Special Year Seminar I

We will present recent applications of enumerative algebra to the study of stationary states in physics. Our point of departure are classical Newtonian differential equations with nonlinear potential. It turns out that the study of their stationary states is closely related to solving systems of special polynomial equations. As observed by physicists, the number of solutions is smaller than predicted by the Bezout bound or its celebrated refinement due to Bernstein and Kushnirenko.

We will present a generalization of these two bounds based on the theory of Khovanskii bases. Our results give the correct, expected numbers for the polynomial systems that inspired us to look at the more general theory.