Mayr-Meyer’82]

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Complexity

The worst case time complexity of Buchberger’s algorithm 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]

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Complexity

The worst case time complexity of Buchberger’s algorithm 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 specific

  • purposes. For example, computing a Gr¨
  • bner basis for

the radical of a zero dimensional Ideal takes randomized O(d), deterministic O(dn) time. [Lakshman ’90]

16/18 08/04/2017 Tushant Mittal Indian Institute of Technology

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Complexity

The worst case time complexity of Buchberger’s algorithm 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 specific

  • purposes. For example, computing a Gr¨
  • bner basis for

the radical of a zero dimensional Ideal takes randomized O(d), deterministic O(dn) time. [Lakshman ’90] Linear Algebra can also be used to compute Gr¨

  • bner

Basis by using Macaulay Matrices [Macaulay 1902].

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Complexity

The worst case time complexity of Buchberger’s algorithm 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 specific

  • purposes. For example, computing a Gr¨
  • bner basis for

the radical of a zero dimensional Ideal takes randomized O(d), deterministic O(dn) time. [Lakshman ’90] Linear Algebra can also be used to compute Gr¨

  • bner

Basis by using Macaulay Matrices [Macaulay 1902]. Faster Algorithms by Jean-Charles Faug´ ere (F4, F5) for a certain (broad) class of systems called regular sequences in singly exponential time. Quite fast in the general case as well, used in computer algebra systems.

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SLIDE 59

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 tree construction

17/18 08/04/2017 Tushant Mittal Indian Institute of Technology

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SLIDE 60

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 Algebraic

  • Geometry. 2008.

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

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