Polyhedral Homotopies in Cox Coordinates

TL;DR

This paper introduces the Cox homotopy algorithm, a novel method for solving sparse polynomial systems on toric varieties by tracking solutions in Cox coordinates, combining advantages of polyhedral and homogeneous homotopies.

Contribution

The paper presents the Cox homotopy algorithm, extending path tracking methods to Cox coordinates on toric varieties, enabling efficient solutions and insights into root count deficiencies.

Findings

Tracks solutions in Cox coordinates, generalizing existing homotopies.

Handles solutions near divisors of toric varieties.

Potentially explains root count deficiencies in certain systems.

Abstract

We introduce the Cox homotopy algorithm for solving a sparse system of polynomial equations on a compact toric variety $X_\Sigma$. The algorithm lends its name from a construction, described by Cox, of $X_\Sigma$ as a GIT quotient $X_\Sigma = (\mathbb{C}^k \setminus Z) // G$ of a quasi-affine variety by the action of a reductive group. Our algorithm tracks paths in the total coordinate space $\mathbb{C}^k$ of $X_\Sigma$ and can be seen as a homogeneous version of the standard polyhedral homotopy, which works on the dense torus of $X_\Sigma$. 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 $G$-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 $X_\Sigma$. 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.