A Polyhedral Homotopy Algorithm For Real Zeros

TL;DR

This paper introduces a homotopy continuation algorithm based on Viro's patchworking to efficiently find and count real zeros of sparse polynomial systems with specific coefficient conditions, operating entirely over the real numbers.

Contribution

It presents a novel polyhedral homotopy method that accurately tracks real solutions of sparse polynomial systems, leveraging the geometry of A-discriminant amoebas.

Findings

Successfully tracks the optimal number of solution paths

Efficiently finds real zeros in unbounded components

Operates entirely over the real numbers

Abstract

We design a homotopy continuation algorithm, that is based on numerically tracking Viro's patchworking method, for finding real zeros of sparse polynomial systems. The algorithm is targeted for polynomial systems with coefficients satisfying certain concavity conditions. It operates entirely over the real numbers and tracks the optimal number of solution paths. In more technical terms; we design an algorithm that correctly counts and finds the real zeros of polynomial systems that are located in the unbounded components of the complement of the underlying A-discriminant amoeba.