Heuristics to accelerate the Dixon resultant

• Authors: Robert H. Lewis
• Venue: Mathematics and Computers in Simulation
• Year: 2008

The Dixon resultant method solves a system of polynomial equations by computing its resultant. It constructs a square matrix whose determinant (det) is a multiple of the resultant (res). The naive way to proceed is to compute det, factor it, and identify res. But often det is too large to compute or factor, even though res is relatively small. In this paper we describe three heuristic methods that often overcome these problems. The first, although sometimes useful by itself, is often a subprocedure of the second two. The second may be used on any polynomial system to discover factors of det without producing the complete determinant. The third applies when res appears as a factor of det in a certain exponential pattern. This occurs in some symmetrical systems of equations. We show examples from computational chemistry, signal processing, dynamical systems, quantifier elimination, and pure mathematics.