Toric Generalized Characteristic Polynomials

TL;DR

This paper introduces an efficient method leveraging toric geometry and sparse resultants to solve polynomial systems with degenerate solutions, providing improved complexity bounds and reliability over previous approaches.

Contribution

It presents a new algorithm for finding isolated points in polynomial systems using sparse resultants and toric geometry, with near-cubic complexity bounds.

Findings

Complexity bounds close to cubic in the degree of the variety

Improved reliability of sparse resultant algorithms

Effective handling of degenerate solution sets

Abstract

We illustrate an efficient new method for handling polynomial systems with degenerate solution sets. In particular, a corollary of our techniques is a new algorithm to find an isolated point in every excess component of the zero set (over an algebraically closed field) of any $n$ by $n$ system of polynomial equations. Since we use the sparse resultant, we thus obtain complexity bounds (for converting any input polynomial system into a multilinear factorization problem) which are close to cubic in the degree of the underlying variety -- significantly better than previous bounds which were pseudo-polynomial in the classical B\'ezout bound. By carefully taking into account the underlying toric geometry, we are also able to improve the reliability of certain sparse resultant based algorithms for polynomial system solving.