Generic root counts and flatness in tropical geometry

Abstract: In this talk, we give new generic root counts of square polynomial systems using methods from tropical and non-archimedean geometry. The main theoretical ingredient is a generalization of a famous theorem by Bernstein, Kushnirenko and Khovanskii, which now says that the behavior of a single well-behaved zero-dimensional tropical fiber spreads to an open dense subset. We use this theorem on modifications of the universal polynomial system to obtain generic root counts of determinantal subvarieties of the universal parameter space. An important role in these generalizations is played by the notion of tropical flatness, which allows us to link a single tropical fiber to fibers over an open dense subset. We also prove a tropical analogue of Grothendieck's generic flatness theorem, saying that a given morphism is tropically flat over a dense open subset.

algebraic geometrycombinatorics

Audience: researchers in the topic

(LAGARTOS) Latin American Real and Tropical Geometry Seminar

Series comments: Please email one of the organizers to receive login info and join our mail list.