In this paper, we tackle the following problem: compute the gcd for several univariate polynomials with parametric coefficients. It amounts to partitioning the parameter space into ``cells'' so that the gcd has a uniform expression over each cell and constructing a uniform expression of gcd in each cell. We tackle the problem as follows. We begin by making a natural and obvious extension of subresultant polynomials of two polynomials to several polynomials. Then we develop the following structural theories about them. 1. We generalize Sylvester's theory to several polynomials, in order to obtain an elegant relationship between generalized subresultant polynomials and the gcd of several polynomials, yielding an elegant algorithm. 2. We generalize Habicht's theory to several polynomials, in order to obtain a systematic relationship between generalized subresultant polynomials and pseudo-remainders, yielding an efficient algorithm. Using the generalized theories, we present a simple (structurally elegant) algorithm which is significantly more efficient (both in the output size and computing time) than algorithms based on previous approaches.

翻译：本文研究如下问题：计算多个带参数系数的一元多项式的最大公因式。该问题需要将参数空间划分为“单元”，使得在每个单元上最大公因式具有统一表达式，并在每个单元中构造最大公因式的统一表达式。我们按以下方式处理该问题：首先，将两个多项式的子结式多项式自然且直观地推广到多个多项式情形。随后，建立关于这些多项式结构的如下理论：1. 将Sylvester理论推广至多个多项式，以建立广义子结式多项式与多个多项式最大公因式之间的优美关系，从而得到简洁的算法；2. 将Habicht理论推广至多个多项式，以建立广义子结式多项式与伪余式之间的系统关系，从而得到高效的算法。基于上述推广理论，我们提出一种简洁（结构优雅）的算法，其在输出规模和计算时间上均显著优于基于先前方法构建的算法。