Parallel Resultant Computation

Abstract

A resultant is a purely algebraic criterion for determining when a finite collection of polynomials have a common zero. It has been shown to be a useful tool in the design of efficient parallel and sequential algorithms in symbolic algebra, computational geometry, computational number theory, and robotics. We begin with a brief history of resultants and a discussion of some of their important applications. Next we review some of the mathematical background in commutative algebra that will be used in subsequent sections. The Nullstellensatz of Hilbert is presented in both its strong and weak forms. We also discuss briefly the necessary background on graded algebras, and define affine and projective spaces over arbitrary fields. We next present a detailed account of the resultant of a pair of univariate polynomials, and present efficient parallel algorithms for its computation. The theory of subresultants is developed in detail, and the computation of polynomial remainder sequences is derived. A resultant system for several univariate polynomials and algorithms for the gcd of several polynomials are given. Finally, we develop the theory of multivariate resultants as a natural extension of the univariate case. Here we treat both classical results on the projective (homogeneous) case, as well as more recent results on the affine (inhomogeneous) case. The u-resultant of a set of multivariate polynomials is defined and a parallel algorithm is presented. We discuss the computation of generalized characteristic polynomials and relate them to the decision problem for the theories of real closed and algebraically closed fields.