The mechanization of multires algorithm and computation of mixed volume using mixed subdivision for sparse resultants

Abstract

The method of finding the solutions of a system of non-linear polynomial equations has received a lot of attention since ancient times. Recent active ongoing research related to solving such equations is on the construction and implementations of the method of sparse resultant. The aim of this study is to investigate on the mechanization of the multires algorithm for computing sparse resultant matrix. All the input polynomials are of type over-constrained systems. The basic fact about the number of roots of a polynomial system is given by the mixed volume of the Minkowski sum of convex polytopes. The computation of the mixed volume of a given polynomial system using subdivision and volumes of mixed cells is demonstrated. Maple multires package is used to observe about the size of the sparse resultant matrices for some polynomial systems. The results on mixed volume are compared with the respective results on the size of the resultant matrix.