Generic root counts and flatness in tropical geometry
Paul Alexander Helminck, Yue Ying Ren · Journal of the London Mathematical Society · 2025
Abstract We use tropical and nonarchimedean geometry to study the generic number of solutions of families of polynomial equations over a parameter space . In particular, we are interested in the choices of parameters for which the generic root count is attained. Our families are given as subschemes where is a relative torus over . We generalize Bernstein's theorem from an intersecting family of hypersurfaces to an intersecting family of higher‐codimensional schemes , replacing the mixed volume by a tropical intersection product. Central to our work is the notion of tropical flatness of around a point , which allows us to transfer tropical properties of the fiber over to generic properties. We show that tropical flatness holds over a dense open subset of the Berkovich analytification , and that the tropical intersection number is attained as a root count at all around which the ’s are tropically flat and the tropical prevariety of the fibers 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.