New heuristics and extensions of the Dixon resultant for solving polynomial systems Robert H. Lewis 1 Fordham University, New York, USA rlewis@fordham.edu, https://fordham.academia.edu/RobertLewis Abstract. In this work “solve a polynomial system” means to search for the common roots of a set of multivariate polynomials. Usually there are n variables, n equations, and some parameters. The system is neither over- nor under-determined. We eliminate all but one of the variables, leaving one polynomial in one variable and the parameters – the resultant [2]. The Bezout-Dixon method produces a matrix, M, whose determinant Det[M] is a multiple of the resultant. Dixon-EDF [7,9] is a way to com- pute the resultant without finding the entire determinant. In this work we present four new significant acceleration techniques and extensions to EDF. Ordering of variables. We give a heuristic for the “weight” of a vari- able within the system. The variables should be given precedence using this order, with the heaviest having the highest precedence. Leaving M as a 2 × 2 matrix. Dixon-EDF normalizes M in a special way. When finished, M is the identity matrix. But there are difficult problems where by the 2 × 2 step, the four polynomials are too large to multiply. Simply leaving M in that state can be acceptable. Decomposing into blocks. It has long been noted that Det[M] is often of the form qr k , where q is of no interest and r is the resultant. This suggests that matrix M could be decomposed into k equivalent blocks. We present a fast way to produce this decomposition, if present, and show huge speed-ups are possible. Dealing with more equations than variables. If there are more equations than variables, the Dixon resultant cannot be used even if the solution set is zero-dimensional. We present a method that can be effective in converting the system into one that Dixon can solve. Each new method will be illustrated with examples showing its effective- ness. Keywords: polynomial system, resultant, parameters, Dixon 1 Introduction In this work “solve a polynomial system” means to take a collection of multi- variate polynomials, set each to 0, and search for the common roots.