Polyhedral homotopies in Cox coordinates

Abstract

We introduce the Cox homotopy algorithm for solving a sparse system of polynomial equations on a compact toric variety [Formula: see text]. The algorithm lends its name from a construction, described by Cox, of [Formula: see text] as a GIT quotient [Formula: see text] of a quasi-affine variety by the action of a reductive group. Our algorithm tracks paths in the total coordinate space [Formula: see text] of [Formula: see text] and can be seen as a homogeneous version of the standard polyhedral homotopy, which works on the dense torus of [Formula: see text]. It furthermore generalizes the commonly used path tracking algorithms in (multi)projective spaces in that it tracks a set of homogeneous coordinates contained in the [Formula: see text]-orbit corresponding to each solution. The Cox homotopy combines the advantages of polyhedral homotopies and (multi)homogeneous homotopies, tracking only mixed volume many solutions and providing an elegant way to deal with solutions on or near the special divisors of [Formula: see text]. In addition, the strategy may help to understand the deficiency of the root count for certain families of systems with respect to the BKK bound.