Andreas Steenpass Workshop on Computational Commutative ...math.ipm.ac.ir/conferences/2011/cca2011/SlidesShow-Steenpass.pdf · > NF(g, J); 0 // g is in I Andreas Steenpass Singular

Singular Tutorial

Andreas Steenpass

TU Kaiserslautern

Workshop on Computational Commutative AlgebraIPM, Tehran, IranJuly 2-7, 2011

Table of Contents

1 Getting started with Singular by examples

Andreas Steenpass Singular Tutorial

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2 Basic concepts of the mathematical background


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3 Singular as a Software: Ressources and the Singular Community


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4 Challenges in Computer Algebra


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5 Singular’s modstd.lib


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6 Schreyer’s Algorithm for Syzygies


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7 Sudoku


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7 Sudoku

8 Primary Decomposition with Singular


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7 Sudoku



A first Singular session

Let’s assume that you have installed Singular successfully andthat you can start a Singular session:

SINGULAR / Development

A Computer Algebra System for Polynomial Computations / version 3-1-3

0<

by: W. Decker, G.-M. Greuel, G. Pfister, H. Schoenemann \ March 2011

FB Mathematik der Universitaet, D-67653 Kaiserslautern \

// ** executing /home/steenpas/Singular/trunk/Singular/LIB/.singularrc

>


Rings and Ideals

Almost all Singular functions are ring-dependend, so first of all,Singular needs to know which ring you want to compute in.For Q[x , y , z ] with the lexicographic ordering, simply type

> ring R = 0, (x,y,z), lp;


Rings and Ideals

Almost all Singular functions are ring-dependend, so first of all,Singular needs to know which ring you want to compute in.For Q[x , y , z ] with the lexicographic ordering, simply type

> ring R = 0, (x,y,z), lp;

An ideal in Singular is given by its generators, so for〈x + y + z − 1, x2 + y2 + z2− 1, x3 + y3 + z3− 1〉 ⊂ Q[x , y , z ]we type

> ideal I = x+y+z-1, x2+y2+z2-1, x3+y3+z3-1;


The groebner command

Now compute a Groebner basis of this ideal:

> groebner(I);

_[1]=z3-z2

_[2]=y2+yz-y+z2-z

_[3]=x+y+z-1




> groebner(I);

_[1]=z3-z2

_[2]=y2+yz-y+z2-z

_[3]=x+y+z-1

In the first equation of the new system, the variables x and y areeliminated. In the second equation, x is eliminated. As aconsequence, the solutions can, now, be directly read off:

(1, 0, 0), (0, 1, 0), (0, 0, 1).




> groebner(I);

_[1]=z3-z2

_[2]=y2+yz-y+z2-z

_[3]=x+y+z-1

In the first equation of the new system, the variables x and y areeliminated. In the second equation, x is eliminated. As aconsequence, the solutions can, now, be directly read off:

(1, 0, 0), (0, 1, 0), (0, 0, 1).

Note: Groebner bases computations can produce unexpectedlylarge output!


intersect

Many other computations in modern computer algebra rely onGroebner bases and their several generalisations. An example isintersect:

> ring R = 0, (x,y,z), dp;

> ideal I1 = x, y;

> ideal I2 = y2, z;

> intersect(I1, I2);

_[1]=yz

_[2]=xz

_[3]=y2


factorize

Other algorithms such as factorization do not depend on Groebnerbases. We use Singular to factorize a polynomial in Q[x , y , z ].The second entry of the output indicates the multiplicities of thefactors:

> ring R = 0, (x,y,z), dp;

> poly f = x5y4-x4y5+2x3y6-2x2y7+xy8-y9+x8z-2x6y2z-7x4y4z-4x2y6z

. +2x3y4z2-2x2y5z2+2xy6z2-2y7z2+2x6z3-6x4y2z3-8x2y4z3+xy4z4-y5z4

. +x4z5-4x2y2z5;

> factorize(f);

[1]:

_[1]=1

_[2]=xy4-y5+x4z-4x2y2z

_[3]=x2+y2+z2

[2]:

1,1,2


factorize

Other algorithms such as factorization do not depend on Groebnerbases. We use Singular to factorize a polynomial in Q[x , y , z ].The second entry of the output indicates the multiplicities of thefactors:

> ring R = 0, (x,y,z), dp;

> poly f = x5y4-x4y5+2x3y6-2x2y7+xy8-y9+x8z-2x6y2z-7x4y4z-4x2y6z

. +2x3y4z2-2x2y5z2+2xy6z2-2y7z2+2x6z3-6x4y2z3-8x2y4z3+xy4z4-y5z4

. +x4z5-4x2y2z5;

> factorize(f);

[1]:

_[1]=1

_[2]=xy4-y5+x4z-4x2y2z

_[3]=x2+y2+z2

[2]:

1,1,2

Note: factorize does not even depend on the monomial ordering.


Ideal Membership

Given an ideal I and a polynomial f , Groebner (or more general:standard) bases can be used to decide wether or not it f iscontained in I .


Ideal Membership

Given an ideal I and a polynomial f , Groebner (or more general:standard) bases can be used to decide wether or not it f iscontained in I .In terms of Singular code, the following holds:The polynomial f is contained in I if and only if NF(f, std(I));

evaluates to 0.


Ideal Membership

Given an ideal I and a polynomial f , Groebner (or more general:standard) bases can be used to decide wether or not it f iscontained in I .In terms of Singular code, the following holds:The polynomial f is contained in I if and only if NF(f, std(I));

evaluates to 0.

> ring R = 0, (x, y), dp;

> ideal I = x10+x9y2, y8-x2y7;

> ideal J = std(I);

> poly f = x2y7+y14;

> poly g = xy13+y12;

> NF(f, J);

-xy12+y8 // f is not in I

> NF(g, J);

0 // g is in I


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Groebner Bases

What is a Groebner Basis?


Groebner Bases


Use monomial orders on K [x1, . . . , xn] which are well-orders.


Groebner Bases


Use monomial orders on K [x1, . . . , xn] which are well-orders.For instance:

xα >lp xβ ⇐⇒ the first nonzero entry of α− β is positive

(lexicographic order).


Groebner Bases


Use monomial orders on K [x1, . . . , xn] which are well-orders.For instance:

xα >lp xβ ⇐⇒ the first nonzero entry of α− β is positive

(lexicographic order). Or:

xα >dp xβ ⇐⇒ deg xα > deg xβ, or (deg xα = deg xβ and thelast nonzero entry of α− β ∈ Zn is negative).

(degree reverse lexicographic order).


Groebner Bases

Given a monomial order on K [x1, . . . , xn], the terms of everypolynomial f ∈ K [x1, . . . , xn] are sorted accordingly.


Groebner Bases

Given a monomial order on K [x1, . . . , xn], the terms of everypolynomial f ∈ K [x1, . . . , xn] are sorted accordingly. For instance:

f = xz + y2 + yz (lp)

orf = y2 + xz + yz . (dp)

We can, then, speak of the leading term L(f ).


Groebner Bases

Given a monomial order on K [x1, . . . , xn], the terms of everypolynomial f ∈ K [x1, . . . , xn] are sorted accordingly. For instance:

f = xz + y2 + yz (lp)

orf = y2 + xz + yz . (dp)

We can, then, speak of the leading term L(f ).

Given an ideal

I = 〈f1, . . . , fr 〉 =

{

r∑

i=1

gi fi | gi ∈ K [x1, . . . , xn]

}

,

we have the leading ideal

L(I ) :=⟨

L(f )∣

∣ f ∈ I⟩

.


Groebner Bases

A set g1, . . . , gs ∈ I of polynomials is called a Groebner basisfor I if the leading terms L(gi ) generate L(I ).


Groebner Bases


The use made of Groebner bases relies on the following two”facts”:


Groebner Bases



L(I ) carries plenty of information on I .


Groebner Bases




Information on I is typically obtained by purely combinatorialmeans.


Groebner Bases




Information on I is typically obtained by purely combinatorialmeans.

For instance, due to a classical result of Macaulay, the monomialsnot contained in L(I ) define a K -basis of the quotient ringK [x1, . . . , xn]/I .


Buchberger’s criterion

Definition

Let f , g ∈ K [x1, . . . , xn] be polynomials and let lcm be the leastcommon multiple of their leading monomials LM(f ) and LM(g).We define the s-polynomial of f and g to be

spoly(f , g) :=lcm

LM(f )· f −

LC(f )

LC(g)·

lcm

LM(g)· g .



Definition


spoly(f , g) :=lcm

LM(f )· f −

LC(f )

LC(g)·

lcm

LM(g)· g .

Theorem (Buchberger)

F := (f1, . . . , fm) ⊂ K [x1, . . . , xn] \ {0} is a Groebner basis of theideal I := 〈f1, . . . , fm〉 iff all s-polynomials spoly(fi , fj) reduce to 0w. r. t. F .



Definition


spoly(f , g) :=lcm

LM(f )· f −

LC(f )

LC(g)·

lcm

LM(g)· g .

Theorem (Buchberger)

F := (f1, . . . , fm) ⊂ K [x1, . . . , xn] \ {0} is a Groebner basis of theideal I := 〈f1, . . . , fm〉 iff all s-polynomials spoly(fi , fj) reduce to 0w. r. t. F . (i. e. their remainder is 0 when divided by F .)


Buchberger’s algorithm

Based on Buchberger’s criterion, there is an rather obviousalgorithm to compute Groebner bases:




Algorithm

Input: A set F := (f1, . . . , fm) ⊂ K [x1, . . . , xn] \ {0} of polys.




Algorithm

Input: A set F := (f1, . . . , fm) ⊂ K [x1, . . . , xn] \ {0} of polys.Output: A Groebner basis G = (f1, . . . , fm, fm+1, . . . , fr ) of theideal I := 〈f1, . . . , fm〉.




Algorithm


1 Compute all s-polynomials spoly(fi , fj) for fi , fj ∈ F .




Algorithm



2 Compute the remainders of these s-polys when divided by F .




Algorithm




3 If there are non-zero remainders, setF := F ∪ {non-zero remainders} and continue with step 1.




Algorithm





4 If all the s-polynomials reduce to zero, set G := F .




Algorithm





4 If all the s-polynomials reduce to zero, set G := F .

This process stops due to K [x1, . . . , xn] being Noetherian.


Normal forms

When dividing a polynomial f by a set of polynomials F as above,the remainder is in general not uniquely determined.


Normal forms

When dividing a polynomial f by a set of polynomials F as above,the remainder is in general not uniquely determined.Groebner bases can help us to overcome this issue:

Proposition

Let I ⊂ K [x1, . . . , xn] be an ideal and letF = (f1, . . . , fm) ⊂ K [x1, . . . , xn] be a Groebner basis of I . Thenfor any polynomial f ∈ K [x1, . . . , xn], division by F yields auniquely determined remainder r ∈ K [x1, . . . , xn] with f − r ∈ I .


Normal Forms

Definition

Let R := K [x1, . . . , xn] and let G be the set of all finite sets G ⊂ R .


Normal Forms

Definition


NF : R × G → R , (f ,G ) 7→ NF(f |G ) ,

is called a normal form on R if, for all f ∈ R and all G ∈ G,


Normal Forms

Definition


NF : R × G → R , (f ,G ) 7→ NF(f |G ) ,


1 NF(0|G ) = 0,


Normal Forms

Definition


NF : R × G → R , (f ,G ) 7→ NF(f |G ) ,


1 NF(0|G ) = 0,

2 NF(f |G ) 6= 0 ⇒ LM(NF(f |G )) /∈ L(G ).


Normal Forms

Definition


NF : R × G → R , (f ,G ) 7→ NF(f |G ) ,


1 NF(0|G ) = 0,

2 NF(f |G ) 6= 0 ⇒ LM(NF(f |G )) /∈ L(G ).

3 If G = {g1, . . . , gs}, then f − NF(f |G ) has a standardrepresentation with respect to G , that is,

f − NF(f |G ) =

s∑

i=1

aigi , ai ∈ R , s ≥ 0 ,


Normal Forms

Definition


NF : R × G → R , (f ,G ) 7→ NF(f |G ) ,


1 NF(0|G ) = 0,

2 NF(f |G ) 6= 0 ⇒ LM(NF(f |G )) /∈ L(G ).

3 If G = {g1, . . . , gs}, then f − NF(f |G ) has a standardrepresentation with respect to G , that is,

f − NF(f |G ) =

s∑

i=1

aigi , ai ∈ R , s ≥ 0 ,

satisfying LM(∑s

i=1 aigi ) ≥ LM(aigi ) for all i with aigi 6= 0.Andreas Steenpass Singular Tutorial

Normal Forms

Thus in the above definition, the remainder r is a normal form of fon R .


Normal Forms

Thus in the above definition, the remainder r is a normal form of fon R .Note, however, that in general there do exist several normal formson R .


Normal Forms


Corollary

It follows directly from the definition that for any Groebner basis Git holds

f ∈ 〈G 〉 ⇔ NF(f |G ) = 0.


Normal Forms


Corollary

It follows directly from the definition that for any Groebner basis Git holds

f ∈ 〈G 〉 ⇔ NF(f |G ) = 0.

We have thus solved the ideal membership problem.


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Singular: Basic Facts

Singular is a computer algebra system for polynomialcomputations, with special emphasis on commutative andnon-commutative algebra, algebraic geometry, and singularitytheory.



Singular is a computer algebra system for polynomialcomputations, with special emphasis on commutative andnon-commutative algebra, algebraic geometry, and singularitytheory.It is free and open-source under the GNU General Public Licence.




Singular consists of

a kernel, written in C/C++, and containing the corealgorithms,






libraries, written in Singular’s programming language withC-like syntax,






libraries, written in Singular’s programming language withC-like syntax, which have greatly augmented the kernelfunctionality, and which make Singular user-extendible,






libraries, written in Singular’s programming language withC-like syntax, which have greatly augmented the kernelfunctionality, and which make Singular user-extendible,

a comprehensive online manual and help function.


Singular: Capabilities

Singular’s main computational objects are ideals and modulesover a large number of baserings. These include

polynomial rings over various ground fields and a few rings(including the integers),





localizations of the above,






a very general class of non-commutative algebras (includingthe exterior algebra and the Weyl algebra),






a very general class of non-commutative algebras (includingthe exterior algebra and the Weyl algebra),

quotient rings of the above.


Singular Developer Teams


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Computer Algebra

Computer algebra systems such as CoCoA or Singular can be,and in fact they are, used quite successfully for theoreticalmathematical research.


Computer Algebra

Computer algebra systems such as CoCoA or Singular can be,and in fact they are, used quite successfully for theoreticalmathematical research.There are also a lot of other scientific disciplines which make useof computational algebra such as e.g. phylogenetics, computerchip design, or cryptography.


Computer Algebra

Computer algebra systems such as CoCoA or Singular can be,and in fact they are, used quite successfully for theoreticalmathematical research.There are also a lot of other scientific disciplines which make useof computational algebra such as e.g. phylogenetics, computerchip design, or cryptography.The computational problems which arise from these areas can bequite challenging; some of them even could not yet be solved byany software on any computer.


Computer Algebra

What is needed to tackle these challenges are not (only) morepowerful computers and more involved implementations of thesame algorithms, but new, more advanced (mathematical)algorithms. This makes computer algebra a branch of mathematicsitself.


Computer Algebra

What is needed to tackle these challenges are not (only) morepowerful computers and more involved implementations of thesame algorithms, but new, more advanced (mathematical)algorithms. This makes computer algebra a branch of mathematicsitself.E.g., in order to make use of computer clusters or modernprocessors which typically contain of several cores, we have todesign parallel algorithms. In the important case of Groebnerbases, this is not at all a trivial task because parallelimplementations of Buchberger’s algorithm are quite inefficient.


Computer Algebra

What is needed to tackle these challenges are not (only) morepowerful computers and more involved implementations of thesame algorithms, but new, more advanced (mathematical)algorithms. This makes computer algebra a branch of mathematicsitself.E.g., in order to make use of computer clusters or modernprocessors which typically contain of several cores, we have todesign parallel algorithms. In the important case of Groebnerbases, this is not at all a trivial task because parallelimplementations of Buchberger’s algorithm are quite inefficient.One possible way to overcome this problem is to use modulartechniques. If we want to compute a Groebner basis incharacteristic 0, we can compute it modulo several primes first andthen lift the result via chinese remaindering.


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Modular Standard Bases

Idea of modStd to compute a standard basis G of an ideal I :

1 compute standard bases Gp modulo several primes p





2 delete unlucky primes:delete p if LM(Ip) 6= LM(Iq) for most primes q 6= p






3 lift the result via Chinese remainder algorithm and Fareyrational map: obtain G







4 pTestSB:test if (G mod p) is a s.b. of Ip for a new random prime p








5 final verification tests:G is a standard basis of 〈G 〉 and I ⊆ 〈G 〉








5 final verification tests:G is a standard basis of 〈G 〉 and I ⊆ 〈G 〉

6 reiterate with further new primes if verification fails



Verification is the hardest part of the algorithm.Singular’s method for this task is due to the following result.




Theorem (Arnold (homogeneous)/ Idrees, Pfister, Steidel (generalfor global & local orderings))

Let G ⊆ Q[X ] be a set of polynomials such that






LM(G ) = LM(Gp) where Gp is a standard basis of Ip for someprime number p,







G is a standard basis of 〈G 〉,








I ⊆ 〈G 〉.








I ⊆ 〈G 〉.

Then I = 〈G 〉.


Modular Standard Bases and Parallel Computing

Remark

The algorithm modStd can be parallelized in the following way:



Remark


1 Compute the standard bases Gp in parallel.



Remark



2 Parallelize the final verification tests:



Remark




Check if I ⊆ 〈G〉 by checking if f ∈ 〈G〉 for each generator.



Remark




Check if I ⊆ 〈G〉 by checking if f ∈ 〈G〉 for each generator.

Check if G is a standard basis of 〈G〉 by checking if everys–polynomial not excluded by well-known criteria, vanishes byreduction w.r.t. G .



Example std modStd modStd∗4 modStd∗9

cyclic8 - 8271 4120 2927

Paris.ilias13 37734 1159 676 580

homog.cyclic7 3343 3436 886 408

- 6 3 3

Table: Total running times (in sec) for computing a standard basis ofexamples chosen from The SymbolicData Project (H.-G. Grabe) via std,modStd and its parallelized variant modStd∗n for n = 4, 9.



Example std modStd modStd∗4 modStd∗9

cyclic8 - 8271 4120 2927

Paris.ilias13 37734 1159 676 580

homog.cyclic7 3343 3436 886 408

- 6 3 3

Table: Total running times (in sec) for computing a standard basis ofexamples chosen from The SymbolicData Project (H.-G. Grabe) via std,modStd and its parallelized variant modStd∗n for n = 4, 9.

Ref.: N. Idrees, G. Pfister, S. Steidel: Parallelization of ModularAlgorithms. Journal of Symbolic Computation 46, 672-684 (2011).


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Syzygies: Known Facts and Definitions

Given any ring R and elements f1, . . . , fr of an R-module M, asyzygy on f1, . . . , fr is a relation

g1f1 + · · ·+ gr fr = 0 ∈ M,

with g1, . . . , gr ∈ R . We think of such a relation as a columnvector (g1, . . . , gr )

t ∈ R r :



Given any ring R and elements f1, . . . , fr of an R-module M, asyzygy on f1, . . . , fr is a relation

g1f1 + · · ·+ gr fr = 0 ∈ M,

with g1, . . . , gr ∈ R . We think of such a relation as a columnvector (g1, . . . , gr )

t ∈ R r :

Definition. Let R be a ring, let M be an R-module, and letf1, . . . , fr ∈ M. A syzygy on f1, . . . , fr is an element of the kernelof the homomorphism

φ : R r → M, ǫi 7→ fi ,

where {ǫ1, . . . , ǫr} is the canonical basis of R r . We call ker φ the(first) syzygy module of f1, . . . , fr , written

Syz(f1, . . . , fr ) = ker φ.



If Syz(f1, . . . , fr ) is finitely generated, we regard the elements of agiven finite set of generators for it as the columns of a matrixwhich we call a syzygy matrix of f1, . . . , fr .




Note that if R is Noetherian, then every submodule of a finitelygenerated R-module is finitely generated again.





Problem. Determine a syzygy matrix of x , y , z ∈ K [x , y , z ].





Problem. Determine a syzygy matrix of x , y , z ∈ K [x , y , z ].

To handle syzygies over polynomial rings, one has to extend theconcept of Grobner bases to free modules. In what follows, letR = K [x1, . . . , xn], and let F be we the free R-module F = R s

with its canonical basis e1, . . . , es .



Definition. A monomial in F is a monomial in R times a basisvector of F , that is, an element of the form xαei .



Definition. A monomial in F is a monomial in R times a basisvector of F , that is, an element of the form xαei . A term in F is amonomial in F times a scalar, that is, an element of type axαei ,where a ∈ K .



Definition. A monomial in F is a monomial in R times a basisvector of F , that is, an element of the form xαei . A term in F is amonomial in F times a scalar, that is, an element of type axαei ,where a ∈ K . A submodule of F which is generated by monomialsis called a monomial submodule.



Definition. A monomial in F is a monomial in R times a basisvector of F , that is, an element of the form xαei . A term in F is amonomial in F times a scalar, that is, an element of type axαei ,where a ∈ K . A submodule of F which is generated by monomialsis called a monomial submodule. A monomial order on F maybe defined in the same way as a monomial ordering on R . That is,it is a total order > on the set of monomials in F satisfying

xαei > xβej =⇒ xγxαei > xγxβej for each γ ∈ Nn.





We require in addition that

xαei > xβei ⇐⇒ xαej > xβej ,

for all i , j = 1, . . . , s.





We require in addition that

xαei > xβei ⇐⇒ xαej > xβej ,

for all i , j = 1, . . . , s. In this way, each monomial ordering on Finduces a unique monomial ordering on R and notions like globaland local carry over to monomial orderings on free modules.



Finally, given a monomial order on F , we define the leading term,the leading coefficient, the leading monomial, and the tail of anelement of F in the same way as for a polynomial in R .



Finally, given a monomial order on F , we define the leading term,the leading coefficient, the leading monomial, and the tail of anelement of F in the same way as for a polynomial in R .

With this basic notation, the whole concept of Grobner basesincluding its fundamental algorithms extend.


Computing Syzygies

Suppose now that G = {g1, , . . . , gt} is a Grobner Basis for anideal of R (in a similar way, what we will do works for submodulesof a free R-module F ).


Computing Syzygies

Suppose now that G = {g1, , . . . , gt} is a Grobner Basis for anideal of R (in a similar way, what we will do works for submodulesof a free R-module F ). Then, if we have standard expressions

spoly(gi , gj )− NF(spoly(gi , gj ) | G ) =t

∑

i=1

aigi , ai ∈ R ,

all remainders NF(spoly(gi , gj ) | G ) are zero by Buchberger’scriterion.


Computing Syzygies



∑

i=1

aigi , ai ∈ R ,

all remainders NF(spoly(gi , gj ) | G ) are zero by Buchberger’scriterion. We, thus, get syzygies G (ij) on g1, . . . , gt .


Computing Syzygies



∑

i=1

aigi , ai ∈ R ,

all remainders NF(spoly(gi , gj ) | G ) are zero by Buchberger’scriterion. We, thus, get syzygies G (ij) on g1, . . . , gt . It turns out,that these syzygies generate all the syzygies on g1, , . . . , gt .


Computing Syzygies



∑

i=1

aigi , ai ∈ R ,

all remainders NF(spoly(gi , gj ) | G ) are zero by Buchberger’scriterion. We, thus, get syzygies G (ij) on g1, . . . , gt . It turns out,that these syzygies generate all the syzygies on g1, , . . . , gt .

In fact, we can say more.


Syzygies: Schreyer ordering

In the situation above, starting from a global monomial orderingon the free R-module F , consider the free R-module F0 = R t withits canonical basis ǫ1, . . . , ǫt .



In the situation above, starting from a global monomial orderingon the free R-module F , consider the free R-module F0 = R t withits canonical basis ǫ1, . . . , ǫt . On F0, consider the inducedmonomial ordering defined as

xαǫi >0 xβǫj :⇐⇒ xαLM(fi) > xβLM(fj), or

(

xαLM(fi ) = xβLM(fj) and i > j)

.

Then >0 is a global monomial ordering on F0.



In the situation above, starting from a global monomial orderingon the free R-module F , consider the free R-module F0 = R t withits canonical basis ǫ1, . . . , ǫt . On F0, consider the inducedmonomial ordering defined as

xαǫi >0 xβǫj :⇐⇒ xαLM(fi) > xβLM(fj), or

(

xαLM(fi ) = xβLM(fj) and i > j)

.

Then >0 is a global monomial ordering on F0.

The syzygies G (ij) and the induced monomial ordering are the keyingredients in Schreyer’s proof of Buchberger’s criterion. Thisproof also yields the following result:


Syzygies: Schreyer’s Algorithm

Theorem (Schreyer). Let g1, . . . , gt ∈ F \ {0} form a Grobnerbasis with respect to a global monomial ordering >. The syzygiesG (ij)∈ F0 arising from Buchberger’s criterion form a Grobner basisfor the syzygies on g1, . . . , gt with respect to the inducedmonomial ordering on F0. In particular, the G (ij) generate allsyzygies on g1, . . . , gt .




If we start from an arbitrary set of generators f1, . . . , fr andcompute a Grobner basis f1, . . . , fr , fr+1, . . . , ft using Buchberger’salgorithm, the syzygies G (ij) either arise from a division withremainder zero or from a division leading to a new Grobner basiselement.




If we start from an arbitrary set of generators f1, . . . , fr andcompute a Grobner basis f1, . . . , fr , fr+1, . . . , ft using Buchberger’salgorithm, the syzygies G (ij) either arise from a division withremainder zero or from a division leading to a new Grobner basiselement.

From the G (ij), the syzygies on the original generators are obtainedby means of linear algebra.


Syzygies: Example

Example.

> ring R = 0, (x,y,z), dp;

> ideal I = x,y,z;

> print(syz(I));

0, -y,-z,

-z,x, 0,

y, 0, x

By iterating the process of computing syzygies, starting from afinitely generated R-module M, we get a free resolution of M:

0 Moo F0ϕ0

oo F1ϕ1

oo . . .oo Fi−1oo Fi

ϕioo Fi+1

ϕi+1oo . . .oo


Syzygies: Example

We continue with our example:

> resolution FI = res(I, 0);

> print(FI);

[1]:

_[1]=z

_[2]=y

_[3]=x

[2]:

_[1]=-y*gen(1)+z*gen(2)

_[2]=-x*gen(1)+z*gen(3)

_[3]=-x*gen(2)+y*gen(3)

[3]:

_[1]=x*gen(1)-y*gen(2)+z*gen(3)


Syzygies: Example

> print(FI[2]);

-y,-x,0,

z, 0, -x,

0, z, y

> print(FI[3]);

x,

-y,

z


Ideal Intersections via Syzygies

Syzygies can also be used to compute ideal intersections and idealquotients.


Ideal Intersections via Syzygies

Syzygies can also be used to compute ideal intersections and idealquotients.

Given ideals I = 〈f1, . . . , fr 〉 and J = 〈g1, . . . , gs〉 of K [x1, . . . , xn],compute the syzygies on the columns of the matrix

(

1 f1 . . . fr 0 . . . 01 0 . . . 0 g1 . . . gs

)

.

The entries of the first row of the resulting syzygy matrix generateI ∩ J.


Quotient Ideals Syzygies

In the same way, we get generators for I : J from the matrix

g1 f1 . . . fr 0 . . . . . . 0g2 0 . . . 0 f1 . . . fr 0 . . . 0...

. . .

gs 0 . . . . . . 0 f1 . . . fr

.


Table of Contents







7 Sudoku



Sudoku

5 86 2 5

6 4 77 9 65 2 6 1

3 6 43 7 4

1 5 86 1


Sudoku

To solve Sudokos, we proceed as follows:


Sudoku


Associate to the places in a Sudoku the variables x1, . . . , x81and to each variable xi the polynomial Fi (xi) =

∏9j=1(xi − j).


Sudoku



∏9j=1(xi − j).

Let E = {(i , j) | i < j and (i , j) corresponds to a placein the same row, column or 3× 3− box}.


Sudoku



∏9j=1(xi − j).


For (i , j) ∈ E , let Gi ,j =Fi−Fj

xi−xj.


Sudoku



∏9j=1(xi − j).



xi−xj.

Let I ⊂ Q[x1, . . . , x81] be the ideal generated by the 891polynomials {Gi ,j}(i ,j)∈E and {Fi}i=1,...,9.


Sudoku



∏9j=1(xi − j).



xi−xj.

Let I ⊂ Q[x1, . . . , x81] be the ideal generated by the 891polynomials {Gi ,j}(i ,j)∈E and {Fi}i=1,...,9.

Then a = (a1, . . . , a81) ∈ V (I ) iff ai ∈ {1, . . . , 9} and ai 6= ajfor (i , j) ∈ E .


Sudoku

A well posed Sudoku has a unique solution.


Sudoku


Let L ⊂ {1, . . . , 81} be the set of pre-assigned places, and let{ai}i∈L be the corresponding numbers of an explicit Sudoku S.


Sudoku



Then IS = I + 〈{xi − ai}i∈L〉 is the ideal associated to theSudoku S.


Sudoku



Then IS = I + 〈{xi − ai}i∈L〉 is the ideal associated to theSudoku S.

The reduced Grobner basis of IS with respect to thelexicographical ordering has the shape x1 − a1, . . . , x81 − a81,and (a1, . . . , a81) is the solution of the Sudoku.


Table of Contents







7 Sudoku



Primary Decomposition

Definition. A proper ideal Q of a ring R is said to be primary iff , g ∈ R, fg ∈ Q and f 6∈ Q implies g ∈ rad Q. In this case,P = rad Q is a prime ideal, and Q is also said to be a P-primaryideal. Given any ideal I of R , a primary decomposition of I is anexpression of I as an intersection of finitely many primary ideals.



Definition. A proper ideal Q of a ring R is said to be primary iff , g ∈ R, fg ∈ Q and f 6∈ Q implies g ∈ rad Q. In this case,P = rad Q is a prime ideal, and Q is also said to be a P-primaryideal. Given any ideal I of R , a primary decomposition of I is anexpression of I as an intersection of finitely many primary ideals.

Now suppose that R is Noetherian. Then every proper ideal I of Rhas a primary decomposition. We can always achieve that such adecomposition I =

⋂ri=1Qi is minimal. That is, the prime ideals

Pi = rad Qi are all distinct and none of the Qi can be left out. Inthis case, the Pi are uniquely determined by I and are referred toas the associated primes of I . If Pi is minimal among P1, . . . ,Pr

with respect to inclusion, it is called a minimal associated primeof I .



The minimal associated primes of I are precisely the minimal primeideals containing I . Their intersection is equal to rad I . Everyprimary ideal occuring in a minimal primary decomposition of I iscalled a primary component of I . The component is said to beisolated if its radical is a minimal associated prime of I .Otherwise, it is said to be embedded. The isolated componentsare uniquely determined by I , the others are far from being unique.




From the definitions, it is clear that there is a number of differenttasks coming with primary decomposition.




From the definitions, it is clear that there is a number of differenttasks coming with primary decomposition. These range fromcomputing radicals via computing the minimal associated primes tocomputing a full primary decomposition.




From the definitions, it is clear that there is a number of differenttasks coming with primary decomposition. These range fromcomputing radicals via computing the minimal associated primes tocomputing a full primary decomposition. A variety ofcorresponding algorithms is implemented in the Singular libraryprimdec.lib.




From the definitions, it is clear that there is a number of differenttasks coming with primary decomposition. These range fromcomputing radicals via computing the minimal associated primes tocomputing a full primary decomposition. A variety ofcorresponding algorithms is implemented in the Singular libraryprimdec.lib. The two main algorithms for computing a fullprimary decomposition are primdecSY and primdecGTZ.



The starting point of the GTZ-algorithm is the following simpleobservation:




Lemma (Splitting Tool). If I ⊂ K [x1, . . . , xn] is an ideal, ifh ∈ K [x1, . . . , xn] is a polynomial, and if m ≥ 1 is an integer suchthat I : 〈h〉∞ = I : 〈h〉m, then

I =(

I : 〈h〉m)

∩ 〈I , hm〉 .




Lemma (Splitting Tool). If I ⊂ K [x1, . . . , xn] is an ideal, ifh ∈ K [x1, . . . , xn] is a polynomial, and if m ≥ 1 is an integer suchthat I : 〈h〉∞ = I : 〈h〉m, then

I =(

I : 〈h〉m)

∩ 〈I , hm〉 .

The key result on which the algorithm is based specifies whichpolynomials h are considered:



Proposition Let I ( K [x] = K [x1, . . . , xn] be a proper ideal, andlet u ⊂ x be a subset of maximal cardinality such thatI ∩ K [u] = {0}. Then:

The ideal I K (u)[x\u] ⊂ K (u)[x\u] is zero-dimensional.

Let > = (>x\u , >u) be a global product ordering on K [x],and let G be a Grobner basis for I with respect to >. Then Gis a Grobner basis for I K (u)[x\u] with respect to themonomial ordering obtained by restricting > to the monomialsin K [x\u]. Further, if h ∈ K [u] is the least common multipleof the leading coefficients of the elements of G (regarded aspolynomials in K (u)[x\u]), then

I K (u)[x\u] ∩ K [x] = I : 〈h〉∞ .



All primary components of the ideal I K (u)[x\u] ∩ K [x] havethe same dimension, namely dim I . Further, ifI K (u)[x\u] = Q1 ∩ . . . ∩ Qr is the minimal primarydecomposition, then

I K (u)[x\u] ∩ K [x] = (Q1 ∩ K [x]) ∩ . . . ∩ (Qr ∩ K [x])

is the minimal primary decomposition, too.



All primary components of the ideal I K (u)[x\u] ∩ K [x] havethe same dimension, namely dim I . Further, ifI K (u)[x\u] = Q1 ∩ . . . ∩ Qr is the minimal primarydecomposition, then

I K (u)[x\u] ∩ K [x] = (Q1 ∩ K [x]) ∩ . . . ∩ (Qr ∩ K [x])

is the minimal primary decomposition, too.

Note that if > is a global monomial ordering on K [x], then everysubset u ⊂ x of maximal cardinality satisfyingLM>(I ) ∩ K [u] = {0} is also a subset of maximal cardinality suchthat I ∩ K [u] = {0}.



By recursion, the proposition allows us to reduce the general caseof primary decomposition to the zero-dimensional case. In turn, ifI ⊂ K [x] is a zero-dimensional ideal “in general position” (withrespect to the lexicographic order satisfying x1 > · · · > xn), and ifhn is a generator for I ∩ K [xn], the minimal primary decompositionof I is obtained by factorizing hn. In characteristic zero, thecondition that I is in general position can be achieved by means ofa generic linear coordinate transformation


Andreas Steenpass Workshop on Computational Commutative ...math.ipm.ac.ir/conferences/2011/cca2011/SlidesShow-Steenpass.pdf · > NF(g, J); 0 // g is in I Andreas Steenpass Singular

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