Reaction networks and polynomial equations

A reaction network is a finite oriented graph embedded in Euclidean space. Dynamical systems generated by reaction networks can have arbitrary polynomial (or rational) right-hand-side. This creates a strong relationship between the study of reaction networks and the study of the set of solutions of some associated polynomial equations. In particular, many important questions about dynamical properties of reaction network models can be translated into questions about polynomial equations generated by the corresponding oriented graph.

For example, if some special polynomial equations have positive solutions, then the associated reaction network has a globally attracting fixed point. Similarly, if a special polynomial is positive everywhere, then the associated reaction network has at most one fixed point.

We will discuss some of these connections and several open problems.