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Gröbner Bases: Ideal Membership and Graph Colouring

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Gröbner Bases: Ideal Membership and Graph Colouring Gr¨obnerBases: Ideal Membership and Graph Colouring by Mitchell Gallinger A project submitted to the Department of Mathematical Sciences in conformity with the requirements for Math 4301 (Honour's Seminar) Lakehead University Thunder Bay, Ontario, Canada copyright c (2013) Mitchell Gallinger Abstract When looking at polynomial ideals, a problem that often arises is how to determine if an arbitrary polynomial is a member of the ideal. In this project we introduce the concept of a Gr¨obnerBasis, which is an algebraic tool used to determine ideal membership. The Gr¨obnerbasis was developed by Bruno Buchberger in 1965 and named for his advisor Wolfgang Gr¨obner. Once we have established a working knowledge of the Gr¨obnerbasis, we will delve into an interesting application in graph theory. By expressing a given graph as a set of polynomials, we will use Gr¨obner bases to determine if the graph is 3-colourable, and if so, determine all possible 3-colourings. i Acknowledgements It comes with no surprise that thanks are in order. I would like to thank: (i) My supervisors, Dr. Adam Van Tuyl and Dr. Jennifer Biermann; their knowledge and resources have been invaluable in my research. (ii) The Algebra Gang, consisting of my supervisors, Dr. Greg Lee, Jesse Krauel, and myself; their input and our weekly meetings have assisted in resolving errors and preparing me to present my research. (iii) My girlfriend Bradey; in listening to my practice presentations, to encouraging me when I felt overwhelmed; she has been essential in my success. ii Contents Abstract i Acknowledgements ii Chapter 1. Introduction 1 Chapter 2. Rings, Polynomials and the Division Algorithm 2 1. Rings 2 2. Polynomials 3 3. Monomial Orderings and The Division Algorithm 4 Chapter 3. Gr¨obnerBases 9 1. Definition and Existence of Gr¨obnerBases 9 2. Computing Gr¨obnerBases: Buchberger's Algorithm 12 Chapter 4. Gr¨obnerBases for Graph Colouring 17 1. Graph Colouring 17 Chapter 5. Concluding Remarks 22 Bibliography 23 iii CHAPTER 1 Introduction In abstract algebra, students are introduced to numerous concepts pertaining to groups and rings. While there are many interesting types of rings and subrings, a particuarly interesting one (and the focus of this project) is the ideal. With additive closure and absorption under multiplication, ideals have interesting and desirable properties. A com- mon problem in abstract algebra is that of determining if an object is a member of a given ideal; essentially, we wish to determine if the object can be expressed as a sum of products of known members of the ideal. This can be a problem of varying difficulty. In the case of ideals in the ring Z or in the polynomial ring in one variable, division will suffice. In the case of polynomial rings in multiple variables, simple division is no longer sufficient. Finding a method to determine ideal membership in this case gives us the following motivating problem: Problem 1.1. Given an ideal I in a polynomial ring R = k[x1; x2; :::; xn] over a field k, how can we determine if an element f of R belongs to I? The project will be structured as follows: In Chapter 2 we will define various algebraic concepts fundamental to better under- standing the ideal membership problem. We begin by reviewing concepts from ring theory. We continue with polynomials, defining the polynomial ring, the leading term, and degree of a polynomial. Next we look at monomial orders and the division algorithm; the concept of a monomial order is necessary to perform division on multiple variable polynomials, as it gives us a means to order terms. In Chapter 3, we present a solution to Problem 1.1 using Gr¨obnerbases. We begin the chapter by defining S-polynomials, essential to compute Gr¨obnerbases. We then define and prove the existence of the Gr¨obnerBasis. We conclude the chapter with a definition and proof of the Buchberger's Algorithm. This algorithm is used to compute the Gr¨obner Basis. In the final chapter we begin to look at an application of Gr¨obnerbases. We investigate a method used to determine if a graph is 3-colourable. We will define the necessary terms to understand the graph theory aspects, and then we will outline the process of determining whether a graph is 3-colourable. This chapter is motivated by the problem: Problem 1.2. Given a graph Γ how can we determine if there exists a vertex 3- colouring of Γ? Our approach will be to translate Problem 1.2 into an ideal membership problem. 1 CHAPTER 2 Rings, Polynomials and the Division Algorithm Before we can define Gr¨obnerBases, we will first review some ideas from abstract algebra. We will start by defining rings, ideals, and the polynomial ring, as well as the necessary related objects. At the end of the chapter, we will define the division algorithm and monomial orderings which are necessary to implement the algorithm for polynomials in one or more variables. Our primary sources for this chapter will be [1, 3]. 1. Rings Recall the following definition from abstract algebra, Definition 2.1. A ring R is a set under two binary operations, · and +, such that for all a; b; c 2 R: (i) a + b = b + a, (commutative addition) (ii) (a + b) + c = a + (b + c), (associative addition) (iii) There exists some 0 2 R such that 0 + a = a for all a 2 R. (iv) For every a 2 R there exists some −a 2 R such that a + (−a) = 0. (v) a(bc) = (ab)c (associative multiplication) (vi) a(b + c) = ab + ac and (b + c)a = ba + ca, (the distributive property) Definition 2.2. A ring R is said to have unity if there exists a multiplicative identity, an element 1 2 R such that 1 · a = a · 1 = a for all a 2 R. Definition 2.3. A ring R is commutative if a · b = b · a for every a; b 2 R. Definition 2.4. A ring R is said to be an integral domain if for every a; b 2 R, if a · b = 0, then either a = 0 or b = 0. Definition 2.5. A ring R is said to be a field if it satisfies the following conditions: (i) R is a commutative ring. (ii) R has unity. (iii) There exists a multiplicative inverse in R, that is, for every a 2 R, a 6= 0, there exists an element a−1 such that a · a−1 = 1. Some important fields include the rational numbers, Q , the real numbers, R , and the complex numbers, C. It is also important to note that all fields are integral domains. Definition 2.6. Let R be a commutative ring. A subset I ⊂ R is said to be an ideal if it satisfies the following: 2 3 (i) 0 2 I (ii) if a; b 2 I then a + b 2 I (iii) if a 2 I and c 2 R, then a · c 2 I. Definition 2.7. An ideal I is said to be finitely generated if there exists h1; h2; :::; hn 2 I, such that every g 2 I can be expressed as g = h1a1 + h2a2 + ::: + hnan; where ai 2 R. We will express this as I = hh1; h2; :::; hni, the ideal generated by h1; h2; :::; hn. Definition 2.8. An ideal I in a ring R is a principal ideal if I = hai for some a 2 R It is useful to note that all ideals in Z are principal ideals. 2. Polynomials Having now reviewed key algebra terms, we review the concept of a polynomial. We will define a monomial, a polynomial, the degree of a monomial and polynomial, and finally the polynomial ring. Definition 2.9. A monomial f on the variables x1; x2; :::; xn is a product of the form α1 α2 αn f = x1 x2 ··· xn where all αi are non-negative integers. Definition 2.10. A monomial ideal is an ideal generated by monomials. α1 α2 αn Definition 2.11. We define the degree of a monomial f = x1 x2 ··· xn , denoted deg(f), as the sum i=n X f = αi: i=1 By convention, when f = 0 we say that f has all degrees. Definition 2.12. A polynomial f on the variables x1; x2; :::; xn over a field k is a finite linear combination of monomials with coefficients in k, such that n X αi f = aix ; i=1 αi αi1 αi2 αin where αi is a finite n-tuple, αi = (αi1; αi2; :::; αin) and x = x1 x2 ··· xn and ai 2 k is the coefficient of xαi . Definition 2.13. We define a set of all polynomials f with the variables x1; x2; :::; xn, over a field k, as the polynomial ring, expressed as k[x1; x2; :::; xn]. If the polynomial ring being studied is of a smaller number of variables, it is a common practice to assign each variable a different symbol. For example, k[x],k[x; y], and k[x; y; z] are the polynomial rings in 1,2, and 3 variables, respectively. 4 The polynomial ring is indeed a ring. To prove this we simply ensure the set meets all the criteria outlined in Definition 2.1. We will forego the proof in interest of preserving space. 3. Monomial Orderings and The Division Algorithm In order to determine ideal membership of some element, it is necessary to see if the element can be expressed as a product of one or more elements of the ideal. In the case of ideals in the polynomial ring, this will require us to perform division on multiple variable polynomials.
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