Parallel Resultant Computation

Abstract

A resultant is a purely algebraic criterion for determining whether a nite collection of polynomials have a common zero. It has been shown to be a useful tool in the design of e cient 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 brie y the necessary background on graded algebras, and de ne a ne and projective spaces over arbitrary elds. We next present a detailed account of the resultant of a pair of univariate polynomials, and present e-cient 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, wedevelop 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 a ne (inhomogeneous) case. The u-resultant of a set of multivariate polynomials is de ned 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 elds. 20.1