Real Tropicalization and Analytification of Semialgebraic Sets

Abstract

Let K be a real closed field with a nontrivial non-archimedean absolute value. We study a refined version of the tropicalization map, which we call real tropicalization map, that takes into account the signs on K⁠. We study images of semialgebraic subsets of Kⁿ under this map from a general point of view. For a semialgebraic set S⊆Kⁿ we define a space S^an_r called the real analytification, which we show to be homeomorphic to the inverse limit of all real tropicalizations of S⁠. We prove a real analogue of the tropical fundamental theorem and show that the tropicalization of any semialgebraic set is described by tropicalization of finitely many inequalities, which are valid on the semialgebraic set. We also study the topological properties of real analytification and tropicalization. If X is an algebraic variety, we show that X^an_r can be canonically embedded into the real spectrum Xr of X⁠, and we study its relation with the Berkovich analytification of X⁠.