Computing the Lowest-Order Element of a Multivariate Elimination Ideal by Using Remainder Sequences

Abstract

Given a set of m+1 multivariate polynomials, with m > 1, in main variables x_1,...,x_m and sub-variables u_1,...,u_n, we can usually eliminate x_1,...,x_m and obtain a polynomial in u_1,...,u_n only. There are basically two methods to perform this elimination. One is the so-called resultant method and the other is the Groebner basis method. The Groebner basis method gives the lowest-order element \haS(u) of the elimination ideal, where (u) = (u_1,...,u_n), but it is often very slow. The resultant method is quite fast, but the resulting polynomial R(u) often contains many more terms than \haS(u). In this paper, we present a simple method of computing \haS(u) by the repeated computation of PRSs (polynomial remainder sequences). The idea is to compute PRSs by changing their arguments systematically and obtain polynomials R_1(u),...,R_k(u), k > 1, in the sub-variables only. Let \baS(u) be the GCD of R_1,...,R_k. Then, our main theorem asserts that \baS(u) is a multiple of \haS(u): \baS(u) = \tie(u)\haS(u). We call \tie(u) the extraneous factor and it often consists of a small number of terms. We present three conditions and one sub-method to remove \tie(u) from \baS(u).