Generic root counts and flatness in tropical geometry

Abstract

We use tropical and non-archimedean geometry to study the generic number of solutions of families of polynomial equations over a parameter space $Y$. In particular, we are interested in the choices of parameters for which the generic root count is attained. Our families are given as subschemes $X\subseteq T$ where $T$ is a relative torus over $Y$. We generalize Bernstein's theorem from an intersecting family of hypersurfaces $X=V(f_1)\cap\dots\cap V(f_n)$ to an intersecting family of higher-codimensional schemes $X=X_1\cap\dots\cap X_k$, replacing the mixed volume by a tropical intersection product. Central to our work is the notion of tropical flatness of $X$ around a point $P\in Y$, which allows us to transfer tropical properties of the fiber over $P$ to generic properties. We show that tropical flatness holds over a dense open subset of the Berkovich analytification $Y^\text{an}$, and that the tropical intersection number is attained as a root count at all $P\in Y^\text{an}$ around which the $X_i$'s are tropically flat and the tropical prevariety of the fibers $\bigcap_{i=1}^k\text{Trop}(X_{i,P})$ is bounded. We then study the generic root count of a wide class of parametrized square polynomial systems. This in particular gives tropical formulas for the volumes of Newton-Okounkov bodies, and the number of complex steady states of chemical reaction networks.