Polyhedral Methods in Numerical Algebraic Geometry

TL;DR

This paper introduces polyhedral methods for numerical algebraic geometry, focusing on certificates for algebraic curves using Puiseux series and tropisms, which can optimize the computation of solution sets.

Contribution

It proposes a novel approach linking tropisms and mixed volumes to improve the computation of algebraic curves in numerical algebraic geometry.

Findings

Tropisms are related to the leading terms of Puiseux series for algebraic curves.

The method reduces computational complexity for systems with few monomials.

Tropisms and Puiseux series can serve as preprocessing for witness set computations.

Abstract

In numerical algebraic geometry witness sets are numerical representations of positive dimensional solution sets of polynomial systems. Considering the asymptotics of witness sets we propose certificates for algebraic curves. These certificates are the leading terms of a Puiseux series expansion of the curve starting at infinity. The vector of powers of the first term in the series is a tropism. For proper algebraic curves, we relate the computation of tropisms to the calculation of mixed volumes. With this relationship, the computation of tropisms and Puiseux series expansions could be used as a preprocessing stage prior to a more expensive witness set computation. Systems with few monomials have fewer isolated solutions and fewer data are needed to represent their positive dimensional solution sets.