The generic geometry of steady state varieties

TL;DR

This paper investigates the geometric properties of reaction networks with power-law kinetics, focusing on steady states, robustness, and multistationarity, using algebraic and linear algebra tools.

Contribution

It provides an ideal-theoretic characterization of robustness and conditions linking multiple steady states to multistationarity in reaction networks.

Findings

Characterizes generic finiteness of steady states.

Provides conditions for robustness and multistationarity.

Introduces a linear algebra criterion for positive nondegenerate zeros.

Abstract

We answer several fundamental geometric questions about reaction networks with power-law kinetics, on topics such as generic finiteness of the number of steady states, robustness, and nondegenerate multistationarity. In particular, we give an ideal-theoretic characterization of generic absolute concentration robustness, as well as conditions under which a network that admits multiple steady states also has the capacity for nondegenerate multistationarity. The key tools underlying our results come from the theory of vertically parametrized systems, and include a linear algebra condition that characterizes when the steady state system has positive nondegenerate zeros.