Bernstein's second theorem and Viro's method for sparse polynomial systems in chemistry

Abstract

We consider a family of sparse polynomial systems denned by a directed graph and a bipartite graph which depend on certain parameters. A convex polyhedral cone serves as a representative of all positive solutions of the family. We study the boundary of this cone with Bernstein's second theorem and Viro's method. In particular we present new results about the parameter regions where several positive solutions appear. Since they are steady states of an underlying dynamical system of mass action kinetics, the resulting multistationarity has important implications for the dynamics of that system. Examples from applications illustrate the theoretical results.