An effective criterion for multiple positive zeros of vertically parametrized polynomial systems

TL;DR

This paper introduces an effective criterion to determine the existence of multiple positive zeros in augmented vertically parametrized polynomial systems, simplifying the analysis to linear feasibility checks and characterizing conditions for their occurrence.

Contribution

The authors develop a new criterion that reduces the problem to linear feasibility and provides a full characterization for systems with certain sparsity structures.

Findings

Provides a sufficient condition for absence of multiple positive zeros.

When the coefficient matrix kernel has specific sparsity, the condition is also necessary.

Offers a complete characterization of multiple zeros in the studied systems.

Abstract

We present an effective criterion for determining whether a (augmented) vertically parametrized polynomial system admits multiple positive zeros for some choice of parameter values. Our method builds on previous algorithms from chemical reaction network theory and reduces the question to checking the feasibility of a linear system of equalities and inequalities. Our criterion provides a sufficient condition for the absence of multiple positive zeros that applies to any augmented vertically parametrized polynomial system, and we show that when the kernel of the coefficient matrix of the system displays a certain sparsity structure, this condition becomes also necessary. We give thereby a full characterization of the existence of multiple zeros for this type of systems.