· Complexity of algorithms computing the Hilbert series of monomial ideals Jamal Hossein Poor Outline Introduction ch.1:Complexity and timing data of algorithms ch.2:Hilbert functio

Complexity ofalgorithms

computing theHilbert seriesof monomial

ideals

Jamal HosseinPoor

Outline

Introduction

ch.1:Complexityand timingdata ofalgorithms

ch.2:Hilbertfunction andHilbert series

ch.3:Analysisof algorithms

ch.4:Hilbertseries of zero-dimensionallex-segmentideal inK[x, y, z]

.

.. ..

.

.

Complexity of algorithms computing theHilbert series of monomial ideals

Jamal Hossein Poor

Institute for Advanced Studies in Basic Sciences (IASBS), Gavazang - Zanjan

19 September 2012



ideals

Jamal HosseinPoor

Outline

Introduction





Outline

Introduction

chapter 1: Analysis of complexity and timing data ofalgorithmschapter 2: Hilbert function and Hilbert series of standardgraded moduleschapter 3: Analysis of algorithms Computating the HilbertSerieschapter 4: Hilbert series of zero-dimensional lex-segmentideal in K[x, y, z]



ideals

Jamal HosseinPoor

Outline

Introduction





Outline

Introductionchapter 1: Analysis of complexity and timing data ofalgorithms

chapter 2: Hilbert function and Hilbert series of standardgraded moduleschapter 3: Analysis of algorithms Computating the HilbertSerieschapter 4: Hilbert series of zero-dimensional lex-segmentideal in K[x, y, z]



ideals

Jamal HosseinPoor

Outline

Introduction





Outline

Introductionchapter 1: Analysis of complexity and timing data ofalgorithmschapter 2: Hilbert function and Hilbert series of standardgraded modules

chapter 3: Analysis of algorithms Computating the HilbertSerieschapter 4: Hilbert series of zero-dimensional lex-segmentideal in K[x, y, z]



ideals

Jamal HosseinPoor

Outline

Introduction





Outline

Introductionchapter 1: Analysis of complexity and timing data ofalgorithmschapter 2: Hilbert function and Hilbert series of standardgraded moduleschapter 3: Analysis of algorithms Computating the HilbertSeries

chapter 4: Hilbert series of zero-dimensional lex-segmentideal in K[x, y, z]



ideals

Jamal HosseinPoor

Outline

Introduction





Outline

Introductionchapter 1: Analysis of complexity and timing data ofalgorithmschapter 2: Hilbert function and Hilbert series of standardgraded moduleschapter 3: Analysis of algorithms Computating the HilbertSerieschapter 4: Hilbert series of zero-dimensional lex-segmentideal in K[x, y, z]



ideals

Jamal HosseinPoor

Outline

Introduction





Introduction

Hilbert Series of a graded module such as M = ⊕∞i=0Mi is

defined by the form of∑∞

i=0 l(Mi)zi , wherel(Mi) is the lengthof the module Mi. In the case ,M is a standard algebra. theseries has a rational form as h(z)

(1− z)d where h(z) is apolynomial in Z[z].

In this thesis, we express some mentions related to analysis ofalgorithms,then we study algebraic concepts like integerfunction,Hilbert function,Laurant series,Hilbert series andetc.After that we compare some algorithms computing theHilbert numerator.Finally ,we introduce a procedure whichcomputes the h-vector of zero-dimensional lex-segment ideal inwith three variables.



ideals

Jamal HosseinPoor

Outline

Introduction





Introduction

Hilbert Series of a graded module such as M = ⊕∞i=0Mi is

defined by the form of∑∞

i=0 l(Mi)zi , wherel(Mi) is the lengthof the module Mi. In the case ,M is a standard algebra. theseries has a rational form as h(z)

(1− z)d where h(z) is apolynomial in Z[z].In this thesis, we express some mentions related to analysis ofalgorithms,then we study algebraic concepts like integerfunction,Hilbert function,Laurant series,Hilbert series andetc.After that we compare some algorithms computing theHilbert numerator.Finally ,we introduce a procedure whichcomputes the h-vector of zero-dimensional lex-segment ideal inwith three variables.



ideals

Jamal HosseinPoor

Outline

Introduction





Algorithm

.Definition 1.1)..

.. ..

.

.

In design of algorithms,a techniqueis not said to language ofprogramming or methods of programming.Infact,we mean themethod of solving a problem.Using a technique to solve aproblem relates to a step by stepmethod which is calledalgorithm.

.Note 1.2)..

.. ..

.

.

When an algorithm performs on a computer,the mention ofTime is said to the number of CPU cycles.



ideals

Jamal HosseinPoor

Outline

Introduction





Algorithm

.Definition 1.1)..

.. ..

.

.

In design of algorithms,a techniqueis not said to language ofprogramming or methods of programming.Infact,we mean themethod of solving a problem.Using a technique to solve aproblem relates to a step by stepmethod which is calledalgorithm.

.Note 1.2)..

.. ..

.

.

When an algorithm performs on a computer,the mention ofTime is said to the number of CPU cycles.



ideals

Jamal HosseinPoor

Outline

Introduction





Algorithm

.Example 1.3)..

.. ..

.

.

(Sequential search)Determine is x in array S or not?

Void seqsearch (int n,Const key type S [ ],Key type x,Index V (location)

{Location = 1;While (location <=n && S [location] ! = x)Location ++,If (location>n)Location = 0;

}



ideals

Jamal HosseinPoor

Outline

Introduction





Algorithm

.Example 1.3)..

.. ..

.

.

(Sequential search)Determine is x in array S or not?Void seqsearch (int n,

Const key type S [ ],Key type x,Index V (location)

{Location = 1;While (location <=n && S [location] ! = x)Location ++,If (location>n)Location = 0;

}



ideals

Jamal HosseinPoor

Outline

Introduction





Algorithm

.Example 1.4)..

.. ..

.

.

(Binary search)Determine is x in array S or not?

Void binsearch (int n,Const key type S [ ],Key type x.Index and location)

{ Index low, high, mid;Low=1; high = n;Location = 0;

While (low <=0 high and location == 0){Mid= [(low + high/2];

If (x==s [mid])Location = mid;

Else if (x<S [mid])high = mid -1;

ElseLow = mid + 1; } }



ideals

Jamal HosseinPoor

Outline

Introduction





Algorithm

.Example 1.4)..

.. ..

.

.

(Binary search)Determine is x in array S or not?Void binsearch (int n,

Const key type S [ ],Key type x.Index and location)

{ Index low, high, mid;Low=1; high = n;Location = 0;

While (low <=0 high and location == 0){Mid= [(low + high/2];

If (x==s [mid])Location = mid;

Else if (x<S [mid])high = mid -1;

ElseLow = mid + 1; } }



ideals

Jamal HosseinPoor

Outline

Introduction





Time complexity

.Definition 1.5)..

.. ..

.

.

We define T(n)as the number of runs of main action performedby the algorithm.T(n) is called the time complexity of thealgorithm normally and determination of that is said Analysis oftime complexity of an algorithmfor a normall call.

.Example 1.6)(Fibonacci sequence)..

.. ..

.

.

Int fib (int n){ If (n<=1)Return;

ElseReturn fib(n − 1) + fib(n − 2) }

T(n) > 2× T(n − 2) > 2× 2× T(n − 4) >2× 2× 2× T(n − 6) > . . . > 2× 2 . . .× 2× T(0) = 2nT(0)



ideals

Jamal HosseinPoor

Outline

Introduction





Time complexity

.Definition 1.5)..

.. ..

.

.



.. ..

.

.



T(n) > 2× T(n − 2) > 2× 2× T(n − 4) >2× 2× 2× T(n − 6) > . . . > 2× 2 . . .× 2× T(0) = 2nT(0)



ideals

Jamal HosseinPoor

Outline

Introduction





Time complexity

.Definition 1.5)..

.. ..

.

.



.. ..

.

.



T(n) > 2× T(n − 2) > 2× 2× T(n − 4) >2× 2× 2× T(n − 6) > . . . > 2× 2 . . .× 2× T(0) = 2nT(0)



ideals

Jamal HosseinPoor

Outline

Introduction





Time complexity

.Theorem 1.7)..

.. ..

.

.

If T(n) be the time complexity of fibonacci algorithm,thenT(n) > 2n/2.

.Note 1.8)..

.. ..

.

.

Some of algorithms just have one complexity;Array [1. . . n] of integer; S := 0;For I:=1 to n doS:=S+A [i];

in this exampleT(n) = n.But the others have several complexity;Array [1…n] of integer;For i:=1 to n doIf (x=A [i]) { Write (’Yes’); Exit ( ); } Write (’NO’);

here W(n) = n(the Worst case),B(n) = 1(the Best case).We defineA(n) =

n∑i=1

in =

1

nn∑

i=1

i = 1

nn(n + 1)

2=

n + 1

2as the Average case.



ideals

Jamal HosseinPoor

Outline

Introduction





Time complexity

.Theorem 1.7)..

.. ..

.

.

If T(n) be the time complexity of fibonacci algorithm,thenT(n) > 2n/2..Note 1.8)..

.. ..

.

.

Some of algorithms just have one complexity;

Array [1. . . n] of integer; S := 0;For I:=1 to n doS:=S+A [i];



n∑i=1

in =

1

nn∑

i=1

i = 1

nn(n + 1)

2=

n + 1




ideals

Jamal HosseinPoor

Outline

Introduction





Time complexity

.Theorem 1.7)..

.. ..

.

.


.. ..

.

.




n∑i=1

in =

1

nn∑

i=1

i = 1

nn(n + 1)

2=

n + 1




ideals

Jamal HosseinPoor

Outline

Introduction





Time complexity

.Theorem 1.7)..

.. ..

.

.


.. ..

.

.


in this exampleT(n) = n.But the others have several complexity;

Array [1…n] of integer;For i:=1 to n doIf (x=A [i]) { Write (’Yes’); Exit ( ); } Write (’NO’);


n∑i=1

in =

1

nn∑

i=1

i = 1

nn(n + 1)

2=

n + 1




ideals

Jamal HosseinPoor

Outline

Introduction





Time complexity

.Theorem 1.7)..

.. ..

.

.


.. ..

.

.




n∑i=1

in =

1

nn∑

i=1

i = 1

nn(n + 1)

2=

n + 1




ideals

Jamal HosseinPoor

Outline

Introduction





Time complexity

.Theorem 1.7)..

.. ..

.

.


.. ..

.

.



here W(n) = n(the Worst case),B(n) = 1(the Best case).

We defineA(n) =

n∑i=1

in =

1

nn∑

i=1

i = 1

nn(n + 1)

2=

n + 1




ideals

Jamal HosseinPoor

Outline

Introduction





Time complexity

.Theorem 1.7)..

.. ..

.

.


.. ..

.

.




n∑i=1

in =

1

nn∑

i=1

i = 1

nn(n + 1)

2=

n + 1




ideals

Jamal HosseinPoor

Outline

Introduction





Order

.Definition 1.9).... ..

.

.f(n) = O(g(n)) ⇐⇒ ∃c ∈ R, n0 ∈ N : ∀n ≥ n0 f(n) ≤ cg(n)

.Example 1.10).... ..

.

.n2 + 10n = O(n2), c = 2, n0 = 10 ⇒ n2 + 10n ≤ 2n2

.Theorem 1.11).... ..

.

.Iff(n) = amnm + . . .+ a1n + a0thenf(n) = O(nm)..Example 1.12)Order =?..

.. ..

.

.

for i=1 to n dofor j=1 to n doX:=X+1 ;Answer:O(n2)



ideals

Jamal HosseinPoor

Outline

Introduction





Order


.

.f(n) = O(g(n)) ⇐⇒ ∃c ∈ R, n0 ∈ N : ∀n ≥ n0 f(n) ≤ cg(n).Example 1.10).... ..

.

.n2 + 10n = O(n2), c = 2, n0 = 10 ⇒ n2 + 10n ≤ 2n2

.Theorem 1.11).... ..

.


.. ..

.

.




ideals

Jamal HosseinPoor

Outline

Introduction





Order


.


.

.n2 + 10n = O(n2), c = 2, n0 = 10 ⇒ n2 + 10n ≤ 2n2

.Theorem 1.11).... ..

.


.. ..

.

.

for i=1 to n dofor j=1 to n doX:=X+1 ;

Answer:O(n2)



ideals

Jamal HosseinPoor

Outline

Introduction





Order


.


.

.n2 + 10n = O(n2), c = 2, n0 = 10 ⇒ n2 + 10n ≤ 2n2

.Theorem 1.11).... ..

.


.. ..

.

.




ideals

Jamal HosseinPoor

Outline

Introduction





Order

.Example 1.13)Order =?..

.. ..

.

.

x:=0;i:=n;while (i>1) do begin

x:=x+1;I:=I div 2;

End;

Answer:O(log2n)

.Note 1.14)The relation between Orders (left>right)..

.. ..

.

.

F Fact Exp Deg2 LogLin Lin Log FixO O(n!) O(2n) O(n2) O(nlog2n) O(n) O(log2n) O(1)



ideals

Jamal HosseinPoor

Outline

Introduction





Order


.. ..

.

.


x:=x+1;I:=I div 2;

End; Answer:O(log2n)


.. ..

.

.




ideals

Jamal HosseinPoor

Outline

Introduction





Order


.. ..

.

.


x:=x+1;I:=I div 2;

End; Answer:O(log2n)


.. ..

.

.




ideals

Jamal HosseinPoor

Outline

Introduction





Order

.Note 1.15)Order of recursive function..

.. ..

.

.

Order Recursion relationO(log2n) T(n) = T(n/2)

O(2log2n) = O(n) T(n) = T(n/2) + T(n/2)O(2n) T(n) = T(n − 1)× T(n − 1)O(n) T(n) = 2T(n − 1)

O(2n/2) T(n) = T(n − 2)× T(n − 2)O(2n) T(n) = T(n − 1) + T(n − 1)

.Example 1.16)..

.. ..

.

.

int fact (int n){ if(n==1)return 1:

return (n ∗ fact(n − 1)): }T(n) = T(n − 1) + C = T(n − 2) + 2C = T(n − 3) + 3C =. . . = T(1) + (n − 1)C = 1 + (n − 1)C = O(n)



ideals

Jamal HosseinPoor

Outline

Introduction





Order


.. ..

.

.




.Example 1.16)..

.. ..

.

.


return (n ∗ fact(n − 1)): }

T(n) = T(n − 1) + C = T(n − 2) + 2C = T(n − 3) + 3C =. . . = T(1) + (n − 1)C = 1 + (n − 1)C = O(n)



ideals

Jamal HosseinPoor

Outline

Introduction





Order


.. ..

.

.




.Example 1.16)..

.. ..

.

.


return (n ∗ fact(n − 1)): }T(n) = T(n − 1) + C = T(n − 2) + 2C = T(n − 3) + 3C =. . . = T(1) + (n − 1)C = 1 + (n − 1)C = O(n)



ideals

Jamal HosseinPoor

Outline

Introduction





Polynomial time

.Definition 1.17)..

.. ..

.

.

We say an algorithm hasPolynomial time,if there is an upperbound by the form of a polynomial for it’s complexity in worstcase.It meansW(n) = O(P(n)) for a polynomial P.

.Example 1.18)..

.. ..

.

.

These examples have polynomial time;2n ∈ O(n), 3n3+4n ∈ O(n3), 5n+n10 ∈ O(n10), nlog2n ∈ O(n2)but 2n, 20.01n, 2

√n, n!dos not have polynomial time.



ideals

Jamal HosseinPoor

Outline

Introduction





Polynomial time

.Definition 1.17)..

.. ..

.

.


.Example 1.18)..

.. ..

.

.

These examples have polynomial time;2n ∈ O(n), 3n3+4n ∈ O(n3), 5n+n10 ∈ O(n10), nlog2n ∈ O(n2)

but 2n, 20.01n, 2√n, n!dos not have polynomial time.



ideals

Jamal HosseinPoor

Outline

Introduction





Polynomial time

.Definition 1.17)..

.. ..

.

.


.Example 1.18)..

.. ..

.

.

These examples have polynomial time;2n ∈ O(n), 3n3+4n ∈ O(n3), 5n+n10 ∈ O(n10), nlog2n ∈ O(n2)but 2n, 20.01n, 2

√n, n!dos not have polynomial time.



ideals

Jamal HosseinPoor

Outline

Introduction





Noncontrollable problems

.Note 1.19)..

.. ..

.

.

In computer scince,a problem is called Noncontrollable if it’ssolving by a polynomial time algorithm be impossible. in thisview,problems are divided into three main groups;

1)The problems which have solved by polynomial timealgorithms.2)The problems which thier noncontrollablity of them has beenproved.3)The problems which thier noncontrollablity of them has notbeen proved,but it has not found any polynomial timealgorithm to solve them .



ideals

Jamal HosseinPoor

Outline

Introduction






.Note 1.19)..

.. ..

.

.

In computer scince,a problem is called Noncontrollable if it’ssolving by a polynomial time algorithm be impossible. in thisview,problems are divided into three main groups;1)The problems which have solved by polynomial timealgorithms.

2)The problems which thier noncontrollablity of them has beenproved.3)The problems which thier noncontrollablity of them has notbeen proved,but it has not found any polynomial timealgorithm to solve them .



ideals

Jamal HosseinPoor

Outline

Introduction






.Note 1.19)..

.. ..

.

.

In computer scince,a problem is called Noncontrollable if it’ssolving by a polynomial time algorithm be impossible. in thisview,problems are divided into three main groups;1)The problems which have solved by polynomial timealgorithms.2)The problems which thier noncontrollablity of them has beenproved.

3)The problems which thier noncontrollablity of them has notbeen proved,but it has not found any polynomial timealgorithm to solve them .



ideals

Jamal HosseinPoor

Outline

Introduction






.Note 1.19)..

.. ..

.

.

In computer scince,a problem is called Noncontrollable if it’ssolving by a polynomial time algorithm be impossible. in thisview,problems are divided into three main groups;1)The problems which have solved by polynomial timealgorithms.2)The problems which thier noncontrollablity of them has beenproved.3)The problems which thier noncontrollablity of them has notbeen proved,but it has not found any polynomial timealgorithm to solve them .



ideals

Jamal HosseinPoor

Outline

Introduction





NP theory

.Definition 1.20)..

.. ..

.

.

Pis the set of all problems which are solvable be polynomialtime algorithms.

.Definition 1.21)..

.. ..

.

.

Deterministic algorithm in polynomial timeis an algotithmwhich it’s verification step be in polynomial time.

.Defenition 1.22)..

.. ..

.

.

NPis the set of all problems which are not solvable bydeterministic algorithms in polynomial time.



ideals

Jamal HosseinPoor

Outline

Introduction





NP theory

.Definition 1.20)..

.. ..

.

.


.Definition 1.21)..

.. ..

.

.


.Defenition 1.22)..

.. ..

.

.




ideals

Jamal HosseinPoor

Outline

Introduction





NP theory

.Definition 1.20)..

.. ..

.

.


.Definition 1.21)..

.. ..

.

.


.Defenition 1.22)..

.. ..

.

.




ideals

Jamal HosseinPoor

Outline

Introduction





NP theory

.Example 1.23)(Traveling Salesman Problem)..

.. ..

.

.

bool verify (weighted – digraph G,number d,claimed – tours)

{if (s is a tour and the total weighted of edges in s is <=d)return true;

elsereturn false;

}



ideals

Jamal HosseinPoor

Outline

Introduction





NP theory

.Example 1.24)Dynamic programming algorithm for thetraveling salesman problem..

.. ..

.

.

void travel (int , nconst number W[ ] [ ]index p[ ] [ ];number and minlength)

{ index I,j,k;number D [1..n] [subset of V- ];for (i=2; i<n ; i++)> D [i] [ ] = w [i] [1];for (k=1; k<=n-2; k++)for (all subsets A=V- containing k vertices)

for (I such that i> 1 and is not in A) {D [i] [A]= minimum (w [i] [j] + D [j] [A- ]);P[i] [A]= value of j that gave the minimum; }

D [1] [v- ]= minimum (W [i] [j]+ D [j] [v- ]);P[1] [v- ]=value of j that gave the minimum;Minlength = D [1] [v- ];}



ideals

Jamal HosseinPoor

Outline

Introduction





NP theory

.Definition 1.25..

.. ..

.

.

If there exiests a polynomial time algorithm for the conversionof problem A to problem B,then A is called reducible to B anddenoted by

A ∝ B

.Theorem 1.26.... ..

.

.Let the problem B ∈ P and A ∝ B,then A ∈ P

.Definition 1.27..

.. ..

.

.

A is called NP-complete ,if first A ∈ NPand secondly for eachother problem B ∈ NP;

A ∝ B



ideals

Jamal HosseinPoor

Outline

Introduction





NP theory

.Definition 1.25..

.. ..

.

.


A ∝ B.Theorem 1.26.... ..

.


.Definition 1.27..

.. ..

.

.


A ∝ B



ideals

Jamal HosseinPoor

Outline

Introduction





NP theory

.Definition 1.25..

.. ..

.

.


A ∝ B.Theorem 1.26.... ..

.


.Definition 1.27..

.. ..

.

.


A ∝ B



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Jamal HosseinPoor

Outline

Introduction





Tn

.Definition 2.1)..

.. ..

.

.

Letn ≥ 1a)A polynomial f ∈ R[x1, . . . , xn] of the form f = xα1

1 . . . xαnnsuch that α1, . . . , αn ∈ N is called a term or power product.The set of all terms of R[x1, . . . , xn] is denoted by Tn.

Tn = {xα11 . . . xαnn | α1, . . . , αn ∈ N}

b)For a termt = xα11 . . . xαnn , the number

deg(t) = α1 + . . .+ αn is called the degree of t.c) The map log : Tn −→ Nn definedbyxα1

1 . . . xαnn 7→ (α1, . . . , αn) is called the logarithm.



ideals

Jamal HosseinPoor

Outline

Introduction





Tn

.Definition 2.1)..

.. ..

.

.

Letn ≥ 1

a)A polynomial f ∈ R[x1, . . . , xn] of the form f = xα11 . . . xαnn

such that α1, . . . , αn ∈ N is called a term or power product.The set of all terms of R[x1, . . . , xn] is denoted by Tn.

Tn = {xα11 . . . xαnn | α1, . . . , αn ∈ N}






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Jamal HosseinPoor

Outline

Introduction





Tn

.Definition 2.1)..

.. ..

.

.



Tn = {xα11 . . . xαnn | α1, . . . , αn ∈ N}






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Jamal HosseinPoor

Outline

Introduction





Tn

.Definition 2.1)..

.. ..

.

.



Tn = {xα11 . . . xαnn | α1, . . . , αn ∈ N}






ideals

Jamal HosseinPoor

Outline

Introduction





Tn

.Definition 2.1)..

.. ..

.

.



Tn = {xα11 . . . xαnn | α1, . . . , αn ∈ N}


deg(t) = α1 + . . .+ αn is called the degree of t.

c) The map log : Tn −→ Nn definedbyxα1




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Jamal HosseinPoor

Outline

Introduction





Tn

.Definition 2.1)..

.. ..

.

.



Tn = {xα11 . . . xαnn | α1, . . . , αn ∈ N}






ideals

Jamal HosseinPoor

Outline

Introduction





Dickson’s Lemma

.Definition 2.2..

.. ..

.

.

Let P = K[x1, . . . , xn].An ideal I ⊆ P is called a monomial Ideal,if it has a system of generators consisting of elements of Tn.

.Lemma 2.3)..

.. ..

.

.

Letn ≥ 1, and let. . . , t2, t1 be a sequence of terms in Tn . Thenthere exists a number N > 0 such that for every i > N the termti is a multiple of one of the terms t1, . . . , tN , i.e. themonoideal (t1, t2, . . .) is generated by {t1, . . . , tN)}. Inparticular, for every ring R, the ideal (t1, t2, . . .) ⊆ R[x1, . . . , xn]is finitely generated.



ideals

Jamal HosseinPoor

Outline

Introduction





Dickson’s Lemma

.Definition 2.2..

.. ..

.

.

Let P = K[x1, . . . , xn].An ideal I ⊆ P is called a monomial Ideal,if it has a system of generators consisting of elements of Tn.

.Lemma 2.3)..

.. ..

.

.

Letn ≥ 1, and let. . . , t2, t1 be a sequence of terms in Tn . Thenthere exists a number N > 0 such that for every i > N the termti is a multiple of one of the terms t1, . . . , tN , i.e. themonoideal (t1, t2, . . .) is generated by {t1, . . . , tN)}. Inparticular, for every ring R, the ideal (t1, t2, . . .) ⊆ R[x1, . . . , xn]is finitely generated.



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Jamal HosseinPoor

Outline

Introduction





Grading

.Definition 2.4..

.. ..

.

.

Let (Γ,+)be a monoid.

a)The ring R is called aΓ-graded ring (or graded overΓ ) ifthere exists a family of additive subgroups {Rγ}γ ∈ Γ such that1)R =

⊕γ∈Γ Rγ

2)RγRγ′ ⊆ Rγ+γ

′ ; γ, γ′ ∈ Γ

b)The elements of Rγ are called homogeneous of degree γ. Forwe write deg(r) = γ .c)If r ∈ R and r =

∑γ∈Γ rγ where rγ ∈ Rγ , then rγ is called the

homogeneous component of degree γof r .



ideals

Jamal HosseinPoor

Outline

Introduction





Grading

.Definition 2.4..

.. ..

.

.

Let (Γ,+)be a monoid.a)The ring R is called aΓ-graded ring (or graded overΓ ) ifthere exists a family of additive subgroups {Rγ}γ ∈ Γ such that1)R =

⊕γ∈Γ Rγ


′ ; γ, γ′ ∈ Γ






ideals

Jamal HosseinPoor

Outline

Introduction





Grading

.Definition 2.4..

.. ..

.

.


⊕γ∈Γ Rγ


′ ; γ, γ′ ∈ Γ

b)The elements of Rγ are called homogeneous of degree γ. Forwe write deg(r) = γ .

c)If r ∈ R and r =∑

γ∈Γ rγ where rγ ∈ Rγ , then rγ is called thehomogeneous component of degree γof r .



ideals

Jamal HosseinPoor

Outline

Introduction





Grading

.Definition 2.4..

.. ..

.

.


⊕γ∈Γ Rγ


′ ; γ, γ′ ∈ Γ






ideals

Jamal HosseinPoor

Outline

Introduction





Grading

.Example 2.5)Standard grading..

.. ..

.

.

Let S be a ring, let n ≥ 1, and let P = S[x1, ..., xn] be a polynomialring over S. If we let

Pd = {f ∈ P | deg(t) = d,∀t ∈ supp(f)}

for d ≥ 0, we make P into an N-graded ring. This grading is calledthe standard grading of P . It satisfies deg(x1) = ... = deg(xn) = 1For d ≥ 0, the elements of Pd are called homogeneous polynomials(or forms) of degree d.

.Definition 2.6..

.. ..

.

.

Let K be a field.A K-algebra R is called a standard graded K-algebraif it is N-graded, satisfies R0 = K and dimK(R1) < ∞, and if R isgenerated by the elements of R1 as a K-algebra.



ideals

Jamal HosseinPoor

Outline

Introduction





Grading


.. ..

.

.




.Definition 2.6..

.. ..

.

.




ideals

Jamal HosseinPoor

Outline

Introduction





Grading


.. ..

.

.




.Definition 2.6..

.. ..

.

.




ideals

Jamal HosseinPoor

Outline

Introduction





Grading


.. ..

.

.




.Definition 2.6..

.. ..

.

.




ideals

Jamal HosseinPoor

Outline

Introduction





Grading

.Definition 2.7) Graded module..

.. ..

.

.

Let (Γ,+)be a monoid, let R be aΓ -graded ring,let M be anR-module. We say that M is a Γ-graded R-module if there is a familyof subgroups {Ms}s∈Γ such that

1)M =⊕

s∈Γ Ms2)Rγ .Ms ⊆ Mγ+s; ∀γ, s ∈ Γ

.Definition 2.8)Graded ideal..

.. ..

.

.

An R-submodule N of the Γ-graded R-module M is called a Γ-gradedR-submodule of M if we have N =

⊕γ∈Γ(N

∩Ms). A Γ-graded

submodule of R is also called a Γ -homogeneous ideal of R, or simplya homogeneous ideal of R if Γ is clear from the context.



ideals

Jamal HosseinPoor

Outline

Introduction





Grading


.. ..

.

.

Let (Γ,+)be a monoid, let R be aΓ -graded ring,let M be anR-module. We say that M is a Γ-graded R-module if there is a familyof subgroups {Ms}s∈Γ such that1)M =

⊕s∈Γ Ms

2)Rγ .Ms ⊆ Mγ+s; ∀γ, s ∈ Γ


.. ..

.

.


⊕γ∈Γ(N

∩Ms). A Γ-graded




ideals

Jamal HosseinPoor

Outline

Introduction





Grading


.. ..

.

.


⊕s∈Γ Ms

2)Rγ .Ms ⊆ Mγ+s; ∀γ, s ∈ Γ


.. ..

.

.


⊕γ∈Γ(N

∩Ms). A Γ-graded




ideals

Jamal HosseinPoor

Outline

Introduction





Grading


.. ..

.

.


⊕s∈Γ Ms

2)Rγ .Ms ⊆ Mγ+s; ∀γ, s ∈ Γ


.. ..

.

.


⊕γ∈Γ(N

∩Ms). A Γ-graded




ideals

Jamal HosseinPoor

Outline

Introduction





Grading

.Definition 2.9)Grading by matrix..

.. ..

.

.

Let m ≥ 1, and let the polynomial ring P = K[x1, . . . , xn] be equippedwith a Zm-grading such that K ⊆ P0 and x1, . . . , xn are homogeneouselements. For j = 1, . . . , n let (w1j, . . . ,wmj) ∈ Zm be the degree ofxj. The matrix W = (wij) ∈ Matm,n(Z) is called the degree matrix ofthe grading.

.Example 2.10)..

.. ..

.

.

Let P = K[x1, x2, x3, x4]be graded by the matrix

W =

1 1 1 11 1 0 01 0 1 0

and let f = x1x4 − x2x3. Then f is homogeneous of degree (2, 1, 1),because W.log(x1x4)tr = W.log(x2x3)tr = (2, 1, 1)tr.



ideals

Jamal HosseinPoor

Outline

Introduction





Grading

.Definition 2.9)Grading by matrix..

.. ..

.

.

Let m ≥ 1, and let the polynomial ring P = K[x1, . . . , xn] be equippedwith a Zm-grading such that K ⊆ P0 and x1, . . . , xn are homogeneouselements. For j = 1, . . . , n let (w1j, . . . ,wmj) ∈ Zm be the degree ofxj. The matrix W = (wij) ∈ Matm,n(Z) is called the degree matrix ofthe grading.

.Example 2.10)..

.. ..

.

.

Let P = K[x1, x2, x3, x4]be graded by the matrix

W =

1 1 1 11 1 0 01 0 1 0

and let f = x1x4 − x2x3. Then f is homogeneous of degree (2, 1, 1),because W.log(x1x4)tr = W.log(x2x3)tr = (2, 1, 1)tr.



ideals

Jamal HosseinPoor

Outline

Introduction





Grading

.Note 2.11)Gradings of positive type..

.. ..

.

.

Let m ≥ 1, let P be graded by a matrix W ∈ Matm,n(Z) of rankm , and let w1, . . . ,wm be the rows of W . We say that thegrading on P given by W is of positive type if there exista1, . . . , am ∈ Z such that all entries of a1w1 + . . .+ amwmarepositive. In this case, we shall also say that W is a matrix ofpositive type. For instance, the matrix W =

(−1 −1

)is of

positive type.



ideals

Jamal HosseinPoor

Outline

Introduction





Integer function

.Definition 2.12..

.. ..

.

.

A map f : Z −→ Z is called an integer function. Given an integerfunction f : Z −→ Z, we define the following operators.

1)The integer function ∆f : Z −→ Z defined by∆f(i) = f(i)− f(i − 1), i ∈ Z for i ∈ Z is called the (first) differencefunction of f .2)Let ∆0f = f. For r ≥ 1, we inductively define an integer function∆rf : Z −→ Z by ∆rf = ∆(∆r−1f) and call it the rth differencefunction of f .3)An integer function f : Z −→ Z is called an integer Laurent functionif there exists a number i0 ∈ Z such that f(i) = 0 for all i < i0.4)Given an integer Laurent function f : Z −→ Z, we define anotherinteger Laurent function Σf : Z −→ Z by Σf(i) = Σj≤if(j) and call itthe summation function of f .



ideals

Jamal HosseinPoor

Outline

Introduction





Integer function

.Definition 2.12..

.. ..

.

.

A map f : Z −→ Z is called an integer function. Given an integerfunction f : Z −→ Z, we define the following operators.1)The integer function ∆f : Z −→ Z defined by∆f(i) = f(i)− f(i − 1), i ∈ Z for i ∈ Z is called the (first) differencefunction of f .

2)Let ∆0f = f. For r ≥ 1, we inductively define an integer function∆rf : Z −→ Z by ∆rf = ∆(∆r−1f) and call it the rth differencefunction of f .3)An integer function f : Z −→ Z is called an integer Laurent functionif there exists a number i0 ∈ Z such that f(i) = 0 for all i < i0.4)Given an integer Laurent function f : Z −→ Z, we define anotherinteger Laurent function Σf : Z −→ Z by Σf(i) = Σj≤if(j) and call itthe summation function of f .



ideals

Jamal HosseinPoor

Outline

Introduction





Integer function

.Definition 2.12..

.. ..

.

.

A map f : Z −→ Z is called an integer function. Given an integerfunction f : Z −→ Z, we define the following operators.1)The integer function ∆f : Z −→ Z defined by∆f(i) = f(i)− f(i − 1), i ∈ Z for i ∈ Z is called the (first) differencefunction of f .2)Let ∆0f = f. For r ≥ 1, we inductively define an integer function∆rf : Z −→ Z by ∆rf = ∆(∆r−1f) and call it the rth differencefunction of f .

3)An integer function f : Z −→ Z is called an integer Laurent functionif there exists a number i0 ∈ Z such that f(i) = 0 for all i < i0.4)Given an integer Laurent function f : Z −→ Z, we define anotherinteger Laurent function Σf : Z −→ Z by Σf(i) = Σj≤if(j) and call itthe summation function of f .



ideals

Jamal HosseinPoor

Outline

Introduction





Integer function

.Definition 2.12..

.. ..

.

.

A map f : Z −→ Z is called an integer function. Given an integerfunction f : Z −→ Z, we define the following operators.1)The integer function ∆f : Z −→ Z defined by∆f(i) = f(i)− f(i − 1), i ∈ Z for i ∈ Z is called the (first) differencefunction of f .2)Let ∆0f = f. For r ≥ 1, we inductively define an integer function∆rf : Z −→ Z by ∆rf = ∆(∆r−1f) and call it the rth differencefunction of f .3)An integer function f : Z −→ Z is called an integer Laurent functionif there exists a number i0 ∈ Z such that f(i) = 0 for all i < i0.

4)Given an integer Laurent function f : Z −→ Z, we define anotherinteger Laurent function Σf : Z −→ Z by Σf(i) = Σj≤if(j) and call itthe summation function of f .



ideals

Jamal HosseinPoor

Outline

Introduction





Integer function

.Definition 2.12..

.. ..

.

.

A map f : Z −→ Z is called an integer function. Given an integerfunction f : Z −→ Z, we define the following operators.1)The integer function ∆f : Z −→ Z defined by∆f(i) = f(i)− f(i − 1), i ∈ Z for i ∈ Z is called the (first) differencefunction of f .2)Let ∆0f = f. For r ≥ 1, we inductively define an integer function∆rf : Z −→ Z by ∆rf = ∆(∆r−1f) and call it the rth differencefunction of f .3)An integer function f : Z −→ Z is called an integer Laurent functionif there exists a number i0 ∈ Z such that f(i) = 0 for all i < i0.4)Given an integer Laurent function f : Z −→ Z, we define anotherinteger Laurent function Σf : Z −→ Z by Σf(i) = Σj≤if(j) and call itthe summation function of f .



ideals

Jamal HosseinPoor

Outline

Introduction





Integer function

.Proposition 12.13..

.. ..

.

.

Let f : Z −→ Z be an integer Laurent function. Then we have

Σ∆f = ∆Σf = f

.Definition 12.14..

.. ..

.

.

A polynomial p ∈ Q[t] is called an integer valued polynomial if wehave p(i) ∈ Z for all i ∈ Z. The set of all integer valued polynomialswill be denoted by IP. Furthermore, for every r ≥ 0, we letIP≤r, r ≥ 0 be the set of all integer valued polynomials of degree ≤ r .



ideals

Jamal HosseinPoor

Outline

Introduction





Integer function

.Proposition 12.13..

.. ..

.

.

Let f : Z −→ Z be an integer Laurent function. Then we have

Σ∆f = ∆Σf = f

.Definition 12.14..

.. ..

.

.

A polynomial p ∈ Q[t] is called an integer valued polynomial if wehave p(i) ∈ Z for all i ∈ Z. The set of all integer valued polynomialswill be denoted by IP. Furthermore, for every r ≥ 0, we letIP≤r, r ≥ 0 be the set of all integer valued polynomials of degree ≤ r .



ideals

Jamal HosseinPoor

Outline

Introduction





Integer function

.Theorem 2.15)(Basic Properties of IntegerValued Polynomials)..

.. ..

.

.

Let a ∈ Z, r ∈ N, and let (a0, a1, a2, . . .) be a sequence of integers.

a)For an integer valued polynomial p, we have deg(p) = r if and onlyif ∆rp(t) ∈ Z \ {0}. If this holds true, we have∆rp(t) = r!LCdeg(p) ∈ Z.b)Let p be an integer valued polynomial of degree r . Then thepolynomial q = p − r!LCDeg(p)

(t+ar)is an integer valued polynomial of

degree < r.c)For every r ≥ 0, the set of polynomials {

(t+aii)| 0 ≤ i ≤ r} is a

Z-basis of IP≤r . Consequently, the set {(t+ai

i)| i ∈ N} is a Z-basis

of IP.d)For a map f : Z −→ Z , the following conditions are equivalent.1)There exists an integer valued polynomial p ∈ IP with f(i) = p(i)for alli ∈ Z2)There exist a number i0 ∈ Z and an integer valued polynomialq ∈ IP such that f(i0) ∈ Z and ∆f(i) = q(i). for all i ∈ Z.



ideals

Jamal HosseinPoor

Outline

Introduction





Integer function


.. ..

.

.

Let a ∈ Z, r ∈ N, and let (a0, a1, a2, . . .) be a sequence of integers.a)For an integer valued polynomial p, we have deg(p) = r if and onlyif ∆rp(t) ∈ Z \ {0}. If this holds true, we have∆rp(t) = r!LCdeg(p) ∈ Z.

b)Let p be an integer valued polynomial of degree r . Then thepolynomial q = p − r!LCDeg(p)



(t+aii)| 0 ≤ i ≤ r} is a






ideals

Jamal HosseinPoor

Outline

Introduction





Integer function


.. ..

.

.

Let a ∈ Z, r ∈ N, and let (a0, a1, a2, . . .) be a sequence of integers.a)For an integer valued polynomial p, we have deg(p) = r if and onlyif ∆rp(t) ∈ Z \ {0}. If this holds true, we have∆rp(t) = r!LCdeg(p) ∈ Z.b)Let p be an integer valued polynomial of degree r . Then thepolynomial q = p − r!LCDeg(p)


degree < r.

c)For every r ≥ 0, the set of polynomials {(t+ai

i)| 0 ≤ i ≤ r} is a






ideals

Jamal HosseinPoor

Outline

Introduction





Integer function


.. ..

.

.




(t+aii)| 0 ≤ i ≤ r} is a



of IP.

d)For a map f : Z −→ Z , the following conditions are equivalent.1)There exists an integer valued polynomial p ∈ IP with f(i) = p(i)for alli ∈ Z2)There exist a number i0 ∈ Z and an integer valued polynomialq ∈ IP such that f(i0) ∈ Z and ∆f(i) = q(i). for all i ∈ Z.



ideals

Jamal HosseinPoor

Outline

Introduction





Integer function


.. ..

.

.




(t+aii)| 0 ≤ i ≤ r} is a






ideals

Jamal HosseinPoor

Outline

Introduction





Term order

.Definition 2.16..

.. ..

.

.

By a term order on Tn we mean a total order < on Tn satisfying thefollowing two conditions;

1)xα < 1for all xβ ∈ Tn, xβ ̸= 1,2) If Xα > Xβ =⇒ XγXα > XγXβ ,for all α, β, γ ∈ Nn

.Example 2.17..

.. ..

.

.

1)Lexicographical order (>lp):

Xα >lp Xβ ⇐⇒ ∃1 ≤ i ≤ n : α1 = β1, ..., αi−1 = βi−1, αi > βi

2)Degree lexicographical order (>Dp):

Xα >Dp Xβ ⇐⇒ degXα > degXβ

or(degXα = degXβ and ∃1 ≤ i ≤ n : α1 = β1, ..., αi−1 = βi−1, αi > βi)



ideals

Jamal HosseinPoor

Outline

Introduction





Term order

.Definition 2.16..

.. ..

.

.

By a term order on Tn we mean a total order < on Tn satisfying thefollowing two conditions;1)xα < 1for all xβ ∈ Tn, xβ ̸= 1,

2) If Xα > Xβ =⇒ XγXα > XγXβ ,for all α, β, γ ∈ Nn

.Example 2.17..

.. ..

.

.








ideals

Jamal HosseinPoor

Outline

Introduction





Term order

.Definition 2.16..

.. ..

.

.

By a term order on Tn we mean a total order < on Tn satisfying thefollowing two conditions;1)xα < 1for all xβ ∈ Tn, xβ ̸= 1,2) If Xα > Xβ =⇒ XγXα > XγXβ ,for all α, β, γ ∈ Nn

.Example 2.17..

.. ..

.

.








ideals

Jamal HosseinPoor

Outline

Introduction





Term order

.Definition 2.16..

.. ..

.

.


.Example 2.17..

.. ..

.

.








ideals

Jamal HosseinPoor

Outline

Introduction





Term order

.Definition 2.16..

.. ..

.

.


.Example 2.17..

.. ..

.

.








ideals

Jamal HosseinPoor

Outline

Introduction





Division algorithm

.Definition 2.18..

.. ..

.

.

Let f = aαXα + aβXβ + . . .+ aγXγ be a polynomial on K[x1, . . . , xn]

and Xα > Xβ > . . . > Xγ .

(1)LM(f) := leadmonomial(f) := Xα

(2)LE(f) := leadexp(f) := α(3)LC(f) := leadcoef(f) := aα(4)LT(f) := leadterm(f) := aαXα

(5)tail(f) := f − leadterm(f) := aβXβ + ...+ aγXγ

.Definition 2.19..

.. ..

.

.

Let f, h and f1, . . . , fs be polynomials in K[x1, . . . , xn], with fi ̸= 0, andlet F = {f1, . . . , fs}. We say that f reduces to h modulo F, denotedf F−→+ h, if and only if there exist i1, i2, . . . , it ∈ {1, 2, . . . , s}andh1, h2, . . . , ht−1 ∈ K[x1, . . . , xn] such that

f fi1−→ h1

fi2−→ h2

fi3−→ . . .fit−1−→ ht−1

fit−→ h



ideals

Jamal HosseinPoor

Outline

Introduction





Division algorithm

.Definition 2.18..

.. ..

.

.


and Xα > Xβ > . . . > Xγ .(1)LM(f) := leadmonomial(f) := Xα



.Definition 2.19..

.. ..

.

.


f fi1−→ h1

fi2−→ h2

fi3−→ . . .fit−1−→ ht−1

fit−→ h



ideals

Jamal HosseinPoor

Outline

Introduction





Division algorithm

.Definition 2.18..

.. ..

.

.



(2)LE(f) := leadexp(f) := α

(3)LC(f) := leadcoef(f) := aα(4)LT(f) := leadterm(f) := aαXα


.Definition 2.19..

.. ..

.

.


f fi1−→ h1

fi2−→ h2

fi3−→ . . .fit−1−→ ht−1

fit−→ h



ideals

Jamal HosseinPoor

Outline

Introduction





Division algorithm

.Definition 2.18..

.. ..

.

.



(2)LE(f) := leadexp(f) := α(3)LC(f) := leadcoef(f) := aα

(4)LT(f) := leadterm(f) := aαXα


.Definition 2.19..

.. ..

.

.


f fi1−→ h1

fi2−→ h2

fi3−→ . . .fit−1−→ ht−1

fit−→ h



ideals

Jamal HosseinPoor

Outline

Introduction





Division algorithm

.Definition 2.18..

.. ..

.

.





.Definition 2.19..

.. ..

.

.


f fi1−→ h1

fi2−→ h2

fi3−→ . . .fit−1−→ ht−1

fit−→ h



ideals

Jamal HosseinPoor

Outline

Introduction





Division algorithm

.Definition 2.18..

.. ..

.

.





.Definition 2.19..

.. ..

.

.


f fi1−→ h1

fi2−→ h2

fi3−→ . . .fit−1−→ ht−1

fit−→ h



ideals

Jamal HosseinPoor

Outline

Introduction





Division algorithm

.Definition 2.18..

.. ..

.

.





.Definition 2.19..

.. ..

.

.


f fi1−→ h1

fi2−→ h2

fi3−→ . . .fit−1−→ ht−1

fit−→ h



ideals

Jamal HosseinPoor

Outline

Introduction





Division algorithm

.Note 2.20)(Multivariable division algorithm)..

.. ..

.

.

INPUT: f, f1, . . . , fs ∈ K[x1, . . . , xn] with fi ̸= 0 (1 ≤ i ≤ s)OUTPUT:u1, ..., us, r such that f = u1f1 + . . .+ usfs + r and

r is reduced with respect to {f1, . . . , fs} andmax(LM(u1)LM(f1), . . . , LM(us)LM(fs), LM(r)) = LM(f).

INITIALIZATION: u1 := 0, u2 := 0, . . . , us := 0, r := 0, h; = fWHILE h ̸= 0 DOIF there exists i such that LM(fi) divides LM(h) THENchoose i least such that LM(fi) divides LM( h)

ui = ui +LT(h)LT(fi)

h := h − LT(h)LT(fi)

fiELSE

r := r + lt(h)h := h − lt(h)



ideals

Jamal HosseinPoor

Outline

Introduction





Grübner basis

.Theorem 2.21)..

.. ..

.

.

Given a set of non-zero polynomials F = {f1, ..., fs} andf ∈ K[x1, ..., xn] , the Division algorithm produces polynomialu1, ..., us ∈ K[x1, ..., xn] such that

f = u1f1 + ...+ usfs + r,

with r reduced with respect to F and

LM(f) = max(max1≤i≤s(LM(ui)LM(fi), LM(r))

.Definition 2.22)..

.. ..

.

.

A set of non-zero polynomials G = {g1, . . . , gt} contained in anideal I, is called a Grübner basis for I if and only if for all f ∈ Isuch that f ̸= 0, there exists i ∈ {1, . . . , t} such thatLM(gi) | LM(f).



ideals

Jamal HosseinPoor

Outline

Introduction





Grübner basis

.Theorem 2.21)..

.. ..

.

.


f = u1f1 + ...+ usfs + r,



.Definition 2.22)..

.. ..

.

.




ideals

Jamal HosseinPoor

Outline

Introduction





Grübner basis

.Theorem 2.21)..

.. ..

.

.


f = u1f1 + ...+ usfs + r,



.Definition 2.22)..

.. ..

.

.




ideals

Jamal HosseinPoor

Outline

Introduction





Grübner basis

.Theorem 2.21)..

.. ..

.

.


f = u1f1 + ...+ usfs + r,



.Definition 2.22)..

.. ..

.

.




ideals

Jamal HosseinPoor

Outline

Introduction





Grübner basis

.Theorem 2.21)..

.. ..

.

.


f = u1f1 + ...+ usfs + r,



.Definition 2.22)..

.. ..

.

.




ideals

Jamal HosseinPoor

Outline

Introduction





S-polynomial

.Defination 2.23)..

.. ..

.

.

Let f and g ∈ k[x1, ..., xn] be nonzero polynomials.Let L = lcm(LM(f), LM(g)), then

S(f, g) = LLT(f) f − L

LT(g)g

.Theorem 2.24)(Buchberger cretition)..

.. ..

.

.

Let G = {g1, ..., gt} be a set of non-zero polynomials inK[x1, ..., xn] . Then G is a Grübner basis for the idealI =< g1, ..., gt > if and only if for all i ̸= j, ,

S(gi, gj)G→+ 0



ideals

Jamal HosseinPoor

Outline

Introduction





S-polynomial

.Defination 2.23)..

.. ..

.

.



LT(g)g


.. ..

.

.


S(gi, gj)G→+ 0



ideals

Jamal HosseinPoor

Outline

Introduction





S-polynomial

.Defination 2.23)..

.. ..

.

.



LT(g)g


.. ..

.

.


S(gi, gj)G→+ 0



ideals

Jamal HosseinPoor

Outline

Introduction





S-polynomial

.Defination 2.23)..

.. ..

.

.



LT(g)g


.. ..

.

.


S(gi, gj)G→+ 0



ideals

Jamal HosseinPoor

Outline

Introduction





Grübner basis

.Note 2.24)Buchberger’s algorithm for ComputingGrübner basis..

.. ..

.

.

INPUT: F = {f1, . . . , fs} ⊆ K[x1, . . . , xn] with fi ̸= 0 (1 ≤ i ≤ s)OUTPUT: G = {g1, . . . , gt}, a Grobner basis for (f1, . . . , fs)INITIALIZATION: G := F, g:={{fi, fj} | fi ̸= fj ∈ G}WHILE g ̸= ϕ DOChoose any {f, f

′} ∈ g

g := g − {{f, g}}S(f, f

′)

G→+ h, where h is reduced with respect to GIF h ̸= 0 THEN

g := g ∪ {{u, h} | for all u ∈ G}G := G ∪ {h}



ideals

Jamal HosseinPoor

Outline

Introduction





Hilbert function

.Definition 2.25)..

.. ..

.

.

Let M be a finitely generated graded P -module. Since thegrading given by W on P is of positive type,we a well-definedmap

HFM : Z −→ Z

i 7→ dimK(Mi)

This map is called the Hilbert function of M.

.Proposition 2.26).... ..

.

.

For every i ∈ N, we have HFP(i) =(i+n−1

n−1

)



ideals

Jamal HosseinPoor

Outline

Introduction





Hilbert function

.Definition 2.25)..

.. ..

.

.


HFM : Z −→ Z

i 7→ dimK(Mi)



.

.


n−1

)



ideals

Jamal HosseinPoor

Outline

Introduction





Hilbert function

.Definition 2.25)..

.. ..

.

.


HFM : Z −→ Z

i 7→ dimK(Mi)



.

.


n−1

)



ideals

Jamal HosseinPoor

Outline

Introduction





Hilbert function

.Definition 2.25)..

.. ..

.

.


HFM : Z −→ Z

i 7→ dimK(Mi)



.

.


n−1

)



ideals

Jamal HosseinPoor

Outline

Introduction





Hilbert function

.Definition 2.25)..

.. ..

.

.


HFM : Z −→ Z

i 7→ dimK(Mi)



.

.


n−1

)



ideals

Jamal HosseinPoor

Outline

Introduction





Hilbert function

.Theorem 2.27)(Basic properties of Hilbert functions)..

.. ..

.

.

Let M,M′,M

′′be three finitely generated graded P -modules and

M(j)(i) = M(i + j) for any i, j ∈ Z.

a)The Hilbert function of the module M(j) obtained by shiftingdegrees by j is given by HFM(j)(i) = HFM(i + j) for all i, j ∈ Z.b)Given a homogeneous exact sequence of graded P -modules

0 −→ M′−→ M −→ M

′′−→ 0

for all i ∈ Z we have;

HFM(i) = HFM′ (i) + HFM′′ (i) .

c)Given finitely many finitely generated graded P -modulesM1, . . . ,Mr,for all i ∈ Z we have

HFM1

⊕...

⊕Mr(i) = HFM1

(i) + . . .+ HFMr(i).



ideals

Jamal HosseinPoor

Outline

Introduction





Hilbert function


.. ..

.

.

Let M,M′,M


M(j)(i) = M(i + j) for any i, j ∈ Z.a)The Hilbert function of the module M(j) obtained by shiftingdegrees by j is given by HFM(j)(i) = HFM(i + j) for all i, j ∈ Z.

b)Given a homogeneous exact sequence of graded P -modules

0 −→ M′−→ M −→ M

′′−→ 0


HFM(i) = HFM′ (i) + HFM′′ (i) .


HFM1

⊕...

⊕Mr(i) = HFM1

(i) + . . .+ HFMr(i).



ideals

Jamal HosseinPoor

Outline

Introduction





Hilbert function


.. ..

.

.

Let M,M′,M


M(j)(i) = M(i + j) for any i, j ∈ Z.a)The Hilbert function of the module M(j) obtained by shiftingdegrees by j is given by HFM(j)(i) = HFM(i + j) for all i, j ∈ Z.b)Given a homogeneous exact sequence of graded P -modules

0 −→ M′−→ M −→ M

′′−→ 0


HFM(i) = HFM′ (i) + HFM′′ (i) .


HFM1

⊕...

⊕Mr(i) = HFM1

(i) + . . .+ HFMr(i).



ideals

Jamal HosseinPoor

Outline

Introduction





Hilbert function


.. ..

.

.

Let M,M′,M



0 −→ M′−→ M −→ M

′′−→ 0


HFM(i) = HFM′ (i) + HFM′′ (i) .


HFM1

⊕...

⊕Mr(i) = HFM1

(i) + . . .+ HFMr(i).



ideals

Jamal HosseinPoor

Outline

Introduction





Hilbert function


.. ..

.

.

Let M,M′,M



0 −→ M′−→ M −→ M

′′−→ 0


HFM(i) = HFM′ (i) + HFM′′ (i) .


HFM1

⊕...

⊕Mr(i) = HFM1

(i) + . . .+ HFMr(i).



ideals

Jamal HosseinPoor

Outline

Introduction





Hilbert function


.. ..

.

.

Let M,M′,M



0 −→ M′−→ M −→ M

′′−→ 0


HFM(i) = HFM′ (i) + HFM′′ (i) .


HFM1

⊕...

⊕Mr(i) = HFM1

(i) + . . .+ HFMr(i).



ideals

Jamal HosseinPoor

Outline

Introduction





Hilbert function


.. ..

.

.

Let M,M′,M



0 −→ M′−→ M −→ M

′′−→ 0


HFM(i) = HFM′ (i) + HFM′′ (i) .


HFM1

⊕...

⊕Mr(i) = HFM1

(i) + . . .+ HFMr(i).



ideals

Jamal HosseinPoor

Outline

Introduction





Hilbert function


.. ..

.

.

Let M,M′,M



0 −→ M′−→ M −→ M

′′−→ 0


HFM(i) = HFM′ (i) + HFM′′ (i) .


HFM1

⊕...

⊕Mr(i) = HFM1

(i) + . . .+ HFMr(i).



ideals

Jamal HosseinPoor

Outline

Introduction





Hilbert function


.. ..

.

.

Let M,M′,M



0 −→ M′−→ M −→ M

′′−→ 0


HFM(i) = HFM′ (i) + HFM′′ (i) .


HFM1

⊕...

⊕Mr(i) = HFM1

(i) + . . .+ HFMr(i).



ideals

Jamal HosseinPoor

Outline

Introduction





Hilbert function

.Theorem 2.28(The Multiplication sequence)..

.. ..

.

.

Let M be a finitely generated graded P -module, and let f ∈ P be anon-zero homogeneous polynomial of degree d.

a)There is a homogeneous exact sequence of graded P-modules

0 −→ [M/(0 :M (f))](−d) φ−→ M −→ M/fM −→ 0

where the map ϕ is induced by multiplication by fb)For all i ∈ Z, we have

HFM/fM(i) = HFM(i)− HFM/(0:Mf)(i − d)

c) The polynomial f is a non-zerodivisor for the module M if and onlyif

HFM/fM(i) = ∆dHFM(i)

for all For all i ∈ Z .



ideals

Jamal HosseinPoor

Outline

Introduction





Hilbert function


.. ..

.

.

Let M be a finitely generated graded P -module, and let f ∈ P be anon-zero homogeneous polynomial of degree d.a)There is a homogeneous exact sequence of graded P-modules

0 −→ [M/(0 :M (f))](−d) φ−→ M −→ M/fM −→ 0

where the map ϕ is induced by multiplication by f

b)For all i ∈ Z, we have







ideals

Jamal HosseinPoor

Outline

Introduction





Hilbert function


.. ..

.

.


0 −→ [M/(0 :M (f))](−d) φ−→ M −→ M/fM −→ 0








ideals

Jamal HosseinPoor

Outline

Introduction





Hilbert function


.. ..

.

.


0 −→ [M/(0 :M (f))](−d) φ−→ M −→ M/fM −→ 0








ideals

Jamal HosseinPoor

Outline

Introduction





Hilbert function


.. ..

.

.


0 −→ [M/(0 :M (f))](−d) φ−→ M −→ M/fM −→ 0








ideals

Jamal HosseinPoor

Outline

Introduction





Hilbert function


.. ..

.

.


0 −→ [M/(0 :M (f))](−d) φ−→ M −→ M/fM −→ 0








ideals

Jamal HosseinPoor

Outline

Introduction





Macaulay’s basis theorem and Hilbert function

.Theorem 2.29)..

.. ..

.

.

Let K be a field and I be a homogenoues ideal inP = K[x1, . . . , xn] and σ be a term order on Tn . The set of allterms of Tn \ LTσ(I) is denoted by B . B forms a K - basis forP/I .

.Result 2.30)..

.. ..

.

.

Let I be a homogenoues ideal and P and σ be a term order onTn then HFP/I(i) = HFP/LTσ(I)(i) for i ∈ Z.



ideals

Jamal HosseinPoor

Outline

Introduction





Macaulay’s basis theorem and Hilbert function

.Theorem 2.29)..

.. ..

.

.

Let K be a field and I be a homogenoues ideal inP = K[x1, . . . , xn] and σ be a term order on Tn . The set of allterms of Tn \ LTσ(I) is denoted by B . B forms a K - basis forP/I .

.Result 2.30)..

.. ..

.

.

Let I be a homogenoues ideal and P and σ be a term order onTn then HFP/I(i) = HFP/LTσ(I)(i) for i ∈ Z.



ideals

Jamal HosseinPoor

Outline

Introduction





Hilbert series

.Definition 2.31)..

.. ..

.

.

For a finitely generated graded P -module M, the associatedLaurent series of the Hilbert function of M is called the Hilbertseries of M and is denoted by HSM . In other words, theHilbert series of M is the Laurent series

HSM(z) =∑

i≥α HFM(i)zi ∈ Z[[z]]z

, where α = α(M) is the initial degree of M.

.Proposition 2.32)..

.. ..

.

.

The Hilbert series of P = K[x1, . . . , xn] is given by

HSP(z) =1

(1− z)d .



ideals

Jamal HosseinPoor

Outline

Introduction





Hilbert series

.Definition 2.31)..

.. ..

.

.


HSM(z) =∑




.. ..

.

.


HSP(z) =1

(1− z)d .



ideals

Jamal HosseinPoor

Outline

Introduction





Hilbert series

.Definition 2.31)..

.. ..

.

.


HSM(z) =∑




.. ..

.

.


HSP(z) =1

(1− z)d .



ideals

Jamal HosseinPoor

Outline

Introduction





Hilbert series

.Definition 2.31)..

.. ..

.

.


HSM(z) =∑




.. ..

.

.


HSP(z) =1

(1− z)d .



ideals

Jamal HosseinPoor

Outline

Introduction





Hilbert series

.Definition 2.31)..

.. ..

.

.


HSM(z) =∑




.. ..

.

.


HSP(z) =1

(1− z)d .



ideals

Jamal HosseinPoor

Outline

Introduction





Hilbert Series

.Theorem 2.33)..

.. ..

.

.

Let K be a field and P be a standard grading ring. Letf ∈ P \ {0} be a homogeneous polynomial of degree d. Hilbertfunction and Hilbert series of the graded P-module P/⟨f⟩ are ofthe forms

HFP/⟨f⟩(i) = HFP(i)− HFP(i − d)

,

HSP/⟨f⟩(z) =1− zd

(1− z)n



ideals

Jamal HosseinPoor

Outline

Introduction





Hilbert Series

.Theorem 2.33)..

.. ..

.

.



,


(1− z)n



ideals

Jamal HosseinPoor

Outline

Introduction





Hilbert Series

.Theorem 2.33)..

.. ..

.

.



,


(1− z)n



ideals

Jamal HosseinPoor

Outline

Introduction





Hilbert series of monomial ideals

.Theorem 2.34)..

.. ..

.

.

Let P = K[x1, ..., xn] with standard grading and t1, ..., tr be nonzeromonomials in P.Let

| tj |= deg(tj), 1 ≤ j ≤ r ,

| tj1 ∨ tj2 ∨ . . . ∨ tjs |= lcm(deg(tj1), deg(tj2), . . . , deg(tjs))

thenHSP/(t1,...,tr)(z)

=

1 −r∑

j=1z|tj| + · · · + (−1)s ∑

1≤j1≤···≤js≤rz|tj1∨···∨tjs | + . . . + (−1)rz|tj1∨···∨tjr |

(1 − z)n



ideals

Jamal HosseinPoor

Outline

Introduction






.Theorem 2.34)..

.. ..

.

.


| tj |= deg(tj), 1 ≤ j ≤ r ,



=

1 −r∑

j=1z|tj| + · · · + (−1)s ∑


(1 − z)n



ideals

Jamal HosseinPoor

Outline

Introduction






.Theorem 2.34)..

.. ..

.

.


| tj |= deg(tj), 1 ≤ j ≤ r ,



=

1 −r∑

j=1z|tj| + · · · + (−1)s ∑


(1 − z)n



ideals

Jamal HosseinPoor

Outline

Introduction






.Definition 2.35)..

.. ..

.

.

Let M be a non-zero finitely generated graded P -module, and letα(M) = min{i ∈ Z | Mi ̸= 0}. Then the Hilbert series of M has theform

HSM(z) = zα(M)HNM(z)(1− z)n

where HNM(z) ∈ Z[z] and HNM(0) = HFM(α(M)) > 0.Aftereliminating the factor(1− z)from numerator and denumerator,we have

HSM(z) = zα(M)hnM(z)(1− z)d



ideals

Jamal HosseinPoor

Outline

Introduction






.Definition 2.35)..

.. ..

.

.







ideals

Jamal HosseinPoor

Outline

Introduction






.Definition 2.35)..

.. ..

.

.







ideals

Jamal HosseinPoor

Outline

Introduction






.Definition 2.35)..

.. ..

.

.







ideals

Jamal HosseinPoor

Outline

Introduction






.Example 2.36..

.. ..

.

.

Let P = Q[x1, x2, x3] with standard grading andI = (x31x2, x2x32, x32x3, x43). The Hilbert series of P/I is

HSP/(f1,f2,f3,f4(z) =1− 2z3 − z4 + z5 + 2z6 − z7

(1− z)3

=1 + 2z + 3z2 + 2z3 − z5

1− z



ideals

Jamal HosseinPoor

Outline

Introduction






.Example 2.36..

.. ..

.

.

Let P = Q[x1, x2, x3] with standard grading andI = (x31x2, x2x32, x32x3, x43). The Hilbert series of P/I is

HSP/(f1,f2,f3,f4(z) =1− 2z3 − z4 + z5 + 2z6 − z7

(1− z)3

=1 + 2z + 3z2 + 2z3 − z5

1− z



ideals

Jamal HosseinPoor

Outline

Introduction





Bayer-Stillman algorithm

.Theorem 3.1)..

.. ..

.

.

Let I be a monomial ideal, and write I = (J,XA) for a monomial idealJ and a monomial XA . Let = p(z) denote the numerator g(z)of the Hilbert series for P/I, and let | A | denote the total degree ofthe monomial XA . Then

1) < XA >= 1− z|A|

2) < J ∩ (XA) >= 1− z|A| + z|A| < J : XA >3) =< J > −z|A| < J : XA >

.Theorem 3.2..

.. ..

.

.

Let I be a monomial ideal. Suppose that the variables x0, ..., xn of Pcan be partitioned into disjoint sets V1 ∪ .... ∪ Vj , such that eachgenerator of I belongs to the subring K[Vi] for some i . DefineIi = I ∩ k[Vi] . Then

=∏

< Ii >



ideals

Jamal HosseinPoor

Outline

Introduction






.Theorem 3.1)..

.. ..

.

.

Let I be a monomial ideal, and write I = (J,XA) for a monomial idealJ and a monomial XA . Let = p(z) denote the numerator g(z)of the Hilbert series for P/I, and let | A | denote the total degree ofthe monomial XA . Then1) < XA >= 1− z|A|


.Theorem 3.2..

.. ..

.

.


=∏

< Ii >



ideals

Jamal HosseinPoor

Outline

Introduction






.Theorem 3.1)..

.. ..

.

.


2) < J ∩ (XA) >= 1− z|A| + z|A| < J : XA >

3) =< J > −z|A| < J : XA >

.Theorem 3.2..

.. ..

.

.


=∏

< Ii >



ideals

Jamal HosseinPoor

Outline

Introduction






.Theorem 3.1)..

.. ..

.

.



.Theorem 3.2..

.. ..

.

.


=∏

< Ii >



ideals

Jamal HosseinPoor

Outline

Introduction






.Theorem 3.1)..

.. ..

.

.



.Theorem 3.2..

.. ..

.

.


=∏

< Ii >



ideals

Jamal HosseinPoor

Outline

Introduction






.Theorem 3.1)..

.. ..

.

.



.Theorem 3.2..

.. ..

.

.


=∏

< Ii >



ideals

Jamal HosseinPoor

Outline

Introduction






.Note 3.2..

.. ..

.

.

input :A monomial ideal I minimally generated by (xA1 , ..., xAr)output:The numerator of the Hilbert series for S/Ibegin(*a) rearrange(xA1 , ..., xAr) so that they are in ascendinglexicographic order on the reversed set of variables xn, ..., x0(b) ifr = 0then set h(t) := 1else set h(t) := 1− t|A1| .(c) for i := 2 to r dodefine xBj to be the least monomial such that x|Ai|x|Bj|

is a multiple ofx|Aj| ,for j = 1, ..., i − 1.set J = (xA1 , ..., xAi−1 : xAi) := (xB1 , ..., xBi−1),and reduce to a minimal set of generators .Compute < J > by one of the following three variants .



ideals

Jamal HosseinPoor

Outline

Introduction






.Note 3.2)Continue of the algorithm..

.. ..

.

.

variant A :set (J1, xj1, ..., xjk) := J,where J1 contains no linear monomials .compute < J1 > by a recursive call to this procedure .set < J >:= (1− t)k < J1 >.variant B :partition the variables x0, ..., xn into disjoint setsV1 ∪ ... ∪ Vp such that each generator of Jbelongs to the subring k[Vj],for somej .for each j = 1, ..., p doset Jj := J ∩ k[Vj]compute < Jj > by a recursive call to this procedure.end for;set < J >:=< J1 >< J2 > ... < Jp > .variant C:compute < J > by a recursive call to this procedure.set h(t) := h(t)− t|Ai|(J); end for;return := h(t) ;end.



ideals

Jamal HosseinPoor

Outline

Introduction






.Theorem 3.3..

.. ..

.

.

Let T(m, n) denote an upper bound for the number of recursiveprocedure calls made by Bayer-Stillman algorithm , to compute theHilbert series of a monomial ideal I with m generators in the n + 1variables of Pn . Then

(i) if step (a) is omitted, we can take

T(m, n) = 2m−1

(ii) if step (a) is not omitted, we can take

T(m, n) ={ (m

n)+( m

n−2

)+ ...+

(m0

)if n ≥ 0 , n = 2k(m

n)+( m

n−2

)+ ...+

(m1

)if n ≥ 1 , n = 2k + 1

for any m ≥ 1 so for m >> n,

T(m, n) ≈ mn

n!



ideals

Jamal HosseinPoor

Outline

Introduction






.Theorem 3.3..

.. ..

.

.

Let T(m, n) denote an upper bound for the number of recursiveprocedure calls made by Bayer-Stillman algorithm , to compute theHilbert series of a monomial ideal I with m generators in the n + 1variables of Pn . Then(i) if step (a) is omitted, we can take

T(m, n) = 2m−1


T(m, n) ={ (m

n)+( m

n−2

)+ ...+

(m0

)if n ≥ 0 , n = 2k(m

n)+( m

n−2

)+ ...+

(m1

)if n ≥ 1 , n = 2k + 1


T(m, n) ≈ mn

n!



ideals

Jamal HosseinPoor

Outline

Introduction






.Theorem 3.3..

.. ..

.

.


T(m, n) = 2m−1


T(m, n) ={ (m

n)+( m

n−2

)+ ...+

(m0

)if n ≥ 0 , n = 2k(m

n)+( m

n−2

)+ ...+

(m1

)if n ≥ 1 , n = 2k + 1


T(m, n) ≈ mn

n!



ideals

Jamal HosseinPoor

Outline

Introduction






.Theorem 3.3..

.. ..

.

.


T(m, n) = 2m−1


T(m, n) ={ (m

n)+( m

n−2

)+ ...+

(m0

)if n ≥ 0 , n = 2k(m

n)+( m

n−2

)+ ...+

(m1

)if n ≥ 1 , n = 2k + 1


T(m, n) ≈ mn

n!



ideals

Jamal HosseinPoor

Outline

Introduction






.Theorem 3.3..

.. ..

.

.


T(m, n) = 2m−1


T(m, n) ={ (m

n)+( m

n−2

)+ ...+

(m0

)if n ≥ 0 , n = 2k(m

n)+( m

n−2

)+ ...+

(m1

)if n ≥ 1 , n = 2k + 1


T(m, n) ≈ mn

n!



ideals

Jamal HosseinPoor

Outline

Introduction






.Theorem 3.3..

.. ..

.

.


T(m, n) = 2m−1


T(m, n) ={ (m

n)+( m

n−2

)+ ...+

(m0

)if n ≥ 0 , n = 2k(m

n)+( m

n−2

)+ ...+

(m1

)if n ≥ 1 , n = 2k + 1


T(m, n) ≈ mn

n!



ideals

Jamal HosseinPoor

Outline

Introduction






.Theorem 3.3..

.. ..

.

.


T(m, n) = 2m−1


T(m, n) ={ (m

n)+( m

n−2

)+ ...+

(m0

)if n ≥ 0 , n = 2k(m

n)+( m

n−2

)+ ...+

(m1

)if n ≥ 1 , n = 2k + 1


T(m, n) ≈ mn

n!



ideals

Jamal HosseinPoor

Outline

Introduction





A note about algorithms computing Hilbert function

.Proposition 3.4..

.. ..

.

.

The following problem is NP-complete : Given a monomialideal J ⊂ K[x0, . . . , xn], and an integer K, is the codimension ofJ < K?

One might wonder whether there is an algorithm for Hilbertfunctions which executes in polynomial time in the size of theinput .The proposition shows that such an algorithm cannotexist (unless P = NP).



ideals

Jamal HosseinPoor

Outline

Introduction





A note about algorithms computing Hilbert function

.Proposition 3.4..

.. ..

.

.

The following problem is NP-complete : Given a monomialideal J ⊂ K[x0, . . . , xn], and an integer K, is the codimension ofJ < K?One might wonder whether there is an algorithm for Hilbertfunctions which executes in polynomial time in the size of theinput .The proposition shows that such an algorithm cannotexist (unless P = NP).



ideals

Jamal HosseinPoor

Outline

Introduction





Implementation notes about Bayer-Stillman algorithm

.Note 3.5..

.. ..

.

.

Finding the Hilbert function of a homogeneous ideal involves twosteps . A Grübner basis for the ideal is first computed, and theHilbert function of the monomial ideal of initial terms is then found .In small numbers of variables, the Grübner basis computation takesmuch longer than the corresponding Hilbert function computation,even if Macaulay’s Hilbert function algorithm is used . For morevariables, we found the Hilbert function algorithm of Macaulay totake much longer than the corresponding Grübner basis computation.



ideals

Jamal HosseinPoor

Outline

Introduction






.Example 3.6)..

.. ..

.

.

Let M be a generic n by n matrix in a polynomial ring S with n2

variables xij, let I ⊂ P be the ideal generated by the entries of thematrix M2,and let > be the reverse lexicographic order such thatx11 > x12 > . . . > x1n > x21 > . . .The monomial ideal which is theinput to the Hilbert function algorithms is the initial ideal J = in(I),with respect to this order when n = 5

= (1− z)25HSP/I(z)=1-25z2 + 24z3 + 275z4 − 576z5 − 1400z6 + 6024z7 + 503z8-51176z9 + 179220z10 − 393000z11 + 671803z12 − 963728z13

+1161279z14 − 1105688z15 + 687275z16 + 18472z17 − 740749z18+1182944z19 − 1222811z20 + 960600z21 − 600774z22 + 303600z23-123971z24 + 40480z25 − 10350z26 + 2000z27 − 275z28 + 24z29 − z30



ideals

Jamal HosseinPoor

Outline

Introduction






.Example 3.6)..

.. ..

.

.

Let M be a generic n by n matrix in a polynomial ring S with n2

variables xij, let I ⊂ P be the ideal generated by the entries of thematrix M2,and let > be the reverse lexicographic order such thatx11 > x12 > . . . > x1n > x21 > . . .The monomial ideal which is theinput to the Hilbert function algorithms is the initial ideal J = in(I),with respect to this order when n = 5

= (1− z)25HSP/I(z)=1-25z2 + 24z3 + 275z4 − 576z5 − 1400z6 + 6024z7 + 503z8-51176z9 + 179220z10 − 393000z11 + 671803z12 − 963728z13

+1161279z14 − 1105688z15 + 687275z16 + 18472z17 − 740749z18+1182944z19 − 1222811z20 + 960600z21 − 600774z22 + 303600z23-123971z24 + 40480z25 − 10350z26 + 2000z27 − 275z28 + 24z29 − z30



ideals

Jamal HosseinPoor

Outline

Introduction






.Example 3.6)..

.. ..

.

.

Example 1:(n=4);16 var,161 gen;26 gen in radicalAlgorithm time(mm:ss) spaceMacaulay Grübner basis 0:15 126 kA 0:01 -A1 0:01 -B 0:01 -Macaulay Hilbert fcn 0:36 -codimension = 8 0:00 -

Example 1:(n=5);25 var,1372 gen;91 gen in radicalMacaulay Grübner basis 1:04:15 1950 kA 3:06 252 kA1 3:47 252 kB 3:08 755 kMacaulay Hilbert fcn 14:32:24 504 kcodimension = 13 0:02 126 k



ideals

Jamal HosseinPoor

Outline

Introduction






.Example 3.6)..

.. ..

.

.

Example 1:(n=4);16 var,161 gen;26 gen in radicalAlgorithm time(mm:ss) spaceMacaulay Grübner basis 0:15 126 kA 0:01 -A1 0:01 -B 0:01 -Macaulay Hilbert fcn 0:36 -codimension = 8 0:00 -

Example 1:(n=5);25 var,1372 gen;91 gen in radicalMacaulay Grübner basis 1:04:15 1950 kA 3:06 252 kA1 3:47 252 kB 3:08 755 kMacaulay Hilbert fcn 14:32:24 504 kcodimension = 13 0:02 126 k



ideals

Jamal HosseinPoor

Outline

Introduction





Bigatti, Caboara and Robbiano algorithm

.Lemma 3.7)..

.. ..

.

.

Let L be a set of terms, I := (L) and Xaa term. Then

1)< XaI >= 1− z|a| + z|a| = 1 + z|a|( −1) < (Xa, I) >= −z|a| 

2)< (Xa, I) >= −z|a| 

.Definition 3.8)Decomposition of I with respect to xi...

.. ..

.

.

Let L be a set of terms, which is minimal in the sense that no elementof I := (L) is multiple of any other element of I := (L). Let I := (L)and let xi be one of the indeterminates. Then we may uniquely decompose L = xai

i La1 ∪ xa2i La2∪, ...,∪xam−1

i Lam−1 ∪ xami Lam where

aj ∈ N,j = 1, . . . ,m;0 ≤ a1 < a2 < . . . < am and the Laj are minimalsets of terms in {x1, . . . , x̂i, . . . , xn}. Hence we get a well-definedexpression I = (xa1

i Ia1 , xa2i Ia2 , . . . , x

am−1

i Iam−1, xam

i Iam) where Iaj = Laj .



ideals

Jamal HosseinPoor

Outline

Introduction






.Lemma 3.7)..

.. ..

.

.

Let L be a set of terms, I := (L) and Xaa term. Then1)< XaI >= 1− z|a| + z|a| = 1 + z|a|( −1) < (Xa, I) >

= −z|a| 

2)< (Xa, I) >= −z|a| 


.. ..

.

.


i La1 ∪ xa2i La2∪, ...,∪xam−1



i Ia1 , xa2i Ia2 , . . . , x

am−1

i Iam−1, xam




ideals

Jamal HosseinPoor

Outline

Introduction






.Lemma 3.7)..

.. ..

.

.


= −z|a| 2)< (Xa, I) >= −z|a| 


.. ..

.

.


i La1 ∪ xa2i La2∪, ...,∪xam−1



i Ia1 , xa2i Ia2 , . . . , x

am−1

i Iam−1, xam




ideals

Jamal HosseinPoor

Outline

Introduction






.Lemma 3.7)..

.. ..

.

.


= −z|a| 2)< (Xa, I) >= −z|a| 


.. ..

.

.


i La1 ∪ xa2i La2∪, ...,∪xam−1



i Ia1 , xa2i Ia2 , . . . , x

am−1

i Iam−1, xam




ideals

Jamal HosseinPoor

Outline

Introduction






.Theorem 3.9)..

.. ..

.

.

Let L be a set of terms in x1, ..., xn,I = (L); let

I = (xa1i Ia1 , xa2

i Ia2 , . . . , xam−1

i Iam−1 , xami Iam).

be the decomposition of I w.r. to xi and let Jar = Ia1 + ...+ Iar

r = 1, ...,m.Then1)= 1− z|a| + (za1 − za2) < Ja1 > +...+ (zam−1 − zam)

< Jam−1 > +zam < Jam > .2)= 1 + za1(< Ja1 > −1) + za2(< Ja2 > − < Ja1) + . . .+ zam

< Jam > − < Jam−1 > .



ideals

Jamal HosseinPoor

Outline

Introduction






.Theorem 3.9)..

.. ..

.

.


I = (xa1i Ia1 , xa2

i Ia2 , . . . , xam−1





< Jam > − < Jam−1 > .



ideals

Jamal HosseinPoor

Outline

Introduction






.Theorem 3.9)..

.. ..

.

.


I = (xa1i Ia1 , xa2

i Ia2 , . . . , xam−1





< Jam > − < Jam−1 > .



ideals

Jamal HosseinPoor

Outline

Introduction






.Theorem 3.9)..

.. ..

.

.


I = (xa1i Ia1 , xa2

i Ia2 , . . . , xam−1



r = 1, ...,m.Then

1)= 1− z|a| + (za1 − za2) < Ja1 > +...+ (zam−1 − zam)< Jam−1 > +zam < Jam > .

2)= 1 + za1(< Ja1 > −1) + za2(< Ja2 > − < Ja1) + . . .+ zam

< Jam > − < Jam−1 > .



ideals

Jamal HosseinPoor

Outline

Introduction






.Theorem 3.9)..

.. ..

.

.


I = (xa1i Ia1 , xa2

i Ia2 , . . . , xam−1




< Jam−1 > +zam < Jam > .

2)= 1 + za1(< Ja1 > −1) + za2(< Ja2 > − < Ja1) + . . .+ zam

< Jam > − < Jam−1 > .



ideals

Jamal HosseinPoor

Outline

Introduction






.Theorem 3.9)..

.. ..

.

.


I = (xa1i Ia1 , xa2

i Ia2 , . . . , xam−1





< Jam > − < Jam−1 > .



ideals

Jamal HosseinPoor

Outline

Introduction






.Definition 3.10..

.. ..

.

.

Let I be a monomial ideal in K[x1, ..., xn], given by its uniqueminimal basis. We denote as usual by ν(I) the cardinality ofsuch a basis. If I = (xa1

i Ia1 , xa2i Ia2 , . . . , x

am−1

i Iam−1 , xami Iam) is the

decomposition of I w.r.to xi and ifJar = Ia1 + . . .+ Iar , r = 1, ...,m, then we denote by νxi(I) theinteger νxi(I) =

∑mr=1 ν(Jar)

.Note 3.11)The algorithm(B.C.R)..

.. ..

.

.

Input : A minimal set L of terms in K[x1, ..., xn]Output : where I := (L)step(1) Choose an optimal xistep(2) Decompose I w.r.to xistep(3) Compute recursively < Ja1 >, ..., < Jar >step(4) Compute via theorem 3.9(1)



ideals

Jamal HosseinPoor

Outline

Introduction






.Definition 3.10..

.. ..

.

.

Let I be a monomial ideal in K[x1, ..., xn], given by its uniqueminimal basis. We denote as usual by ν(I) the cardinality ofsuch a basis. If I = (xa1

i Ia1 , xa2i Ia2 , . . . , x

am−1

i Iam−1 , xami Iam) is the

decomposition of I w.r.to xi and ifJar = Ia1 + . . .+ Iar , r = 1, ...,m, then we denote by νxi(I) theinteger νxi(I) =

∑mr=1 ν(Jar)

.Note 3.11)The algorithm(B.C.R)..

.. ..

.

.

Input : A minimal set L of terms in K[x1, ..., xn]Output : where I := (L)step(1) Choose an optimal xistep(2) Decompose I w.r.to xistep(3) Compute recursively < Ja1 >, ..., < Jar >step(4) Compute via theorem 3.9(1)



ideals

Jamal HosseinPoor

Outline

Introduction






.Definition 3.12)..

.. ..

.

.

Let I be a monomial ideal in K[x1, ..., xn]. Then we definedecomposition cost of I, shortly dc(I) = 1, by dc(I) := 1 if I isprincipal and recursively dc(I) =

∑mr=1 dc(Jar) if

I = (xa1i Ia1 , xa2

i Ia2 , . . . , xam−1

i Iam−1 , xami Iam) and xi is the chosen optimal

indeterminate. We may regard dc(I) as the intrinsic complexity of Iw.r.to Algorithm(BCR).


.. ..

.

.

Let I be a monomial ideal in K[x1, . . . , xn]. Thena)combcost(l) =O(n3dc(I)2)b)algcost(I) = O(2ndc(l))



ideals

Jamal HosseinPoor

Outline

Introduction






.Definition 3.12)..

.. ..

.

.


∑mr=1 dc(Jar) if

I = (xa1i Ia1 , xa2

i Ia2 , . . . , xam−1




.. ..

.

.

Let I be a monomial ideal in K[x1, . . . , xn]. Then

a)combcost(l) =O(n3dc(I)2)b)algcost(I) = O(2ndc(l))



ideals

Jamal HosseinPoor

Outline

Introduction






.Definition 3.12)..

.. ..

.

.


∑mr=1 dc(Jar) if

I = (xa1i Ia1 , xa2

i Ia2 , . . . , xam−1




.. ..

.

.

Let I be a monomial ideal in K[x1, . . . , xn]. Thena)combcost(l) =O(n3dc(I)2)

b)algcost(I) = O(2ndc(l))



ideals

Jamal HosseinPoor

Outline

Introduction






.Definition 3.12)..

.. ..

.

.


∑mr=1 dc(Jar) if

I = (xa1i Ia1 , xa2

i Ia2 , . . . , xam−1




.. ..

.

.

Let I be a monomial ideal in K[x1, . . . , xn]. Thena)combcost(l) =O(n3dc(I)2)b)algcost(I) = O(2ndc(l))



ideals

Jamal HosseinPoor

Outline

Introduction






.Example 3.14)..

.. ..

.

.

1)A = (x1, x2, x3, x4, x5, x6)52)B = (x1, x2, x3, x4, x5, x6)63)C = (x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12)34)D = (x2, xy, xz, xt, y3, y2z)75)E = (x5, x4y, x3y2, x2y3, xy4, y5, x4z, x3yz, x2y2z, x3z2, x2yz2)7

N0 of terms Macaulay CoCoAExample A 252 26 s 1.2 sExample B 462 65 s 7.3 sExample C 364 250 s 8.9 sExample D 288 34 s 4.7 sExample E 323 40 s 2.5 s



ideals

Jamal HosseinPoor

Outline

Introduction






.Example 3.14)..

.. ..

.

.

1)A = (x1, x2, x3, x4, x5, x6)52)B = (x1, x2, x3, x4, x5, x6)63)C = (x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12)34)D = (x2, xy, xz, xt, y3, y2z)75)E = (x5, x4y, x3y2, x2y3, xy4, y5, x4z, x3yz, x2y2z, x3z2, x2yz2)7

N0 of terms Macaulay CoCoAExample A 252 26 s 1.2 sExample B 462 65 s 7.3 sExample C 364 250 s 8.9 sExample D 288 34 s 4.7 sExample E 323 40 s 2.5 s



ideals

Jamal HosseinPoor

Outline

Introduction





Bigatti algorithm

.Definition 3.15)..

.. ..

.

.

we will say it is a n-list if it contains n power-products which arenot simple. This fact extends what is called variant (A) in [BS].

.Note 3.16)Base cases..

.. ..

.

.

1)(0-base case)If I = (xan1n1 , . . . , xansns ) then

< xan1n1 , . . . , xansns >=∏

i=1,...,s(1− tdegxani

i ) < 0 >=∏


i ).

2)(1-base case)If p = xp11 . . . xpnn and I = (p, xan1n1 , . . . , xansns ) then

< p, xan1n1 , . . . , xansns >=∏

i=1,...,s(1− tani )− tdeg p

( ∏i=1,...,s

(1− tani−pni )

)



ideals

Jamal HosseinPoor

Outline

Introduction





Bigatti algorithm

.Definition 3.15)..

.. ..

.

.



.. ..

.

.


< xan1n1 , . . . , xansns >=∏


i ) < 0 >=∏


i ).


< p, xan1n1 , . . . , xansns >=∏


( ∏i=1,...,s

(1− tani−pni )

)



ideals

Jamal HosseinPoor

Outline

Introduction





Bigatti algorithm

.Definition 3.15)..

.. ..

.

.



.. ..

.

.


< xan1n1 , . . . , xansns >=∏


i ) < 0 >=∏


i ).


< p, xan1n1 , . . . , xansns >=∏


( ∏i=1,...,s

(1− tani−pni )

)



ideals

Jamal HosseinPoor

Outline

Introduction





Bigatti algorithm

.Definition 3.15)..

.. ..

.

.



.. ..

.

.


< xan1n1 , . . . , xansns >=∏


i ) < 0 >=∏


i ).


< p, xan1n1 , . . . , xansns >=∏


( ∏i=1,...,s

(1− tani−pni )

)



ideals

Jamal HosseinPoor

Outline

Introduction





Bigatti algorithm

.Definition 3.15)..

.. ..

.

.



.. ..

.

.


< xan1n1 , . . . , xansns >=∏


i ) < 0 >=∏


i ).


< p, xan1n1 , . . . , xansns >=∏


( ∏i=1,...,s

(1− tani−pni )

)



ideals

Jamal HosseinPoor

Outline

Introduction





Bigatti algorithm

.Definition 3.15)..

.. ..

.

.



.. ..

.

.


< xan1n1 , . . . , xansns >=∏


i ) < 0 >=∏


i ).


< p, xan1n1 , . . . , xansns >=∏


( ∏i=1,...,s

(1− tani−pni )

)



ideals

Jamal HosseinPoor

Outline

Introduction





Bigatti algorithm

.Note 3.17)Bigatti algorithm..

.. ..

.

.

input :I = (T1, . . . ,Ts)with Ti power products in A = K[X1, . . . ,XN]output :Function HPNum(I)beginI is a base-case then return else if I is a spliting-case then return HPNum(I1) · . . . · HPNum(Ir)elsechoose a pivot P;return HPNum(I,P) + tdegPHPNum(I : P);

end.



ideals

Jamal HosseinPoor

Outline

Introduction





Bigatti algorithm

.Note 3.18)Choosing the pivot..

.. ..

.

.

0 −→ P/(I : p)(−d) .p−→ P/I −→ P/(I, p) −→ 0.

HSP/I(z) = HSP/(I,p)(z) + zdHSP/(I:p)(z)

1)(Indeterminate-pivot)p := x

< xz3, x2y2z, xy3z, x3yzw >=< x > +t < z3, xy2z, y3z, x2yzw > .

2)(Generator-pivot)p := x3yzw

< xz3, x2y2z, xy3z, x3yzw >=< xz3, x2y2z, xy3z > +t6 < z2, y >

3)(GCD-pivot)p := xyz

< xz3, x2y2z, xy3z, x3yzw >=< xyz, xz3 > +t3 < z2, xy, y2, x2z >

4)(Simple power-pivot)p := z3< xz3, x2y2z, xy3z, x3yzw >=< x2y2z, xy3z, x3yzw, z3 > +t3 < x >



ideals

Jamal HosseinPoor

Outline

Introduction





Bigatti algorithm


.. ..

.

.

0 −→ P/(I : p)(−d) .p−→ P/I −→ P/(I, p) −→ 0.











ideals

Jamal HosseinPoor

Outline

Introduction





Bigatti algorithm


.. ..

.

.

0 −→ P/(I : p)(−d) .p−→ P/I −→ P/(I, p) −→ 0.











ideals

Jamal HosseinPoor

Outline

Introduction





Bigatti algorithm


.. ..

.

.

0 −→ P/(I : p)(−d) .p−→ P/I −→ P/(I, p) −→ 0.











ideals

Jamal HosseinPoor

Outline

Introduction





Bigatti algorithm


.. ..

.

.

0 −→ P/(I : p)(−d) .p−→ P/I −→ P/(I, p) −→ 0.











ideals

Jamal HosseinPoor

Outline

Introduction





Bigatti algorithm


.. ..

.

.

0 −→ P/(I : p)(−d) .p−→ P/I −→ P/(I, p) −→ 0.











ideals

Jamal HosseinPoor

Outline

Introduction





Bigatti algorithm


.. ..

.

.

0 −→ P/(I : p)(−d) .p−→ P/I −→ P/(I, p) −→ 0.











ideals

Jamal HosseinPoor

Outline

Introduction





Bigatti algorithm


.. ..

.

.

0 −→ P/(I : p)(−d) .p−→ P/I −→ P/(I, p) −→ 0.











ideals

Jamal HosseinPoor

Outline

Introduction





Bigatti algorithm


.. ..

.

.

0 −→ P/(I : p)(−d) .p−→ P/I −→ P/(I, p) −→ 0.








4)(Simple power-pivot)p := z3

< xz3, x2y2z, xy3z, x3yzw >=< x2y2z, xy3z, x3yzw, z3 > +t3 < x >



ideals

Jamal HosseinPoor

Outline

Introduction





Bigatti algorithm


.. ..

.

.

0 −→ P/(I : p)(−d) .p−→ P/I −→ P/(I, p) −→ 0.











ideals

Jamal HosseinPoor

Outline

Introduction





Bigatti algorithm

.Result 3.19)Comparing indeterminate-pivot, simplepower-pivot and GCD-pivot..

.. ..

.

.

Example/Indet’s/Mon pivot-cases timeind/SP/GCD ind/SP/GCD

mayr12/21/ 444 586/614/483 0.2/0.2/0.2mayr13/21/610 1242/921/752 0.5/0.4/0.4prod 4 /32/500 1036/1036/ 685 0.4/0.4/0.3square 5/25/1371 1880/1824/1493 0.9/0.9/0.9mayr 22/31/3204 8045/6766/5254 4.5/4.1/4.1mayr 23/31/8100 40518/22706/18465 21.7/14.7/25.8prod 5/50/4785 39709/39709/35227 39.9/38.9/41.1power10,50 /10/10000 3175/3094/3095 7.1/5.5/8.2power20,50/20/10000 1528/1484/1483 6.3/4.8/6.3power20,20 /20 /10000 1498/1484/1483 5.2/4.7/6.3



ideals

Jamal HosseinPoor

Outline

Introduction





Comparing Bigatti algorithm with Bayer-Stillman algorithm

.Result 3.20)..

.. ..

.

.

Example Ind Mon CoCoA B-Smayr12 21 444 0/24/0.36 21mayr13 21 610 0.37/0.53 45prod 4 32 500 0.44 /0.55 35square 5 25 1371 0.91/1.17 186mayr 22 31 3204 4.11/4.86 3056mayr 23 31 8100 14.67/16.65 22013prod 5 50 4785 38.88/39.77 10403



ideals

Jamal HosseinPoor

Outline

Introduction





Zero-dimensional ideal

.Theorem 4.1)(Finiteness Criterion)..

.. ..

.

.

Let K be a field, I = (f1, . . . , fs) ⊂ P,and P = K[x1, . . . , xn]. Sdenotes the system of polynomial equations;

f1(x1, . . . , xn) = 0...fs(x1, . . . , xn) = 0

σ be a term ordering on Tn.The following conditions are equivalent;a) The system of equations S has only finitely many solutions.b)The ideal IP is contained in only finitely maximal ideals of P .c)For i = 1, . . . , n, we have I ∩ K[xi] ̸= 0.d)The K-vector space K[x1,,xn]

I is finite-dimensional.e)The set Tn\LTσ(I) is finite.f )For every i ∈ {1, 2, . . . , n}, there exists a number αi ≥ 0 such thatwe have xαi

i ∈ LTσ(I)



ideals

Jamal HosseinPoor

Outline

Introduction







.. ..

.

.


f1(x1, . . . , xn) = 0...fs(x1, . . . , xn) = 0



i ∈ LTσ(I)



ideals

Jamal HosseinPoor

Outline

Introduction







.. ..

.

.


f1(x1, . . . , xn) = 0...fs(x1, . . . , xn) = 0

σ be a term ordering on Tn.The following conditions are equivalent;

a) The system of equations S has only finitely many solutions.b)The ideal IP is contained in only finitely maximal ideals of P .c)For i = 1, . . . , n, we have I ∩ K[xi] ̸= 0.d)The K-vector space K[x1,,xn]


i ∈ LTσ(I)



ideals

Jamal HosseinPoor

Outline

Introduction







.. ..

.

.


f1(x1, . . . , xn) = 0...fs(x1, . . . , xn) = 0

σ be a term ordering on Tn.The following conditions are equivalent;a) The system of equations S has only finitely many solutions.

b)The ideal IP is contained in only finitely maximal ideals of P .c)For i = 1, . . . , n, we have I ∩ K[xi] ̸= 0.d)The K-vector space K[x1,,xn]


i ∈ LTσ(I)



ideals

Jamal HosseinPoor

Outline

Introduction







.. ..

.

.


f1(x1, . . . , xn) = 0...fs(x1, . . . , xn) = 0

σ be a term ordering on Tn.The following conditions are equivalent;a) The system of equations S has only finitely many solutions.b)The ideal IP is contained in only finitely maximal ideals of P .

c)For i = 1, . . . , n, we have I ∩ K[xi] ̸= 0.d)The K-vector space K[x1,,xn]


i ∈ LTσ(I)



ideals

Jamal HosseinPoor

Outline

Introduction







.. ..

.

.


f1(x1, . . . , xn) = 0...fs(x1, . . . , xn) = 0

σ be a term ordering on Tn.The following conditions are equivalent;a) The system of equations S has only finitely many solutions.b)The ideal IP is contained in only finitely maximal ideals of P .c)For i = 1, . . . , n, we have I ∩ K[xi] ̸= 0.

d)The K-vector space K[x1,,xn]I is finite-dimensional.

e)The set Tn\LTσ(I) is finite.f )For every i ∈ {1, 2, . . . , n}, there exists a number αi ≥ 0 such thatwe have xαi

i ∈ LTσ(I)



ideals

Jamal HosseinPoor

Outline

Introduction







.. ..

.

.


f1(x1, . . . , xn) = 0...fs(x1, . . . , xn) = 0


I is finite-dimensional.

e)The set Tn\LTσ(I) is finite.f )For every i ∈ {1, 2, . . . , n}, there exists a number αi ≥ 0 such thatwe have xαi

i ∈ LTσ(I)



ideals

Jamal HosseinPoor

Outline

Introduction







.. ..

.

.


f1(x1, . . . , xn) = 0...fs(x1, . . . , xn) = 0


I is finite-dimensional.e)The set Tn\LTσ(I) is finite.

f )For every i ∈ {1, 2, . . . , n}, there exists a number αi ≥ 0 such thatwe have xαi

i ∈ LTσ(I)



ideals

Jamal HosseinPoor

Outline

Introduction







.. ..

.

.


f1(x1, . . . , xn) = 0...fs(x1, . . . , xn) = 0



i ∈ LTσ(I)



ideals

Jamal HosseinPoor

Outline

Introduction






.Definition 4.2)..

.. ..

.

.

An ideal I = (f1, . . . , fs) in P = K[x1, . . . , xn] is calledzero-dimensional if it satisfies the equivalent conditions of theFiniteness Criterion.

.Example 4.3)..

.. ..

.

.

P = Q[x, y] ,I =< x2y − y + x, xy2 − x > .

G = {g1, g2, g3} = {x2y − y + x,−y2 + xy + x2, x3 + y − 2x}

LTσ(I) =< LTσ(g1), LTσ(g2), LTσ(g3) >=< x2y, y2, x3 >



ideals

Jamal HosseinPoor

Outline

Introduction






.Definition 4.2)..

.. ..

.

.

An ideal I = (f1, . . . , fs) in P = K[x1, . . . , xn] is calledzero-dimensional if it satisfies the equivalent conditions of theFiniteness Criterion..Example 4.3)..

.. ..

.

.

P = Q[x, y] ,I =< x2y − y + x, xy2 − x > .





ideals

Jamal HosseinPoor

Outline

Introduction






.Definition 4.2)..

.. ..

.

.

An ideal I = (f1, . . . , fs) in P = K[x1, . . . , xn] is calledzero-dimensional if it satisfies the equivalent conditions of theFiniteness Criterion..Example 4.3)..

.. ..

.

.

P = Q[x, y] ,I =< x2y − y + x, xy2 − x > .





ideals

Jamal HosseinPoor

Outline

Introduction





Lex segment Space

.Definition 4.4)..

.. ..

.

.

Let P = ⊕i∈Z+Pi.Let d ∈ N,and t ∈ Tn be a term of degree d;a)A set of terms of the form

{t′ ∈ Tn | deg(t′) = d, t′ ≥Lex t}

is called a Lex-segment.b)A K-vector subspace V of

Pd = {f ∈ P | deg(t) = d for all t ∈ Supp(f)}

is called a Lex-segment space if V ∩ Tn is both a K-basis of V and aLex-segment. In this case we denote the K-basis V ∩ Tn by T(V)..Example 4.5)..

.. ..

.

.

if n = 3 and d = 2 then V = Kx21 + Kx1x2 + Kx1x3 + Kx22 is theLex-segment space such that t′ ≥Lex t = x22.



ideals

Jamal HosseinPoor

Outline

Introduction





Lex segment Space

.Definition 4.4)..

.. ..

.

.


{t′ ∈ Tn | deg(t′) = d, t′ ≥Lex t}




.. ..

.

.




ideals

Jamal HosseinPoor

Outline

Introduction





Lex segment Space

.Definition 4.4)..

.. ..

.

.


{t′ ∈ Tn | deg(t′) = d, t′ ≥Lex t}

is called a Lex-segment.

b)A K-vector subspace V of



.. ..

.

.




ideals

Jamal HosseinPoor

Outline

Introduction





Lex segment Space

.Definition 4.4)..

.. ..

.

.


{t′ ∈ Tn | deg(t′) = d, t′ ≥Lex t}




.. ..

.

.




ideals

Jamal HosseinPoor

Outline

Introduction





Lex segment Space

.Definition 4.4)..

.. ..

.

.


{t′ ∈ Tn | deg(t′) = d, t′ ≥Lex t}




.. ..

.

.




ideals

Jamal HosseinPoor

Outline

Introduction





Lex segment Space

.Definition 4.4)..

.. ..

.

.


{t′ ∈ Tn | deg(t′) = d, t′ ≥Lex t}



is called a Lex-segment space if V ∩ Tn is both a K-basis of V and aLex-segment. In this case we denote the K-basis V ∩ Tn by T(V).

.Example 4.5)..

.. ..

.

.




ideals

Jamal HosseinPoor

Outline

Introduction





Lex segment Space

.Definition 4.4)..

.. ..

.

.


{t′ ∈ Tn | deg(t′) = d, t′ ≥Lex t}




.. ..

.

.




ideals

Jamal HosseinPoor

Outline

Introduction





Lex segment ideal

.Definition 4.6)..

.. ..

.

.

If for any i ≥ 1 ,I = ⊕i∈Z+ Ii ⊆ P = ⊕i∈Z+Pi be a Lex-segmentspace ,then I is called a Lex-segment ideal.

.Lemma 4.7)Lex segment zero-dimensional ideal inK[x, y, z]..

.. ..

.

.

Let I be a lex-segment zero-dimensional ideal in k[x, y, z]. Then

I =<xa, xa−1yb1 , xa−1yb1−1z, ..., xa−1zb1 ,

xa−2yb2 , xa−2yb2−1z, ...., xa−2zb2 , ...

xyba−1−1, xyba−1−1z, ..., xyzba−1−1, xzba−1

, yba , yba−1zc1 , yba−2zc2 , ..., y2zcb−2 , yzcb−1 , zcb >

where b1 < b2 < ... < ba−1 < ba = b,c1 < c2 < ... < cb−1 < cb = cand a < ba < cb.



ideals

Jamal HosseinPoor

Outline

Introduction





Lex segment ideal

.Definition 4.6)..

.. ..

.

.

If for any i ≥ 1 ,I = ⊕i∈Z+ Ii ⊆ P = ⊕i∈Z+Pi be a Lex-segmentspace ,then I is called a Lex-segment ideal..Lemma 4.7)Lex segment zero-dimensional ideal inK[x, y, z]..

.. ..

.

.



xa−2yb2 , xa−2yb2−1z, ...., xa−2zb2 , ...






ideals

Jamal HosseinPoor

Outline

Introduction





Lex segment ideal

.Definition 4.6)..

.. ..

.

.


.. ..

.

.



xa−2yb2 , xa−2yb2−1z, ...., xa−2zb2 , ...






ideals

Jamal HosseinPoor

Outline

Introduction





Lex segment ideal

.Definition 4.6)..

.. ..

.

.


.. ..

.

.



xa−2yb2 , xa−2yb2−1z, ...., xa−2zb2 , ...






ideals

Jamal HosseinPoor

Outline

Introduction





Lex segment ideal

.Definition 4.6)..

.. ..

.

.


.. ..

.

.



xa−2yb2 , xa−2yb2−1z, ...., xa−2zb2 , ...






ideals

Jamal HosseinPoor

Outline

Introduction





h-vector of Lex segment zero-dimensional ideal in K[x, y, z]

.Lemma 4.8)..

.. ..

.

.

If I be a Lex-segment zero-dimensional ideal with threevariables,then

hvP/I =(

(2

2

),

(3

2

), ...,

(a + 1

2

),H(a), ...,H(ba − 1),

ba, ..., ba︸︷︷︸c1−1

, ..., ba − i, ..., ba − i︸︷︷︸ci+1−ci−1

, ..., 1, ..., 1︸︷︷︸cb−cb−1−1

)

H(a) =

(a + 2

2

)− (1), H(a + 1) =

(a + 3

2

)− (1 + 2), . . .

H(a + bj − j) =

(a + bj − j + 2

2

)− (1 + 2 + . . . + bj + 1), . . .

H(a + bj+1 − (j + 2)) =

(a + bj+1 − j2

)− (1 + 2 + . . . + bj+1 − 1)

such that j = 1, 2, · · · , a − 1 and bj+1 ≥ bj + 2.



ideals

Jamal HosseinPoor

Outline

Introduction






.Lemma 4.8)..

.. ..

.

.


hvP/I =(

(2

2

),

(3

2

), ...,

(a + 1

2

),H(a), ...,H(ba − 1),

ba, ..., ba︸︷︷︸c1−1

, ..., ba − i, ..., ba − i︸︷︷︸ci+1−ci−1

, ..., 1, ..., 1︸︷︷︸cb−cb−1−1

)

H(a) =

(a + 2

2

)− (1), H(a + 1) =

(a + 3

2

)− (1 + 2), . . .

H(a + bj − j) =

(a + bj − j + 2

2

)− (1 + 2 + . . . + bj + 1), . . .

H(a + bj+1 − (j + 2)) =

(a + bj+1 − j2

)− (1 + 2 + . . . + bj+1 − 1)




ideals

Jamal HosseinPoor

Outline

Introduction






.Lemma 4.8)..

.. ..

.

.


hvP/I =(

(2

2

),

(3

2

), ...,

(a + 1

2

),H(a), ...,H(ba − 1),

ba, ..., ba︸︷︷︸c1−1

, ..., ba − i, ..., ba − i︸︷︷︸ci+1−ci−1

, ..., 1, ..., 1︸︷︷︸cb−cb−1−1

)

H(a) =

(a + 2

2

)− (1), H(a + 1) =

(a + 3

2

)− (1 + 2), . . .

H(a + bj − j) =

(a + bj − j + 2

2

)− (1 + 2 + . . . + bj + 1), . . .

H(a + bj+1 − (j + 2)) =

(a + bj+1 − j2

)− (1 + 2 + . . . + bj+1 − 1)




ideals

Jamal HosseinPoor

Outline

Introduction






.Lemma 4.8)..

.. ..

.

.


hvP/I =(

(2

2

),

(3

2

), ...,

(a + 1

2

),H(a), ...,H(ba − 1),

ba, ..., ba︸︷︷︸c1−1

, ..., ba − i, ..., ba − i︸︷︷︸ci+1−ci−1

, ..., 1, ..., 1︸︷︷︸cb−cb−1−1

)

H(a) =

(a + 2

2

)− (1), H(a + 1) =

(a + 3

2

)− (1 + 2), . . .

H(a + bj − j) =

(a + bj − j + 2

2

)− (1 + 2 + . . . + bj + 1), . . .

H(a + bj+1 − (j + 2)) =

(a + bj+1 − j2

)− (1 + 2 + . . . + bj+1 − 1)




ideals

Jamal HosseinPoor

Outline

Introduction






.Note 4.9)A procedure for computing the h-vector byorder O(n)..

.. ..

.

.

LIB ”general.lib”;proc Hilbertlex(intvec v){ list a ; int i=0;int j=1;while(i<v[1]){bigintb = binomial(i + 2, 2); a=a+list(b); i=i++; }

a=a+list(binomial(v[1]+2,2)-1); list c=a;if(3<=v[2]){while(j <= (v[2]− 2)){intk = (j2 + 3 ∗ j + 2)div2;c=c+list(bigint(binomial(v[1]+j+2,2))-k);j++; } } list m=c; int s=0;



ideals

Jamal HosseinPoor

Outline

Introduction






.Note 4.9)Continue..

.. ..

.

.

while (s < (v[1]− 1)){int h=v[s+2]-(s+1);if((v[s + 2] + 2) <= v[s + 3]) }while (h <= v[s + 3]− (s + 3)){int l = ((h + s + 1)2 + 3 ∗ (h + s + 1) + 2)div2;m = m + list(bigint(binomial(v[1] + h + 2, 2)− l));h++;} } s ++; }listt = m; inte = v[1] + 2; intf = v[e − 1]; intx = v[e];while((x − 1) <> 0){t = t + list(f); x −−; } listr = t; intq; into = 1;while(o <= v[e − 1]− 1){for(q = 1; q <= (v[v[1] + o + 2]− v[v[1] + o + 1]− 1); q ++){r = r + list(f − o); }o ++; }return(r); }



ideals

Jamal HosseinPoor

Outline

Introduction






.Example 4.10)..

.. ..

.

.

x7

x6y4, x6y3z, x6y2z2, x6yz3, x6z4,

x5y5, x5y4z, x5y3z2, x5y2z3, x5yz4, x5z5

x4y6, x4y5z, x4y4z2, x4y3z3, x4y2z4, x4yz5, x4z6

x3y7, x3y6z, x3y5z2, x3y4z3, x3y3z4, x3y2z5, x3yz6, x3z7

x2y9, x2y8z, x2y7z2, x2y6z3, x2y5z4, x2y4z5, x2y3z6, x2y2z7, x2yz8, x2z9

xy12, xy11z, xy10z2, xy9z3, xy8z4, xy7z5, xy6z6, xy5z7, xy4z8, xy3z9, xy2z10, xyz11, xz12

y16, y15z2, y14z3, y13z5, y12z6, y11z9, y10z10, y9z11, y8z12, y7z13, y6z16, y5z17, y4z20, y3z23

y2z24, yz26, z30

ring r=0,(x,y,z),lp;intvec v=7,4,5,6,7,9,12,16,2,3,5,6,9,10,11,12,13,16,17,20,23,24,26,30;Hilbertlex(v);

([1] : 1, [2] : 3, [3] : 6, [4] : 10, [5] : 15, [6] : 21, [7] : 28, [8] : 35, [9] : 42, [10] : 49, [11] :

30, [12] : 23, [13] : 25, [14] : 14, [15] : 15, [16] : 16, [17] : 16, [18] : 14, [19] : 12, [20] : 12, [21] :

7, [22] : 7, [23] : 5, [24] : 5, [25] : 4, [26] : 4, [27] : 2, [28] : 1, [29] : 1, [30] : 1)



ideals

Jamal HosseinPoor

Outline

Introduction






.Example 4.10)..

.. ..

.

.

x7

x6y4, x6y3z, x6y2z2, x6yz3, x6z4,







y2z24, yz26, z30


([1] : 1, [2] : 3, [3] : 6, [4] : 10, [5] : 15, [6] : 21, [7] : 28, [8] : 35, [9] : 42, [10] : 49, [11] :

30, [12] : 23, [13] : 25, [14] : 14, [15] : 15, [16] : 16, [17] : 16, [18] : 14, [19] : 12, [20] : 12, [21] :

7, [22] : 7, [23] : 5, [24] : 5, [25] : 4, [26] : 4, [27] : 2, [28] : 1, [29] : 1, [30] : 1)



ideals

Jamal HosseinPoor

Outline

Introduction






.Example 4.10)..

.. ..

.

.

x7

x6y4, x6y3z, x6y2z2, x6yz3, x6z4,







y2z24, yz26, z30


([1] : 1, [2] : 3, [3] : 6, [4] : 10, [5] : 15, [6] : 21, [7] : 28, [8] : 35, [9] : 42, [10] : 49, [11] :

30, [12] : 23, [13] : 25, [14] : 14, [15] : 15, [16] : 16, [17] : 16, [18] : 14, [19] : 12, [20] : 12, [21] :

7, [22] : 7, [23] : 5, [24] : 5, [25] : 4, [26] : 4, [27] : 2, [28] : 1, [29] : 1, [30] : 1)



ideals

Jamal HosseinPoor

Outline

Introduction






.Example 4.10)..

.. ..

.

.

x7

x6y4, x6y3z, x6y2z2, x6yz3, x6z4,







y2z24, yz26, z30


([1] : 1, [2] : 3, [3] : 6, [4] : 10, [5] : 15, [6] : 21, [7] : 28, [8] : 35, [9] : 42, [10] : 49, [11] :

30, [12] : 23, [13] : 25, [14] : 14, [15] : 15, [16] : 16, [17] : 16, [18] : 14, [19] : 12, [20] : 12, [21] :

7, [22] : 7, [23] : 5, [24] : 5, [25] : 4, [26] : 4, [27] : 2, [28] : 1, [29] : 1, [30] : 1)



ideals

Jamal HosseinPoor

Outline

Introduction





References

Adams, W.W., Loustaunau.P., An Introduction on Gr¨obner Bases, American Mathematical Society,1996A. Nasrollah Nejad and R. Zaare-Nahandi, Hilbert series of monomial algebras, Proceedings of 39thAnnual Iranian Mathematics Conference, University of Kerman, 24-27 August 2008.Atiyah, M. F., Macdonald, I. G. Introduction to Commutative Algebra. Addison- Wesley publishingco., 1969. 1Bigatti,A.,Computation of Hilbert Poincare series.AMS Mathematical SubjectPrimary:13D40,13-04,13P99.Secondary13P10,68Q40.Bigatti, A., Caboara, M., Robbiano, L., On the computation of Hilbert-Poincar series, Appl. AlgebraEngrg. Comm. Comput. 2 (1991), no. 1, 21–33.

Capani, A., Niesi, G., Robbiano, L. CoCoA a system for doing computations in CommutativeAlgebra.(1998) Availble via anonymous ftp from cocoa.dima.unige.it.

G.Pfister, G.M.Greuel,A Singular introduction to commutative Algebra , Springer-Verlag 2000.Kreuzer,M., Robiano,L.,Computational Commutative Algebra 1, Springer- Verlag, 2000.Kreuzer,M., Robiano,L.,Computational Commutative Algebra 2, Springer- Verlag, 2005.M.Stillman,D.Bayer, Computation of Hilbert functions, J. Symbolic Computation 14, 3 1 -50,1992.R. M.Karp, Reducibility among combinatorial problems, in R .E. Miller and J .W.Thatcher (eds),”Complexity of computer computations”, Plenum Press, New York, 85-103 ,(1972).

R.Nipolitan,Design of algorithms, translated by Naeemi Poor and Jafar Nejad, 2008.S.faghfouri, J.Hossein Poor,R.Zaar Nahandi, Hilbert function of Lex-segment zero-dimensionalmonomial ideal in K[x,y,z],IASBS, Zanjan, accepted in 43 rd annual Iranian mathematiciansconferences.



ideals

Jamal HosseinPoor

Outline

Introduction





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ideals

Jamal HosseinPoor

Outline

Introduction





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ideals

Jamal HosseinPoor

Outline

Introduction





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ideals

Jamal HosseinPoor

Outline

Introduction





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ideals

Jamal HosseinPoor

Outline

Introduction





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ideals

Jamal HosseinPoor

Outline

Introduction





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ideals

Jamal HosseinPoor

Outline

Introduction





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ideals

Jamal HosseinPoor

Outline

Introduction





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ideals

Jamal HosseinPoor

Outline

Introduction





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Jamal HosseinPoor

Outline

Introduction





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Jamal HosseinPoor

Outline

Introduction





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Jamal HosseinPoor

Outline

Introduction





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Jamal HosseinPoor

Outline

Introduction

Documents

· Complexity of algorithms computing the Hilbert series of monomial ideals Jamal Hossein Poor Outline Introduction ch.1:Complexity and timing data of algorithms ch.2:Hilbert functio