Complexity of sparse polynomial solving: homotopy on toric varieties and the condition metric

TL;DR

This paper analyzes the complexity of solving sparse polynomial systems using homotopy methods on toric varieties, introducing new condition numbers and convergence criteria that generalize existing results.

Contribution

It introduces a homotopy algorithm on toric varieties with new condition metrics and convergence analysis, extending prior work to sparse polynomial systems.

Findings

Homotopy on toric varieties effectively solves sparse polynomial systems.

New condition numbers are invariant under group actions related to the momentum map.

The algorithm's complexity is linear in the condition length of the homotopy path.

Abstract

This paper investigates the cost of solving systems of sparse polynomial equations by homotopy continuation. First, a space of systems of $n$-variate polynomial equations is specified through $n$ monomial bases. The natural locus for the roots of those systems is known to be a certain toric variety. This variety is a compactification of $(\mathbb C\setminus\{0\})^n$, dependent on the monomial bases. A toric Newton operator is defined on that toric variety. Smale's alpha theory is generalized to provide criteria of quadratic convergence. Two condition numbers are defined and a higher derivative estimate is obtained in this setting. The Newton operator and related condition numbers turn out to be invariant through a group action related to the momentum map. A homotopy algorithm is given, and is proved to terminate after a number of Newton steps which is linear on the condition length of the lifted homotopy path. This generalizes a result from Shub (2009).