A Polyhedral Method for Sparse Systems with many Positive Solutions

TL;DR

This paper introduces a polyhedral approach based on regular triangulations and positive decorability to construct polynomial systems with many positive solutions, providing new lower bounds and insights into their asymptotic behavior.

Contribution

It develops a novel duality concept of positive decorability and applies it to cyclic polytopes, yielding improved bounds on positive solutions of polynomial systems.

Findings

Established new lower bounds for positive solutions count

Identified a log-concavity property in the asymptotics

Produced large positively decorable subcomplexes of cyclic polytopes

Abstract

We investigate a version of Viro's method for constructing polynomial systems with many positive solutions, based on regular triangulations of the Newton polytope of the system. The number of positive solutions obtained with our method is governed by the size of the largest positively decorable subcomplex of the triangulation. Here, positive decorability is a property that we introduce and which is dual to being a subcomplex of some regular triangulation. Using this duality, we produce large positively decorable subcomplexes of the boundary complexes of cyclic polytopes. As a byproduct we get new lower bounds, some of them being the best currently known, for the maximal number of positive solutions of polynomial systems with prescribed numbers of monomials and variables. We also study the asymptotics of these numbers and observe a log-concavity property.