Generic root counts and flatness in tropical geometry

TL;DR

This paper extends Bernstein's theorem to higher codimension schemes using tropical geometry, introducing tropical flatness to analyze generic root counts and their relation to tropical intersection theory.

Contribution

It generalizes Bernstein's theorem to higher codimension schemes via tropical intersection products and introduces tropical flatness as a key concept for understanding generic root counts.

Findings

Tropical flatness holds over a dense open subset of the Berkovich analytification.

The tropical intersection number equals the root count at points with tropical flatness.

Provides formulas for volumes of Newton-Okounkov bodies and steady states in chemical networks.

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.