Newton-Okounkov bodies of chemical reaction systems

TL;DR

This paper applies Newton-Okounkov body theory to chemical reaction networks, introducing a new upper bound on the number of positive steady states and demonstrating its effectiveness through examples.

Contribution

It introduces the Newton-Okounkov body bound as a novel upper bound for positive steady states in chemical reaction systems, improving upon existing bounds.

Findings

Newton-Okounkov body bound often outperforms mixed volume bounds.

Explicit examples demonstrate the effectiveness of the new bound.

The approach links algebraic geometry with chemical reaction network analysis.

Abstract

Despite their noted potential in polynomial-system solving, there are few concrete examples of Newton-Okounkov bodies arising from applications. Accordingly, in this paper, we introduce a new application of Newton-Okounkov body theory to the study of chemical reaction networks, and compute several examples. An important invariant of a chemical reaction network is its maximum number of positive steady states, which is realized as the maximum number of positive real roots of a parametrized polynomial system. Here, we introduce a new upper bound on this number, namely the `Newton-Okounkov body bound' of a chemical reaction network. Through explicit examples, we show that the Newton-Okounkov body bound of a network gives a good upper bound on its maximum number of positive steady states. We also compare this Newton-Okounkov body bound to a related upper bound, namely the mixed volume of a chemical reaction network, and find that it often achieves better bounds.