Parametrized systems of generalized polynomial inequalitites via linear algebra and convex geometry

TL;DR

This paper introduces a geometric and linear algebraic framework to analyze solutions of parametrized generalized polynomial inequalities, revealing new insights into their structure and applications in reaction networks and fewnomials.

Contribution

It establishes a novel geometric approach linking polynomial inequalities to polytopes and subspaces, enabling explicit solution parametrizations and complexity analysis.

Findings

Solution set characterized by binomial equations on polytopes

Explicit bijection between original and geometric solution sets

Applications to reaction networks and polynomial root analysis

Abstract

We provide fundamental results on positive solutions to parametrized systems of generalized polynomial $\textit{inequalities}$ (with real exponents and positive parameters), including generalized polynomial $\textit{equations}$. In doing so, we also offer a new perspective on fewnomials and (generalized) mass-action systems. We find that geometric objects, rather than matrices, determine generalized polynomial systems: a bounded set/"polytope" $P$ (arising from the coefficient matrix) and two subspaces representing monomial differences and dependencies (arising from the exponent matrix). The dimension of the latter subspace, the monomial dependency $d$, is crucial. As our main result, we rewrite $\textit{polynomial inequalities}$ in terms of $d$ $\textit{binomial equations}$ on $P$, involving $d$ monomials in the parameters. In particular, we establish an explicit bijection between the original solution set and the solution set on $P$ via exponentiation. (i) Our results apply to any generalized polynomial system. (ii) The dependency $d$ and the dimension of $P$ indicate the complexity of a system. (iii) Our results are based on methods from linear algebra and convex/polyhedral geometry, and the solution set on $P$ can be further studied using methods from analysis such as sign-characteristic functions (introduced in this work). We illustrate our results (in particular, the relevant geometric objects) through three examples from real fewnomial and reaction network theory. For two mass-action systems, we parametrize the set of equilibria and the region for multistationarity, respectively, and even for univariate trinomials, we offer new insights: We provide a "solution formula" involving discriminants and "roots".