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We use tropical and non-archimedean geometry to study generic root counts offamilies of polynomial equations. These families are given as morphisms ofschemes $X\to{Y}$ that factor through a closed embedding into a relative torusover a parameter space $Y$. We prove a generalization of Bernstein's theoremfor these morphisms, showing that the root count of a single well-behavedtropical fiber spreads to an open dense subset of $Y$. We use this to expressthe generic root count of a wide class of square systems in terms of thematroidal degree of an explicit variety. This in particular gives tropicalformulas for the birational intersection indices and volumes of Newton-Okounkovbodies defined by Kaveh and Khovanskii, and the generic root counts of thesteady-state equations of chemical reaction networks. An important role inthese theorems is played by the notion of tropical flatness, which allows us toinfer generic properties of $X\to{Y}$ from a single tropical fiber. We showthat the tropical analogue of the generic flatness theorem holds, in the sensethat $X\to{{Y}}$ is tropically flat over an open dense subset of the Berkovichanalytification of $Y$.

Generic root counts and flatness in tropical geometry