  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.
