Gröbner Bases for Solving Multivariate Polynomial Equations

Abstract

In this chapter we give an introduction to Grobner basis theory which was developed by Buchberger [1965, 1985] and has recently received considerable attention in the field of symbolic computation and computational algebra. In algebra the famous ideal membership problem states that given f in the set k[x 1 ,..., x n ] of all polynomials in n variables with coefficients in a field k and an ideal I = of k[x 1 ,..., x n ], determine whether f is in I or not. Buchberger [1965] proposed a remarkable algorithm for solving this problem completely. Briefly, his algorithm can be described as follows. Given a nonzero ideal I, first find a standard basis G,called a Grobner basis, for the ideal I. Then by a reduction process, Buch-berger has shown that f belongs to I if and only if f can be reduced to zero by members of G. The whole process can be carried out within a finite number of steps. Here we should be aware that this short description does not properly convey the true significance of Buchberger’s contribution. The reader is referred to the excellent books by Cox, Little and O’Shea [1996] and Adams and Loustaunau [1994] for more details on the impact of Buchberger’s work. We also refer to Wu [1984] and Chou [1984, 1988] for Wu’s method on the automated proving of elementary geometry theorems based on systems of polynomial equations. Wu’s method is somehow related to Buchberger’s method but different. In this chapter we will particularly apply Grobner basis theory to solve systems of multivariate polynomial equations over the complex number field ℂ. First we will be able to determine whether a system of polynomials has a solution or not. Second we will be able to find all solutions of the polynomial system when the system has a finite number of solutions. If the system has an infinite number of solutions, we can apply a homotopy algorithm to find as finitely many solutions as we wish to have. Third we will be able to solve a new class of complementarity problems in which the underlying functions are polynomials.