Software for the Gale transform of fewnomial systems and a Descartes rule for fewnomials

TL;DR

This paper introduces software for computing the Gale transform of fewnomial systems and establishes a Descartes'-like bound on positive solutions, advancing numerical methods for solving such systems.

Contribution

It develops algorithms and software for the Gale transform of fewnomial systems and presents a new bound on positive solutions under specific conditions.

Findings

Software implementation for Gale transform

New Descartes'-like bound for fewnomials

Integration into Khovanskii-Rolle continuation method

Abstract

We give a Descartes'-like bound on the number of positive solutions to a system of fewnomials that holds when its exponent vectors are not in convex position and a sign condition is satisfied. This was discovered while developing algorithms and software for computing the Gale transform of a fewnomial system, which is our main goal. This software is a component of a package we are developing for Khovanskii-Rolle continuation, which is a numerical algorithm to compute the real solutions to a system of fewnomials.