# Sign conditions for the existence of at least one positive solution of a sparse polynomial system

## TL;DR

This paper establishes sign-based criteria on the support and coefficients of sparse polynomial systems that ensure at least one positive real solution, using degree theory and Gale duality, with connections to toric ideals.

## Contribution

It introduces new sign conditions for positive solutions of sparse polynomial systems, linking algebraic and geometric methods like Gale duality and toric ideals.

## Key findings

- Sign conditions guarantee positive solutions in sparse systems
- Conditions relate to Gale duality and degree theory
- Connections made to algebraic properties of toric ideals

## Abstract

We give sign conditions on the support and coefficients of a sparse system of d generalized polynomials in d variables that guarantee the existence of at least one positive real root, based on degree theory and Gale duality. In the case of integer exponents, we relate our sufficient conditions to algebraic conditions that emerged in the study of toric ideals.