How to count efficiently all affine roots of a polynomial system

Abstract

Polynomials are ubiquitous in a variety of applications. A relatively recent theory exploits their sparse structure by associating a point configuration to each polynomial system: however, it has so far mostly dealt with roots having nonzero coordinates. We shift attention to arbitrary affine roots, and improve upon the existing algorithms for counting them and computing them numerically. The one existing approach is too expensive in practice because of the usage of recursive liftings of the given point configuration. Instead, we define a single lifting which yields the desired count and defines a homotopy continuation for computing all solutions. We enhance the numerical stability of the homotopy by establishing lower bounds on the lifting values and prove that they can be derived dynamically to obtain the lowest possible values. Our construction may be regarded as a generalization of the dynamic lifting algorithm for the computation of mixed cells.