gröbner basis - what, why and how?
TRANSCRIPT
![Page 1: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/1.jpg)
Grobner basis - What, Whyand How?
Tushant Mittal
![Page 2: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/2.jpg)
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
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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 ?
3/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 4: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/4.jpg)
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 ?
3/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 5: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/5.jpg)
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 ?
3/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 6: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/6.jpg)
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 ?
3/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 7: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/7.jpg)
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
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 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
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 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
4/18 08/04/2017 Tushant Mittal Indian Institute of Technology
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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 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.
5/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 11: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/11.jpg)
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.
5/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 12: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/12.jpg)
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.
5/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 13: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/13.jpg)
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.
6/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 14: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/14.jpg)
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.
6/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 15: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/15.jpg)
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.
6/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 16: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/16.jpg)
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 )
7/18 08/04/2017 Tushant Mittal Indian Institute of Technology
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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 )
7/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 18: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/18.jpg)
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 )
7/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 19: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/19.jpg)
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 )
7/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 20: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/20.jpg)
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 )
7/18 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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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 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
y + 1
1 → x + y
0 → x + y + 1
9/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 24: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/24.jpg)
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 + 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
0 → x + y + 1
9/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 25: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/25.jpg)
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
y + 1
1 → x + y
0 → x + y + 1
9/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 26: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/26.jpg)
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
y + 1
1 → x + y
0 → x + y + 1
9/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 27: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/27.jpg)
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
y + 1
1 → x + y
0 → x + y + 1
9/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 28: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/28.jpg)
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
y + 1
1 → x + y
0 → x + y + 1
9/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 29: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/29.jpg)
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
y + 1
1 → x + y
0 → x + y + 1
9/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 30: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/30.jpg)
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
y + 1
1 → x + y
0 → x + y + 1
9/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 31: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/31.jpg)
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
y + 1
1 → x + y
0 → x + y + 1
9/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 32: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/32.jpg)
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
y + 1
1 → x + y
0 → x + y + 1
9/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 33: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/33.jpg)
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
y + 1
1 → x + y
0 → x + y + 1
9/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 34: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/34.jpg)
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 ;
10/18 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 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 ?
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 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 ?
11/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 37: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/37.jpg)
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 ?
11/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 38: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/38.jpg)
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 ?
11/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 39: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/39.jpg)
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 ?
11/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 40: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/40.jpg)
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 .
12/18 08/04/2017 Tushant Mittal Indian Institute of Technology
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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 .
12/18 08/04/2017 Tushant Mittal Indian Institute of Technology
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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 ci fi , where ci ∈ k and multideg(fi ) = α. Ifmultideg(
∑ni=1 ci fi ) < α , then
n∑i=1
ci fi =n∑
i=1
c ′ijS(fi , fj)
13/18 08/04/2017 Tushant Mittal Indian Institute of Technology
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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 ci fi , where ci ∈ k and multideg(fi ) = α. Ifmultideg(
∑ni=1 ci fi ) < α , then
n∑i=1
ci fi =n∑
i=1
c ′ijS(fi , fj)
13/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 44: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/44.jpg)
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 ci fi , where ci ∈ k and multideg(fi ) = α. Ifmultideg(
∑ni=1 ci fi ) < α , then
n∑i=1
ci fi =n∑
i=1
c ′ijS(fi , fj)
13/18 08/04/2017 Tushant Mittal Indian Institute of Technology
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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 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
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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 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
14/18 08/04/2017 Tushant Mittal Indian Institute of Technology
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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
4)
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.
15/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 48: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/48.jpg)
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
4)
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.
15/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 49: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/49.jpg)
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
4)
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.
15/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 50: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/50.jpg)
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
4)
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.
15/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 51: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/51.jpg)
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
4)
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.
15/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 52: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/52.jpg)
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
4)
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.
15/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 53: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/53.jpg)
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.
16/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 54: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/54.jpg)
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.
16/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 55: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/55.jpg)
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.
16/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 56: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/56.jpg)
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.
16/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 57: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/57.jpg)
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.
16/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 58: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/58.jpg)
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.
16/18 08/04/2017 Tushant Mittal Indian Institute of Technology
![Page 59: Gröbner basis - What, Why and How?](https://reader030.vdocument.in/reader030/viewer/2022012107/61dc5550670ee3555b0c6774/html5/thumbnails/59.jpg)
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
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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).
18/18 08/04/2017 Tushant Mittal Indian Institute of Technology