Topic: Grobner Basis Over Other Fields

UNIVERSITY OF NAIROBI

The Department of Mathematics

Groebner Bases over finite Fields and over finite extensions of Q

Maingi M Damian

I/56/7485/2003

Academic year 2003/2005

Supervised by Claudio Achola

1 Declaration

This project, as presented in this report is my original work and has not been presented for any other University award in any university.

Maingi Muindi Damian (I/56/7485/2003)

Signature

Signed by Supervisor:

Claudio Achola

Signature

Authenticated by the Chairman

Prof John W Owino

Signature

A Research Project submitted to the Department of Mathematics, University of Nairobi in partial

fulfillment of the requirements for the award of a Degree in Master of Science in Pure Mathematics.

2018 年 5 月 8 日

2 Dedication

To the great Mathematicians of all time like David Hilbert, Cauchy, Paul Erdos and all men of good will. Let us push on in search for the depths of truth!

3 Acknowledgement To God who has enabled me to undertake this work. To my Dad and Mom who have been such a great source of inspiration and support. To my supervisor, Claudio Achola; how could I forget you? Who has been to me more than a supervisor, you have instilled in me all I could say I know about algebra and many more things about life in general, I will be eternally grateful. I really appreciate the kindness extended to me by Professor Buchberger himself; he assisted me despite the distance and never having met me.

My sincere thanks go to ICTP and ISP for had it not been there organizing and funding the conferences we have had in Nairobi and Kampala I would not have had an inspiration, I thank Profs

Rikaard Bogvad and Enrico Rogora who spend their time reading my dissertation and actually sat down with me to show the corrections I had to make. I also thank Profs Jennifer Morse, Claudio

Procesi and Mike Zabrocki since they were the people on ground conducting the conferences.

I also remember to thank the Professor Owino for his support as well, sincere thanks go to Dr Patrick

Weke for his concern and care for me, I shall not forget people like Dan Odera and Ken Odhiambo,

Peter Maina, Linet Muhati, Ngare Odhiambo my former classmates, sincere thanks to Cecilia, Irene and Winnie, to all especially the department members and to all who contributed to the success of this project. How could I ever thank all of you?

4 Table of contents

UNIVERSITY OF NAIROBI...... 1

Dedication...... 3

1. Introduction...... 6

2. Problem statement...... 8

3. Abbreviations and Symbols...... 9

4. Concepts and Theorems used...... 10

4.1 Monomial Ideals and Dickson’s Lemma...... 12

4.2 Definitions on monomial orderings ...... 12

4.3 Hilbert basis theorem...... 15

4.4 The notion of “Groebner Bases”...... 16

4.5 The notion of "S-polynomials''...... 18

5 Groebner Bases Computation and the Theory behind...... 19

5.1 Buchberger’s Algorithm...... 19

5.2 Buchberger’s Algorithm(Another version)-...... 20

5.3 Groebner bases for polynomials ideals with coefficients in Z p ...... 23

5.4 Groebner bases for polynomials ideals with coefficients in Q(i) ...... 24

5.5 Groebner bases for polynomials ideals with coefficients in Q(a ) ...... 26

5.6 Groebner Basis for polynomial ideals with coefficients in Q() ...... 29

5.6.1 Theorem on simple extensions of fields. Herstein ...... 32

5.6.2 Theorem on Groebner basis over finite extensions of Q ...... 33

6 Appendix...... 36

7 References...... 37

5 1. Introduction

The concept of Groebner Bases was introduced by Buchberger an Austrian, in 1965 in the context of his work on performing algorithmic computations in residue classes of polynomial rings. Buchberger's algorithm for computing Groebner Bases is a powerful tool for solving many important problems in polynomial ideal theory. The algorithm was named after Wolfgang Groebner who was the Ph.D. Advisor to Buchberger and who stimulated the research on the subject. Groebner Bases orderings on monomials. [1]

Wolfgang Groebner proposed the problem by asking how a multiplicative table for the associative algebra, formed by the residue ring modulo a polynomial ideal can be constructed algorithmically and Buchberger showed that it sufficed to adjoin the differences of each critical pair (S-Polynomials) to get a certain basis that solves this problem in finite number of steps. [5]

The basic idea of Groebner Bases is that starting from a finite set say A, of polynomials that generate an Ideal, I, say then there is a method that is used to transform the set A into a certain standard form S, this is by following Buchberger’s algorithm.

A Closely related concept i.e. that of “standard bases” was discovered in 1964 by Hironaka. He did this independently, but he only gave an unconstructive existence proof. Whereas Buchberger gave a proof for their existence and proceeded to give an algorithmic way of constructing them.

In the thesis dissertation of Buchberger, we find that he succeeded in establishing the existence of Groebner basis and gave an algorithm for their computation with coefficients in the any field. Our aim is to see what happens in the case of finite fields mainly integers modulo a prime number p i.e. Z p in relation to the Groebner basis over a rational number field Q and also whether it is possible to make arithmetic in the computation of the Groebner basis with coefficients in the finite extensions of the rational number field. This project is more theoretical oriented in the sense that we do not aim at actually computing Groebner basis since that has already been done see[5] but ours is to facilitate the process and to establish some kind of theorem using the theory of fields and especially field extensions to do this.

6 After their introduction by Buchberger in 1965, there have been many studies carried out relating to this subject, some of them carried by Buchberger himself with some of his colleagues for instance he gives a better version of his algorithm in [6].

Some of the studies that have been carried out relating to this area are: For example in the work of D Kapur and Y Y Cai [7], where they give an algorithm for computing a Groebner Bases of an ideal of polynomials whose coefficients are taken from a ring with zero divisors, i.e. rings like Z n and Z n [i] , where n is not a prime number. The algorithm is patterned after Buchberger’s algorithm for computing a Groebner Bases of a polynomial ideal whose coefficients are from a field and its extension developed by Kandri-Rody and Kapur when the coefficients appearing in the polynomials are from a Euclidean domain. The algorithm works as Buchberger’s algorithm when a polynomial ideal is over a field and as Kandri-Rody and Kapur’s algorithm when a polynomial ideal is over a Euclidean domain. There have been many publications and texts on Groebner basis, bearing in mind that they were developed in 1965 and Buchberger gave many papers on their improvement and again many people have studied them and this we can get an idea by looking at the webpage of Buchberger where we find that has co-published many papers on the subject.

7 2. Problem statement As we will see how to compute Groebner Basis, we will find that their computation over finite fields i.e. having their coefficients in fields is easier than in their computation over rational number fields and it gets more complicated as the field grows or as we extend the field for in instance when we have finite extensions of Q for instance Q()[x, y, z] where a is irrational. Then manipulation of polynomials over such a field would be agony even for the computer it would take a lot of memory space and thus the speed would be quite compromised. Thus we set out to investigate what could be done to solve these problems and thus to facilitate dealings with happens in the case of Groebner Basis computation over more general fields for example finite extensions Q() of the rational number field, Q .

Now what drove us to such areas of study?

We had a school on Algebraic Geometry and Commutative Algebra in August 2004 some of the striking areas at least for me were the part II of the course which were covered by Jennifer Morse and

Rikard Bogvad i.e. Ideals, varieties and algorithms and the computer implementation by Mike

Zabrocki. Under this we covered;

 Division algorithm

 Monomial ideals and Dickson’s lemma

 Hilbert basis theorem and Groebner Basis

 Buchberger’s algorithm

 Ideal membership problem, implication problem and elimination theory

After having attended the conference, I consulted with my supervisor, who gave me a list of projects from [2]. I read the appendix and came across a possible project on improvement of Buchberger’s algorithm for the computation of Groebner basis.

8 3. Abbreviations and Symbols

K …………………………………………Field

K[x] ………………………………………Polynomial ring over K

I …………………………………………Ideal

R …………………………………………Ring (Our usual ring is K[x] )

I ◁R …………………………………… I is an ideal of R

x  {x1 , x2, ...xn }………………………… x bar stands for n variables

 f1 , f 2 ,... f n  ……………………………An Ideal generated by f1, f2 ,... fn

Z P ………………………………………Integers modulo p

2 Z P [i]  {x  iy / x, y  Z p i 1  0}……Gaussian Integers

Q …………………………………………Rational number field

Q(i)  {x  iy / x, y  Q i 2 1  0}………Gaussian Rational numbers

Q[x, y, z]…………………………………Polynomial ring over x, y, z with coefficients in Q

Q()[x, y, z]……………………………. Polynomial ring over x, y, z with coefficients in

Q() , where Q() is an extension field of Q

w.r.t ………………………………………with respect to

9 4. Concepts and Theorems used

The ideas we present here are applicable to any polynomial ring

R= K[ x ] = K [ x1 , x 2 ,..., xk ] where K is a field, and I ◁R , i.e. I is an ideal of the ring R .

Finite extensions [8] A field K is said to be an extension field (or field extension, or extension), denoted K/ F , of a field F if F is a subfield of K . For example, the complex numbers are an extension field of the real numbers, and the real numbers are an extension field of the rational numbers. The extension field degree (or relative degree, or index) of an extension field K/ F , denoted [K : F ], is the dimension of K as a vector space over F , i.e.,

[K : F ]= dim K F

Given a field F , there are a couple of ways to find an extension field. If F is contained in a larger field, F F ' . Then by picking some elements a F 'not in F, one defines F(a ) to be the smallest subfield of F ' containing F and the a . For instance, the rational numbers can be extended by the complex number i , yielding Q( i ) . If there is only one new element, the extension is called a simple extension. The process of adding a new element is called 'adjoining.'

Let I ◁R

Now the ideal I can be generated by some finite number of elements gi 's where i  1,2,...n

I  g1, g2 ,...gn  , if R is Noetherian, for example the polynomial ring that we will consider it.

Definition Noetherian Ring [9]

A ring is called left (respectively, right) Noetherian if it does not contain an infinite ascending chain of left (respectively, right) ideals. In this case, the ring in question is said to satisfy the ascending chain condition on left (respectively, right) ideals.

10 A ring is said to be Noetherian if it is both left and right Noetherian. If a ring R is Noetherian, then the following are equivalent.

1. R satisfies the ascending chain condition on ideals.

2. very ideal of R is finitely generated.

3. Every set of ideals contains a maximal element.

2 n 2n+ 1 If R  {a0  a1 x  a2 x  ...  an x / ai  K} and I={ a0 x + a 1 x + ... + an x }, Here we have an ideal whose generating set is {x } Example: I  x then I is an ideal generated by x We shall use I as our default symbol representing an Ideal.

Polynomial ring Let R be a ring. The polynomial ring over R in one variable x is the set R[x] of all

sequences in R with only finitely many nonzero terms. If (a0 , a1 , a2 ,...) is an element in R[x],

with a n  n for all n  N , then we usually write this element as

N n 2 N  an x  a0 a1 x  a2 x  ...  a N x n0 Elements of R[x] are called polynomials in the indeterminate x with coefficients in . The

ring elements (a0 , a1 , a2 ,...) are called coefficients of the polynomial, and the degree of a

polynomial is the largest natural number N for which a N  0 , if such an N exists. When a polynomial has all of its coefficients equal to 0, its degree is usually considered to be undefined, although some people adopt the convention that its degree is   . A monomial is a polynomial with exactly one nonzero coefficient. Similarly, a binomial is a polynomial with exactly two nonzero coefficients, and a trinomial is a polynomial with exactly three nonzero coefficients.

11 Proofs for the following propositions and the definitions are contained in David Cox, et al [2]

4.1 Monomial Ideals and Dickson’s Lemma

Definition 1 [2 P.67]:

An Ideal I ◁K[x] of a ring of polynomials is a monomial ideal if there’s a subset

n A  Z 0 such that I consists of all polynomials which are finite sums of the form

h x  A  . i.e. I  x :  A

1 2 n e.g. I  x  {a0 x  a1x  a2 x ... an x ...;ai  K}

Monomial Ideals occur as ideals of “leading terms” of Groebner Basis.

Lemma 1: [2 P.68]

Let I be a monomial Ideal and let f  K[x] .

Then the following are equivalent;

i) f  I

ii) Every term of f lies in I

i i (If f   i x by term of f above we mean any i x ) iii) f is linear combination of the monomials in I

Lemma 2 (Dickson’s Lemma): [2 P.69]

A monomial Ideal I  x :  A can be written the form

I  x (1) , x (2) ,..., x (s)  , where (1),(2),...,(s)  A .

i.e. the ideal has finite basis.

12 4.2 Definition 2 of monomial orderings [2 P.53]

 n A monomial ordering on K[x]  [x1, x2 ,..., xn ] is any relation on  0 or  n equivalently, any relation on the set of monomials x ,  0 satisfying:

n i)  is total (or linear) ordering on  0

n ii) If    and   0       

n n iii)  is well–ordering on  0 . This means that every non-empty subset of  0 has a smallest element under  .

Thus given any pair of monomials e.g. x and x with x  x  then

x .x  x  .x for some monomial x .

There are three commonly used ways of ordering monomials;

Definition 3: Lexicographic Order-lex

n Let   (1 , 2 ,..., n ) and   (1 ,  2 ,...,  n )  0 . We say that  lex  if, in

n the vector difference     0 the left most entry is non-zero and we write   x lex x .

Examples:

(1,2,0) lex (0,3,4) since     (1,1,4) and (3,2,4) lex (3,4,1) since     (0,0,3)

Definition 4: Graded Lex Order-grlex

n Let ,   0 . We say that   grlex  if

n n   i     j or    and  lex  i1 j 1

Examples:

(1,2,3)  grlex (3,2,0) since     (1,1,4) (1,2,4)  (1,1,5) and (1,2,4) lex (1,1,5) .

13 Definition 4: Graded Reverse Lex Order-grevlex

n    Let ,   0 . We say that grevlex if

n n   i     j or    and the right most non-zero entry is i1 j 1 negative.

E.g. (4,7,1)  grevlex (4,2,3)

More definitions 5

i Let f  ai x be non-zero polynomial in be K[x1 , x2 ,....xn ] and let  be monomial order

n i. The multidegree of f is multideg( f )=max (  0 : a  0)

ii. The leading coefficient of f is LC( f )  amulti deg( f )  K iii. The leading monomial of f is LM( f )  x multi deg ree( f ) iv. The leading term of f is LT( f )= LC( f ).LM( f )

Definition 5:

Let I ◁K[x] then

We denote by LT (I) , the set of leading terms of elements of I

 LT (I) {cx ; f  I with LT ( f )  cx }

 LT (I) is the ideal generated by elements of LT(I)

Note that if I   f1 , f 2 ,..., f n  then

LT ( f1 ), LT ( f2 ),..., LT ( fn ) may be different from LT (I) in fact

LT ( f1), LT ( f2 ),..., LT ( fn )  LT (I) e.g.

14 3 Let I   f1, f2  , where f1  x  2xy

2 2 f 2  x  2y  x

The order we are using is Lex order.

Then x.(x 2  2y 2  x)  y.(x3  2xy)  x 2

So that x 2  I , since I is an Ideal, and x and y belong to I thus x 2  I .

i.e. x 2  LT (x 2 )  LT (I)

2 3 2 However, x is not divisible by LT ( f1 )  x nor LT ( f 2 )  x y

2 i.e. x  LT ( f1 ), LT ( f 2 )

4.3 Hilbert basis theorem

The Hilbert Basis theorem guarantees the existence of a finite basis generating any ideal.

It can be stated as: “Every Ideal I ◁K[x] has a finite generating set i.e.

I  g1, g2 ,..., gs  for some g1, g2 ,..., gs  I ”

Proposition 1

Let I ◁K[x] then

1. LT (I) ◁K[x]

2.  g1, g2 ,..., g s  I such that LT (I)  LT (g1), LT (g2 ),..., LT (gs )

Such g1, g2 ,..., g s  I are what we call Groebner basis.

Definition 6

Fix a monomial order. A finite subset G  {g1, g2 ,..., gs } of an Ideal I is said to be a

Groebner Basis if LT (I)  LT (g1), LT (g2 ),..., LT (gs )

15 Corollary

Every (monomial) Ideal has a Groebner Basis.

4.4 The notion of “Groebner Bases”[3]

A Groebner basis for a system of polynomials has very useful properties, for example, that a polynomial f in an ideal generated by the system is a combination of those in the system if and only if the remainder of f with respect to the system is 0. (Here, the division algorithm requires an order of a certain type on the monomials). It’s amazing how useful theses bases are: For example

a) Equality of ideals Reduced Groebner bases can be shown to be unique for any given ideal and monomial ordering, Meaning that if we start with two different bases or more and we will end up with the same Groebner basis.

b) Ideals membership (f “belongs to” I) The reduction of a polynomial f by the multivariate division algorithm for an ideal using a Groebner Bases will yield 0 if and only if f is in the ideal. (This is not true in general for polynomials in more than one variable). This gives a test for determining whether or not a polynomial is in an ideal with a given set of generators. c) Solving equations Given a set of simultaneous polynomial equations, we move the solution, i.e. the constant to the left side and have the solution being homogeneous i.e. all of them yielding to zero. Then we take each equation as a basis element for an Ideal and then using Buchberger’s algorithm we reduce them to a Groebner basis. E.g. Solve x  y  z  1 7z  y  2x  8 2y  x  5z  5 Solution

16 Let I  x  y  z 1, 7z  y  2x  8, 2y  x  5z  5 On reducing using Buchberger algorithm, we get the Groebner basis as

3 3 I  x  y  z 1,  2 y  2 z 1, 3z  2

2 This leads us to say 3z  2  0  z  3

3 3 2  2 y  ( 2 )( 3 ) 1  0  y  0 finally

2 5 x  y  z 1  0  x  1 3  3 On using any other method you’ll arrive at our solution e.g. Gaussian elimination method we’d

5  1 1 1 1  1 0 0 3      start from  2 1 7 8  and end up with 0 1 0 0 1 2  5  5 0 0 1 2     3 

17 Proposition 2

Let G  {g1, g2 ,..., gt } be a Groebner Basis for an Ideal I ◁K[x] and let

f  K[x] then there exists a unique remainder r  K[x] such that

i) No term of r is divisible by any of LT (g i ), i  1,2,...,t ii) There exists a  I such that f  a  r

Corollary

Let G  {g1, g2 ,..., gt }be a Groebner Basis for an Ideal I ◁K[x] and let

f  K[x] then f  I if and only if r  0 on division by G .

We should bear in mind to compute the Groebner basis order matters i.e. we could use lexicographic, graded lexicographic or whatever we choose.

4.5 The notion of "S-polynomials''

The "S-polynomials'' are computed in two variables i.e. functions for that matter i.e.

S= S( f , g ) .

If multideg ( f )   and multideg (g)   then let   ( 1 , 2 ,..., n ) where  i  max( i ,  i )

We call x the LCM (Least Common Multiple) of LM(f) and LM(g). The S-polynomial of f, g is

x x S( f , g ) = f  g LT ( f ) LT (g)

Example

In Q[x, y, z]

We are given Let f  4x2 z  7y2 and g  xyz 2  3xz 4

We use Lexicographic order

  (2,0,1) and   (1,1,2) thus   (2,1,2)

18 x2 yz x2 yz S( f , g)  (4x2 z  7y 2 )  (xyz 2  3xz 4 ) 4x2 z xyz2

7 3 4   4 y  3xz

If we used reversed lexicographic order we would obtain:

7 3 2 1 2 2 S( f , g)   4 z y  3 z yx

Theorem [2 P.82]

Let I be a polynomial Ideal. Then a basis G  {g1, g2 ,..., gt } for I is Groebner basis for

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

5 Groebner Bases Computation and the Theory behind

Let us look at how to compute Groebner Bases over the fields using Buchberger’s algorithm.

5.1 Buchberger’s Algorithm The general form Consider the ring R .

Suppose I ◁R with I   f1, f2 ,..., fn  , with fi  R i  1,2,...,n

We form the S-polynomials i.e. S( fi , f j )  i  j

Now divide each S( fi , f j ) by { f1 , f 2 ,..., f n }

If the remainder is zero in all cases then { f1 , f 2 ,..., f n } is actually a Groebner Basis.

Otherwise include the remainders in { f1 , f 2 ,..., f n } so that we have a new set

{ f1 , f 2 ,..., f n ,r1 ,r2 ,...,rs }  A This new set A must be a Groebner Basis but not reduced i.e. there are many redundancies. The following theorem will help to eliminate them.

Theorem [2 P.86]

Let I   f1, f2 ,..., fn   {0} ◁R

19 Then a Groebner Basis for I can be constructed in a finite number of steps by the following algorithm

Input: F  ( f1, f2 ,..., fn ) Output: a Groebner Basis

G  {g1 ,g 2,..., g s ) for I with F  G G : F REPEAT G ' : G For each pair {p,q}, p  q in G ' DO

G' S : S( p,q)

If S  0 then G : G {S} UNTIL G : G '

5.2 Buchberger’s Algorithm(Another version)-[10]

Let I  g1 ,g 2,..., g s  assign A  {g1 ,g 2,..., g s } Find the Groebner basis S for the Ideal I

1. Compute S(gi , g j )  i  j

2. Divide S(gi , g j ) by A  {g1 ,g 2,..., g s } If r  0 (the remainder) then A : A  r End all r  0 , S  A This version gives a reduced Groebner basis i.e. it removes any repeated basis elements or any redundancies like basis elements that are linear combinations of other basis elements and the end result is almost unique.

Notice below that on computation using CoCoA the basis we get is actually a reduced Groebner basis. Example 1

In Q[x, y, z]

Let I  x2  y, y  z and A  {x 2  y, y  z}

x2 y x2 y S(x2  y, y  z)  (x2  y)  (y  z) x2 x2

20  y 2  xz

And on dividing S(x2  y, y  z) by A , as shown below we obtain a Zero remainder and

thus we conclude that A constitutes the Groebner Basis of I.

z  y x2  y  y 2  xz y  z  y 2  yz x2 z  yz x2 z  yz 0

We can check this using a computer.

Verification on the computer using CoCoA

Use R :: Q[x, y, z] --Declaring the ring environment we are operating

I : Ideal(x2  y, y  z) --Ideal whose Groebner Basis we'll compute

GB.Start_GBasis (I) --Start computing

GB.Complete (I) --Finish computation and finally

I. Basis --display the Groebner basis

These are the Groebner Basis:

{x2  y, y  z}  A .

Example 2

Let I  x2 + y2 + z2 -1, x2  z2 - y, x - z

Find the Groebner Basis and verify your answer. The order we are using is grlex i.e. graded lexicographic order, which works in the following way; x  y if      or

A  {x2 + y2 + z2 -1, x2  z2 - y,x - z}

21 We now form the S-polynomials:

2 2 2 2 2 2 S1(x  z  y, x  y  z 1)  y  y 1

2 2 2 2 2 S2 (x  z, x  y  z 1)  xz  y  z 1

2 2 2 S3 (x  z, x  z  y)  xz  y  z

And on dividing Si 's by A we get the remainders as:

2 For S1 r  -y - y + 1

2 2 S2 r  -y - 2z + 1

2 S3 r  -2z + y

Thus the reduced Groebner basis is A  {x - z, - y2  y 1,2z2  y}

Use R :: Q[x, y, z] ; --Declaring the ring environment we are operating

I : Ideal(x2 + y2 + z2 -1, x2  z2 - y,x - z) ;

--Definition of the Ideal whose Groebner Basis we'll compute

GB.Start_GBasis(I); --Start computing

GB.Complete(I); --Finish computation and finally

I.GBasis; --display the Groebner basis

These are the Groebner Basis:

{x  z,2z 2  y,y 2  y 1}

The cases we tackled above were for when the coefficients come from the rational number field Q . We are now going to investigate the behavior of the Groebner Bases of a polynomial ideal whose coefficients are taken from any fields other than Q . In particular we shall look at Groebner Bases over finite fields like Z p , i.e. integers modulo p, a prime number like 2,3,5 etc and Groebner Bases over infinite like Q(i) , where i 2 1  0 . Then we look at the general cases, where Q() is any finite extension of Q ,  being algebraic and then we say something about computation of Groebner Bases

22 over finitely generated extensions of Q i.e. the Groebner basis of polynomial ideals with coefficients in Q()[x] and Q()[x] e.g. Q( 2)[x] , Q( 2, 3. 5)[x, y, z], etc

We are using some field theory to study Groebner Bases computation over other fields other thanQ , the field of rational numbers.

5.3 Groebner bases for polynomials ideals with coefficients in Z p

In an example above we found that the ideal

I=� x2+y 2 +z 2 -1, x 2 裉 z 2 -y, x -z Q[ x , y , z ]

has Groebner Basis A={ x - z , - y2 - y + 1, - 2 z 2 + y } over Q but if it was

over Z 2 then the Groebner Basis would be A2 = {1}.

2 2 Over Z 3 we get A3 ={ x - z , z + y , - y - y + 1} but over Z 5 and above we get

2 2 the same as over Q i.e. A5 ={ x - z , - y - y + 1, - 2 z + y }

We got this by use of CoCoA i.e.

Use R :: Z /( p)[x, y, z] our ring then you just replace p with any prime number and

there you have your Z p . Then use CoCoA commands for computing the Groebner Basis as usual.

We’ve found out that if

2 2 2 2 2 Ip=� x+y +z -1, x 裉 z -y, x -z Z p [ x , y , z ]

if p = 2 , then the Groebner basis will be A = {1}, meaning that the ideal is equal to the

whole ring. If p = 3 we get the Groebner basis {x- z , z2 + y , - y 2 - y + 1} =

{x- z , - y2 - y + 1, - 2 z 2 + y } (recall calculations in mod 3). Same for

p = 5 .

Notice what is happening!

23 A2 Amod 2

A3 = Amod3

A5 = Amod5

We also notice that the larger the prime p that determines the field Z p , the closer the

Groebner basis are when we are getting them over the rational number field Q .

There is a relationship that: for integers modulo 2, if we have the Groebner basis for I over Q,

by reducing them modulo 2 does not guarantee getting A2 .

Thus we have the following relation Ap = Amod p for all p 3

In the following pages, we show how tedious and awful it can be to compute

Groebner basis when we get out of the rational number field. What we are doing in

each class is the same algorithm to reduce the arithmetic outside Q into

Q _arithmetic. This is shown even in the use of the computers in particular in

CoCoA.

5.4 Groebner bases for polynomials ideals with coefficients in Q(i)

To compute the Groebner bases for a polynomial ideal with coefficients in a field of Gaussian rational numbers i.e. Q(i)  {a  ib / a,b  Q, i 2 1  0}.

I  xy  i, x  i let A={ xy + z , x + z , z 2 + 1}

Let i= z� z 2 =1 0

Now let I=� xy + z, x + z , z 2 1 with x> y > z Lex order xy xy S= S( xy + z , x + z ) = ( xy + z ) - ( x + z ) 1 xy x

24 =xy + z - xy - yz = -yz + z

xyz2 xyz 2 S= S( xy + z , z2 + 1) = ( xy + z ) - ( z 2 + 1) 2 xy x =xyz2 + z 3 - xyz 2 - xy = -xy + z3

xz2 xz 2 S= S( x + z , z2 + 1) = ( x + z ) - ( z 2 + 1) 3 x z 2 =xz2 + z 3 - xz 2 - x = -x + z3

2 Now we divide the Si ' s by A={ xy + z , x + z , z + 1}

S1 by A 0 0 xy+ z 0 x+ z - yz + z z2 +1 r:= - yz + z

S2 by A

25 -1 0 xy+ z z x+ z - xy + z3 z2 +1 -xy - z z3 + z z3 + z r := 0

S3 by A 0 -1 xy+ z z x+ z - x + z3 z2 +1 -x - z z3 + z z3 + z r := 0

The Groebner basis therefore is A={ - yz + z , xy + z , x + z , z 2 + 1} and on substituting z back we get A={ - yi + z , xy + i , x + i } This is the same if we went on and computed directly.

An Example of a complicated problem is Consider an Ideal I=�2 wxyi + 2 x2 i / 3, iw - ix / 3 2 y 2 / 5 Now it’s clear that this would take forever to do it if we apply B’s algorithm directly. Let z= i� z2 =1 0 , the ideal whose Groebner basis we are looking for now is I=�2 wxyz + 2 x2 z / 3, zw - zx / 3 + 2 y 2 / 5, z 2 1 By use of CoCoA we get

26 [z2 + 1, -2/5y 2 + wz + 1/3xz - 2, wxy + 1/3x 2 , -5/2w 2 xz - 5/6wx2 z - 1/3x 2 y + 5wx, -2/15x 2 yz + w 2 x + 1/3wx 2 + 2wxz, 15/2w3 x + 5/2w 2 x 2 - 5wx 2 z + 1/3x 2 z - 2x 2 y + 30wx]

This is not exciting at all.

5.5 Groebner bases for polynomials ideals with coefficients in Q(a )

To compute the Groebner bases for a polynomial ideal with coefficients in a field Q()  {a  b / a,b  Q} ,  satisfies a minimal polynomial p()  0

Example:1

Consider an Ideal I:=� x2 3 - 2, xy 3 1

Working manually i.e. direct use of Buchberger’s algorithm

Let A={ x2 3 + 2, xy 3 - 1} and f= x2 3 + 2 , g= xy 3 - 1

Now we have a = (2,0) and b = (1,1)

x2 y x 2 y S( f , g )= ( x2 3 + 2) - ( xy 3 - 1) x2 3 xy 3

2 3y x 3 = + 3 3 2 3y x 3 Dividing S by A we obtain + as the remainder 3 3

Now our new A is 2 2 3y x 3 which is supposed to be {x 3+ 2, xy 3 - 1,3 + 3 } the Groebner Basis.

22 y x 2 But �x-2, xy + 1,3 窆 3 � - x 2, xy 1 since their Groebner Bases over are not the same i.e. [-x - 2 y , - 2 y 2 - 1] and -4 respectively.

27 Now we compute the Groebner Basis for I:=� x2 3 - 2, xy 3 1 in the following way

Let z = 3 then z2 -3 = 0 .

Now I:=� x2 z - 2, xyz - 1, z 2 3 Now using CoCoA we get: [z2- 3, - 3 x - 6 y , - 2 y 2 - z / 3] and on substituting 3 back our Groebner Basis is

[- 3x - 6 y , - 2 y2 - 3 / 3]

Let us check whether [- 3x - 6 y , - 2 y2 - 3 / 3] = [ x 2 3 + 2, xy 3 - 1] maybe z If it is the same then we are home. We need to verify this using COCOA. When we check using CoCoA it is the same i.e. [- 3x - 6 y , - 2 y2 - 3 / 3] = [ x 2 3 + 2, xy 3 - 1] i.e. they yield the same Groebner Basis and the Groebner Basis is [- 3x - 6 y , - 2 y2 - 3 / 3] .

Thus life becomes better in Q(a ) .

Example 2

Consider I=� x2 + y , x2 y 1 compute the Groebner basis.

The ring R:= Q ( 2)[ x , y ]

Direct i.e. using Buchberger’s algorithm as it is;

Let f= x2 + y and g= x2 y +1

x2 y x 2 y Now S( f , g )= ( x 2 + y ) - ( x2 y + 1) x2 x 2

xy2 2 = -1 2

y3 On dividing S( f , g ) by {f , g } we get r = - -1 2

y3 Thus the Groebner basis is {x 2  y, x2 1, 1} 2

28 Working the Groebner basis directly we get x2 y x 2 y S( x 2+ y , x2 y + 1) = ( x 2 + y ) - ( x 2 y + 1) x 2 x2 y

xy2 2 = -1 2 Divide S by A

2 a : y 2 b : x2 + y xy2 2 -1 x2 y +1 2 xy2 2 3 + y 2 2 3 -y -1 2

3 The reminder is:= -y -1 2

3 The Groebner basis is {x 2+ y , x2 y + 1, -y - 1} 2

Here we let t = 2 and take the minimal polynomial p( t ) for which p( 2)= 0 is p( t )= t 2 - 2 = 0 with x> y > t as the order we are using

Now our ring is R:= Q ( 2)[ x , y , t ]

The Ideal is I=� x2 + y , x2 y - 1, t 2 2

On computing the Groebner basis we obtain

{z2- 2, xz + y , - yz - 2 x , - 2 x 2 + y 2 ,1/ 2 y 3 + 1,2 xy 2 - 2 z }

And on substituting z back the Groebner basis is

29 {x 2+ y , - y 2 - 2 x , - 2 x2 + y 2 ,1/ 2 y 3 + 1,2 xy 2 - 2 2} and on comparing this with

y3 the basis {x 2  y, x2 1, 1} we see that we arrive at the required basis very fast and 2 with facility.

5.6Groebner Basis for polynomial ideals with coefficients in Q() Using Buchberger’s best survey on Groebner Bases from one of his papers done in the year 1998, we apply the improved version which explains his algorithm for the construction of Groebner Basis including the use of his criteria for improving the algorithm.

We look at Q() e.g. Q( 2, 3) Consider an Ideal I= { x2 ,xy2 3 }

Now Q( 2, 3)= Q ( 2 + 3) since Q( 2, 3) is simple.

Let z =2 + 3 then

z =2 + 3

1 1 1 2- 3 = = . z 2+ 3 2 + 3 2 - 3

2- 3 = =3 - 2 2- 3

From this we have

z =3 + 2

1 =3 - 2 z

1骣 1 1骣 1 � 2 - 琪z and 3 =琪z + 2 桫 z 2 桫 z

30 Now z =2 + 3 then z -2 = 3 and on squaring both sides

2 ( z -2) = 3

z2 -2 z 2 + 2 = 3

2 2 (z2 -1) = ( 2 z 2 )

z4-2 z 2 + 1 = 8 z 2

z4-10 z 2 + 1 = 0

The minimal polynomial of z =2 + 3 is z4-10 z 2 + 1 = 0

2 禳x骣1 xy 骣 1 4 2 I=睚琪 z + , 琪 z - ,z -10 z + 1 铪2桫z 2 桫 z

Now computing the Groebner basis for I using CoCoA we get

[z4 - 10z 2 + 1, -1/9x]

Let us look at another example of this type! i.e. Q() e.g. Q( 5, 7)

Let I  x 2 7  y, xy 5

Let z  5  7 then

31 1 1 1 5  7 5  7 5  7 1   .     7  5 z 5  7 5  7 5  7 5  7  2 2

Thus we have z  5  7 and

2  7  5 z

1  2    z    7 and 2  z 

1  2   z    5 and the minimal polynomial obtained by squaring both sides and 2  z  taking the radicals to their side appropriately we obtain

p(z)  z 4  24z 2  4  0

Now computing using CoCoA we the Groebner basis as

2 28 1 1 2 155 56 {z 4  24z 2  4, xy, x 2 z  yz 2  2y, yz 3  x 2  yz,26y 2 , x3} 11 13 13 28 13 182 13

On replacing back the z we get

2 28 1 1 2 155 56 {-xy , x2 ( 5 + 7) - y ( 5 + 7) 2 + 2 y , y ( 5 + 7) 3 - x 2 - y ( 5 + 7), - 26 y 2 , x 3 } 11 13 13 28 13 182 13

Which is very easy to compute.

Groebner bases over any finitely generated extensions of Q

We shall make use a theorem from that states that:

5.6.1 Theorem on simple extensions of fields. Herstein [5]

If F is of characteristic 0, and if a and b are algebraic over F, then there exists an element c  F(a,b) such that F(a,b)  F(c)

32 When we talk of F above, we mean a Field (i.e. a commutative additive and multiplicative group) and the characteristic refers to the number of times 111 .... 1  0 if at all it ever adds up to zero. If it never adds up to zero like in the case of rational numbers Q, then the characteristic is zero.

We shall not prove the theorem but use it’s corollary i.e.

Corollary

Any finite extension of a field of characteristic 0 is a simple extension.

Examples:

Q( 2) , Q( 3) , Q( 5) , Q( 2, 3) , even Q( 2, 3, 5) are finite extensions of Q which has a characteristic 0 they therefore simple.

5.6.2 Theorem on Groebner basis over finite extensions of Q

This theorem is enables us to compute Groebner basis of any polynomial ideal with coefficients from any field, using only arithmetic over Q , instead of from the field.

- - - - Let be an Ideal and let I= 狁 f1(a , x ), f 2 ( a , x ), f 3 ( a , x ),..., fn ( a , x )

    have a Groebner basis I   f1 (t, x), f 2 (t, x), f 3 (t, x),..., f n (t, x), p(t)

     I  g1(t, x), g2 (t, x), g3 (t, x),..., gs (t, x), gs1(t, x) Then

 1. gs1(t, x)  p(t)

    2. is the Groebner Basis for I {g1(, x), g2 (, x), g3 (, x),..., gs (, x)}

Proof:

- - - - Given an Ideal , where the fi 's are any I= 狁 f1( x ), f 2 ( x ), f 3 ( x ),..., fn ( x )◁ K [ x ]

basis for I then I has a unique basis g1, g 2, ..., g s in I such that

33    LT(I)  LT(g ), LT(g ),..., LT (g ) and see 1 2 s I  g1(x), g2 (x),..., gs (x) proposition 1 and definition 4 and the Hilbert basis theorem [2].

    Now, in the case where we I   f1(, x), f2 (, x), f3 (, x),..., fn (, x) introduce a variable to take care of the  i.e. let   t with a monomial ordering, say

p x1> x 2 >... xn > t and p()  0 , this such polynomial is the minimal.

    Now we have let and include p(t) in I   f1(t, x), f2 (t, x), f3 (t, x),..., fn (t, x)

    the basis so that we have by the same corollary we I   f1 (t, x), f 2 (t, x), f3 (t, x),..., f n (t, x), p(t)

    can arrive at a Groebner basis say {g1(t, x), g2 (t, x),,..., gs (t, x), gs1(t, x)}

By the introduction/definition of the polynomial p(t) we will always have an extra basis element on

replacement of  with t as required prescribed. Thus g1, g2 ,..., g s  I is a Groebner basis and

p(t)  g s1 . This is so since on division no other polynomial’s leading term divides the leading term of p(t) . Cox[2 P.86]

A basis element disappears from the basis given, when two or more basis elements have their leading terms dividing one another.

Example1:

Given I  x3 - 2xy, x2 y  2y 2  x,x2 ,2xy,2y2  x

If we compute the Groebner basis we shall get A  {x2 ,2xy,2y2  x}

Now LT (x 3 - 2xy)  x3  xLT (x 2 ) thus x3 - 2xy disappears from the basis element set

2 2 2 1 2 2 Similarly, LT (x y  2y  x)  x y  ( 2 x)LT (2xy) , thus x y  2y  x disappears also.

34 Example2:

Let I  x2 + y2 + z2 -1, x2  z2 - y,x - z

We found out that the Groebner basis is A  {x - z, - y2  y 1,2z 2  y}

These two basis elements, x 2 + y 2 + z 2 -1 and x2  z2 - y disappeared from the basis since their sum yields a simpler basis element - y2  y 1. This actually disappears on applying the division algorithm.

35 6 Appendix

Some functions from CoCoA

CoCoA stands for Computations in Computer Algebra which facilitates computations so that we apply our theorems without having to spent much time in computations.

CoCoA is a program used for computing numbers and polynomials.

It works on many operating systems. It was developed by a small group of mathematicians at the University of Genova Italy and there as been collaboration with mathematicians from Universität Regensburg for computers equipped with the Windows operating system. It can be downloaded at http://CoCoA.dima.unige.it/

In our project we made use of CoCoA especially in the following:

Division Algorithm Define the environment i.e. R::= Q [ x , y , z ]; Then give the function you want to divide F:= x2 + xy + ......

Now give the divisors L:= [ xy - 1, y2 - 1,...] The output we get are some functions that on multiplying with L plus a remainder we get F.

Computation of Groebner Basis Again define the environment i.e. R::= Z /(a )[ x , y , z ]; Then give the Ideal whose Groebner Basis you want to compute I:= Ideal ( polynomial - generators ) Call the function to begin the process GB. Start _ GBasis ( I ); Call a function to stop the process GB. Complete ( I ); Display the Groebner Basis I. GBasis ;

36 7 References

[1.] Http://www.symbolicnet.org/areas/GB_and_CS/groebner.html

A webpage that hosts most of the Symbolic Journals of computations.

[2.] Ideals, Varieties, and Algorithms : An Introduction to Computational Algebraic

Geometry and Commutative Algebra (Undergraduate Texts in Mathematics)

by David Cox, John Little, Donal O'Shea; Springer; 2 edition (1997)

[3.] http://www.mathforum.com

A website in which many contributions by mathematicians are hosted.

[4.] Topics in Algebra, 2nd Edition, Fourteenth Wiley Eastern Reprint,

by I N Herstein , August 1993, Wiley Eastern Limited.

[5.] Buchberger, PhD Thesis. 1965. An Algorithm for finding a basis for the

residue class ring of a non zero-dimensional polynomial ideal.

Doctoral Dissertation . Math Institute University of Innsbruck, Austria.

[6.] Progress, Directions and Open problems in Multidimensional Systems Theory

1985-00-00-A by Bose, Guiver, Kamen, Valenzuela and B Buchberger

[7.] An Algorithm for computing a Groebner basis of a polynomial ideal over a ring

with zero divisors by Deepak Kapur and Yong Yang Cai. Department of Computer

Science, University of Albuquerque, NM 87131, USA {kapur, yycai}@cs.unm.edu

[8.] Http://mathworld.wolfram.com/ExtensionField.html.

[9.] Noetherian Ring -- From MathWorld.htm, taken from Hungerford, T W Algebra 8th ed New

York; Springer verlag, 1997

[10.] East African School 2004-Notes form the summer school organized by ICTP-Trieste-Italy

found at http://garsia.math.yorku.ca/kenya04 (as at April 2005)