Characterization of circuits supporting polynomial systems with the maximal number of positive solutions

TL;DR

This paper characterizes the specific circuits in real n-dimensional space that support polynomial systems with the maximum number of positive solutions, using advanced geometric and combinatorial methods.

Contribution

It identifies all circuits capable of supporting polynomial systems with n+1 positive solutions, advancing understanding of polynomial system solution bounds.

Findings

All such circuits are characterized for any positive integer n.

Restrictions on circuits are derived using Grothendieck's dessins d'enfant.

Constructive examples are provided via Viro's patchworking.

Abstract

A polynomial system with $n$ equations in $n$ variables supported on a set $\mathcal{W}\subset\mathbb{R}^n$ of $n+2$ points has at most $n+1$ non-degenerate positive solutions. Moreover, if this bound is reached, then $\mathcal{W}$ is minimally affinely dependent, in other words, it is a circuit in $\mathbb{R}^n$. For any positive integer number $n$, we determine all circuits $\mathcal{W}\subset\mathbb{R}^n$ which can support a polynomial system with $n+1$ non-degenerate positive solutions. Restrictions on such circuits $\mathcal{W}$ are obtained using Grothendieck's real dessins d'enfant, while polynomial systems with $n+1$ non-degenerate positive solutions are constructed using Viro's combinatorial patchworking.