The steady-state degree and mixed volume of a chemical reaction network

TL;DR

This paper provides a geometric approach to bound the algebraic complexity of steady-states in chemical reaction networks, offering formulas for specific network families' steady-state degree and mixed volume.

Contribution

It introduces a method to bound the steady-state degree using polyhedral geometry and derives explicit formulas for certain network families.

Findings

Upper bounds for steady-state degree using polyhedral geometry

Explicit formulas for steady-state degree in network families

Calculations of mixed volume for polynomial systems associated with networks

Abstract

The steady-state degree of a chemical reaction network is the number of complex steady-states, which is a measure of the algebraic complexity of solving the steady-state system. In general, the steady-state degree may be difficult to compute. Here, we give an upper bound to the steady-state degree of a reaction network by utilizing the underlying polyhedral geometry associated with the corresponding polynomial system. We focus on three case studies of infinite families of networks, each generated by joining smaller networks to create larger ones. For each family, we give a formula for the steady-state degree and the mixed volume of the corresponding polynomial system.