Positive Solutions of Systems of Signed Parametric Polynomial Inequalities

TL;DR

This paper develops a decision procedure for determining the existence of positive solutions in parametric polynomial inequality systems with prescribed coefficient signs, with applications in biological networks and satisfiability solving.

Contribution

It introduces a novel decision procedure that finds parametric positive solutions as rational functions, bridging heuristic methods and formal verification.

Findings

Provides a decision procedure for parametric polynomial inequalities

Characterizes the limitations of heuristic subtropical methods

Enables formal analysis of biological networks and SMT problems

Abstract

We consider systems of strict multivariate polynomial inequalities over the reals. All polynomial coefficients are parameters ranging over the reals, where for each coefficient we prescribe its sign. We are interested in the existence of positive real solutions of our system for all choices of coefficients subject to our sign conditions. We give a decision procedure for the existence of such solutions. In the positive case our procedure yields a parametric positive solution as a rational function in the coefficients. Our framework allows to reformulate heuristic subtropical approaches for non-parametric systems of polynomial inequalities that have been recently used in qualitative biological network analysis and, independently, in satisfiability modulo theory solving. We apply our results to characterize the incompleteness of those methods.