# Parametrized systems of generalized polynomial inequalities via linear algebra and convex geometry

## TL;DR

This paper introduces a new geometric approach to solving polynomial inequalities using linear algebra and convex geometry, offering insights into systems like reaction networks.

## Contribution

The paper introduces a novel geometric framework for analyzing generalized polynomial systems using subspaces and polytopes, enabling explicit bijections between solution sets.

## Key findings

- Geometric objects like polytopes and subspaces determine generalized polynomial systems, with monomial dependency d being crucial.
- Polynomial inequalities can be rewritten as d binomial equations on a polytope P, with a bijection between original and transformed solution sets.
- The framework applies to reaction networks and fewnomials, enabling parametrization of equilibria and multistationarity regions.

## Abstract

We provide fundamental results on positive solutions to parametrized systems of generalized polynomial inequalities (with real exponents and positive parameters), including generalized polynomial 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 polynomial inequalities in terms of d
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”.