Description
The steady states of a chemical reaction network with power-law kinetics can be described by a polynomial system with fixed support and coefficients that depend on parameters called rate constants and total amounts. This so-called vertical parametrization of the coefficients typically introduces dependencies among them, such that the generic root count over the complex numbers drops below the mixed volume bound predicted by the BKK theorem. In this talk, I will give an overview of recent results on the generic geometry of these systems and their incidence varieties. I will also discuss tropical techniques for constructing homotopies that allow solving the systems by tracing an optimal number of paths.
This is based on joint work, partly in progress, with Elisenda Feliu, Paul Helminck, Beatriz Pascual-Escudero, Yue Ren, Benjamin Schröter, and Máté Telek.
