Sparse Polynomial Systems with many Positive Solutions from Bipartite Simplicial Complexes

TL;DR

This paper explores how bipartite simplicial complexes enable the construction of sparse polynomial systems with many positive solutions, advancing understanding of real solutions in algebraic geometry.

Contribution

It establishes a characterization of when all simplices in a triangulation can be positively decorated, linking this to bipartite dual graphs, and applies this to identify maximally positive systems and construct systems with many positive solutions.

Findings

All simplices can be positively decorated iff the triangulation is balanced.

Identifies classical families with maximally positive systems.

Constructs fewnomial systems with many positive solutions.

Abstract

Consider a regular triangulation of the convex-hull $P$ of a set $\mathcal A$ of $n$ points in $\mathbb R^d$, and a real matrix $C$ of size $d \times n$. A version of Viro's method allows to construct from these data an unmixed polynomial system with support $\mathcal A$ and coefficient matrix $C$ whose number of positive solutions is bounded from below by the number of $d$-simplices which are positively decorated by $C$. We show that all the $d$-simplices of a triangulation can be positively decorated if and only if the triangulation is balanced, which in turn is equivalent to the fact that its dual graph is bipartite. This allows us to identify, among classical families, monomial supports which admit maximally positive systems, i.e. systems all toric complex solutions of which are real and positive. These families give some evidence in favor of a conjecture due to Bihan. We also use this technique in order to construct fewnomial systems with many positive solutions. This is done by considering a simplicial complex with bipartite dual graph included in a regular triangulation of the cyclic polytope.