Fast computation of the Bezout and Dixon resultant matrices

Abstract

Efficient algorithms are derived for computing the entries of the Bezout resultant matrix for two univariate polynomials of degree n and for calculating the entries of the Dixon-Cayley resultant matrix for three bivariate polynomials of bidegree (m, n). Standard methods based on explicit formulas require O(n3) additions and multiplications to compute all the entries of the Bezout resultant matrix. Here we present a new recursive algorithm for computing these entries that uses only O(n2) additions and multiplications. The improvement is even more dramatic in the bivariate setting. Established techniques based on explicit formulas require O(m4n4) additions and multiplications to calculate all the entries of the Dixon-Cayley resultant matrix. In contrast, our recursive algorithm for computing these entries uses only O(m2n3) additions and multiplications. © 2002 Academic Press.