An effective criterion for multiple positive zeros of vertically parametrized polynomial 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 problem to checking the feasibility of linear systems of equalities and inequalities. Our criterion provides a necessary condition for the existence 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 particular sparsity structure, this condition also becomes sufficient. This provides a full characterization of the existence of multiple zeros for this type of systems.