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 X Σ . The algorithm lends its name from a construction, described by Cox, of X Σ as a GIT quotient X Σ = (C k \ Z) G of a quasi-affine variety by the action of a reductive group. Our algorithm tracks paths in the total coordinate space C k of X Σ and can be seen as a homogeneous version of the standard polyhedral homotopy, which works on the dense torus of X Σ . 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 Σ . 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.

Introduction to Toric Geometry

“…Exercise 6.12 (A non-simplicial example). Example 3.1 in [13] considers the toric threefold X = X Σ corresponding to a pyramid in R 3 whose normal fan Σ has rays with generators…”

“…Exploiting this toric structure is crucial for the efficiency, but also for the accuracy of numerical solution methods. Examples of such toric solution methods include polyhedral homotopies [13,18], toric resultants [6,15] and toric normal form methods [3,29].…”

“…It has inspired efficient symbolic-numeric methods for solving polynomial equations, such as polyhedral homotopies [18] and sparse resultants [6,15]. The toric interpretation of the BKK theorem given in Theorem 6.29 is important for designing stable numerical algorithms [3,13,29].…”