Generic dimension and optimal start systems in reaction network theory

Parametric polynomial systems with fixed support that arise in applications often have algebraic dependencies between the coefficients, which makes them more intricate to study than sparse systems where the coefficients are completely free. For example, the generic dimension of the solution set might be higher than the one predicted by the supports and number of equations, and in the zero-dimensional case, the generic cardinality might be lower than the one predicted by Bernstein’s theorem.

In this talk, we will look closer at these issues for the steady state equations studied in chemical reaction network theory. In the first part, I will discuss various network-theoretic conditions that ensure that the codimension of the steady state variety generically is the rank of the network, and that it generically intersects the stoichiometric compatibilities classes finitely. In the second part of the talk, I will discuss a tropical generalization of Bernstein’s theorem that allows us to compute the generic number of complex steady states in a stoichiometric compatibility class, by replacing the mixed volume with a tropical intersection number. This, in turn, also gives us optimal start systems for numerically approximating the steady states homotopy continuation, without tracing superfluous paths, and makes it possible to certify that all of them are found.

This is a combination of several joint works with Elisenda Feliu, Paul Helminck, Beatriz Pascual-Escudero, Yue Ren, Benjamin Schröter, and Máté Telek.