Given a set of m+1 multivariate polynomials, with m ≥ 2, in main variables x1,...,xm and sub-variables u1,...,un, we can usually eliminate x1,...,xm and obtain a polynomial in u1,...,un 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) = (u1,...,un), 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 R1(u),...,Rℓ(u), ℓ ≥ 2, in the sub-variables only. Let S̅(u) be the GCD of R1,...,Rℓ. 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).
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机译：给定一组m + 1个多元多项式，其中m≥2，在主变量x中
1
，...，X
m
和子变量u
1
，...，u
n
，我们通常可以消除x
1
，...，X
m
并在u中获得多项式
1
，...，u
n
只要。基本上有两种方法可以执行此消除操作。一种是所谓的合成方法，另一种是Groebner基方法。 Groebner基法给出了消除理想的最低阶元素Ŝ（u），其中（u）=（u
1
，...，u
n
），但通常很慢。生成的方法相当快，但是生成的多项式R（u）包含的项通常比u（u）多得多。在本文中，我们提出了一种通过重复计算PRS（多项式余数序列）来计算Ŝ（u）的简单方法。这个想法是通过系统地改变参数来计算PRS并获得多项式R
1
（u），...，R
ℓ
（u），ℓ≥2，仅在子变量中。令S̅（u）为R的GCD
1
，...，R
ℓ
。然后，我们的主定理断言S̅（u）是Ŝ（u）的倍数：S̅（u）= ℓ̃（u）Ŝ（u）。我们称ℓ̃（u）为无关紧要的因素，它通常由少量的项组成。我们提出了三个条件和一个子方法来从Su（u）中删除ℓ̃（u）。
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