Reaction networks and real algebraic geometry: A fruitful interplay

The study of equilibria of (bio)chemical reaction networks with mass-action kinetics dates back to the 1970s, with foundational work by Horn, Jackson, and Feinberg, who established the field of Chemical Reaction Network Theory (CRNT). A central question in CRNT is whether a given reaction network admits multiple equilibria for some choice of parameter values. Since equilibria correspond to the positive solutions of parametric polynomial systems, the field has seen a significant shift over the past two decades with the integration of ideas, tools, and techniques from (real) algebraic geometry and computational algebra. This interaction has not only deepened our understanding of reaction networks but also revealed that many results initially developed within CRNT have far-reaching applications beyond chemistry.

In this talk, I will start by introducing key concepts and questions from reaction network theory and its algebraic framework in terms of vertically parametrised systems. After this motivation, I will present selected examples of the interplay between the theory of reaction networks and real algebraic geometry.