2

Groebner Bases

§§1Introduction

In Chapter 1, we have seen how the algebra of the polynomial rings k[x1,...,xn] and the geometry of affine algebraic varieties are linked. In this chapter, we will study the method of Groebner bases, which will allow us to solve problems about polynomial ideals in an algorithmic or computational fashion. The method of Groebner bases is also used in several powerful computer algebra systems to study specific polynomial ideals that arise in applications. In Chapter 1, we posed many problems concerning the algebra of polynomial ideals and the geometry of affine varieties. In this chapter and the next, we will focus on four of these problems.

Problems a. The Ideal Description Problem: Does every ideal I ⊂ k[x1,...,xn] have a finite generating set? In other words, can we write I =f1,..., fs for some fi ∈ k[x1,...,xn]? b. The Ideal Membership Problem: Given f ∈ k[x1,...,xn] and an ideal I =f1,..., fs, determine if f ∈ I . Geometrically, this is closely related to the problem of determining whether V( f1,..., fs) lies on the variety V( f ). c. The Problem of Solving Polynomial Equations: Find all common solutions in kn of a system of polynomial equations

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

Of course, this is the same as asking for the points in the affine variety V( f1,..., fs). d. The Implicitization Problem: Let V be a subset of kn given parametrically as

x1 = g1(t1,...,tm), . .

xn = gn(t1,...,tm).

49 50 2. Groebner Bases

If the gi are polynomials (or rational functions) in the variables t j ,thenV will be an affine variety or part of one. Find a system of polynomial equations (in the xi ) that defines the variety.

Some comments are in order. Problem (a) asks whether every polynomial ideal has a finite description via generators. Many of the ideals we have seen so far do have such descriptions—indeed, the way we have specified most of the ideals we have studied has been to give a finite generating set. However, there are other ways of constructing ideals that do not lead directly to this sort of description. The main example we have seen is the ideal of a variety, I(V ). It will be useful to know that these ideals also have finite descriptions. On the other hand, in the exercises, we will see that if we allow infinitely many variables to appear in our polynomials, then the answer to (a) is no. Note that problems (c) and (d) are, so to speak, inverse problems. In (c), we ask for the set of solutions of a given system of polynomial equations. In (d), on the other hand, we are given the solutions, and the problem is to find a system of equations with those solutions. To begin our study of Groebner bases, let us consider some special cases in which you have seen algorithmic techniques to solve the problems given above.

Example 1. When n = 1, we solved the ideal description problem in §5 of Chapter 1. Namely, given an ideal I ⊂ k[x], we showed that I =g for some g ∈ k[x] (see Corollary 4 of Chapter 1, §5). So ideals have an especially simple description in this case.

We also saw in §5 of Chapter 1 that the solution of the Ideal Membership Problem follows easily from the division algorithm: given f ∈ k[x], to check whether f ∈ I = g,wedivideg into f : f = q · g + r, where q, r ∈ k[x] and r = 0ordeg(r)

Example 2. Next, let n (the number of variables) be arbitrary, and consider the prob- lem of solving a system of polynomial equations:

a11x1 +···+a1n xn + b1 = 0, . (1) . am1x1 +···+amnxn + bm = 0, where each polynomial is linear (total degree 1). For example, consider the system

2x1 + 3x2 − x3 = 0, (2) x1 + x2 − 1 = 0, x1 + x3 − 3 = 0. §1. Introduction 51

We row-reduce the matrix of the system to reduced row echelon form: ⎛ ⎞ 1013 ⎝01−1 −2⎠ . 0000

The form of this matrix shows that