Available online at www.sciencedirect.com Mathematics and Computers in Simulation 80 (2010) 1146–1152 Comparing acceleration techniques for the Dixon and Macaulay resultants Robert H. Lewis ∗ Fordham University, New York, NY 10458, USA Received 28 November 2007; received in revised form 24 March 2008; accepted 16 April 2008 Available online 16 May 2008 Abstract The Bezout–Dixon resultant method for solving systems of polynomial equations lends itself to various heuristic acceleration techniques, previously reported by the present author, which can be extraordinarily effective. In this paper we will discuss how well these techniques apply to the Macaulay resultant. In brief, we find that they do work there with some difficulties, but the Dixon method is greatly superior. That they work at all is surprising and begs theoretical explanation. © 2008 IMACS. Published by Elsevier B.V. All rights reserved. Keywords: Resultant; Polynomial system; System of equations; Bezout; Dixon 1. Introduction Using resultants is a well-known method of solving systems of polynomial equations, such as f 1 (x, y, z, . . . , a, b, . . .) = 0 f 2 (x, y, z, . . . , a, b, . . .) = 0 f 3 (x, y, z, . . . , a, b, . . .) = 0 . . . We do not assume homogeneity. Thus, we have n equations in n - 1 variables x, y, z, . . . and a number of parameters a, b, c, . . .. The ground ring is Z, Q, or Z/p. All computations are exact. Solutions are understood to be in an algebraic closure. A resultant is a single polynomial derived from a system of polynomial equations that encapsulates the solution (common zero) to the system. We want to eliminate the variables and have a polynomial in the parameters. A common variation is to have n equations in n variables x i ,i = 1,n. Then one of them, say x 1 , is considered a “parameter” to bring this into the previous form. The resultant is therefore an equation in that one variable; the others have been eliminated. ∗ Tel.: +1 7188173226. E-mail address: rlewis@fordham.edu. 0378-4754/$36.00 © 2008 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.matcom.2008.04.020