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

Given a set of m+1 multivariate polynomials, with m ≥ 2, 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 Ŝ(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 Ŝ(u). In this paper, we present a simple method of computing Ŝ(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 _ℓ (u), ℓ ≥ 2, in the sub-variables only. Let S̅(u) be the GCD of R ₁ ,...,R _ℓ . Then, our main theorem asserts that S̅(u) is a multiple of Ŝ(u): S̅(u) = ℓ̃(u)Ŝ(u). We call ℓ̃(u) the extraneous factor and it often consists of a small number of terms. We present three conditions and one sub-method to remove ℓ̃(u) from S̅(u).