A note on tropicalization in the context of Berkovich spaces

Abstract

Let K be a non-Archimedean non-trivially valued field, and Γ be its value group. Let X be a compact strictly K-analytic space and f: X → \(A_K^{n,an}\) be a morphism of K-analytic spaces. We prove that |f|(X) ∩ (ℝ*+)n is a Γ-rational polyhedral set whose dimension is less than or equal to dim(X). The main ingredient of the proof is a quantifier elimination theorem for subanalytic sets due to Leonard Lipshitz.