Real tropicalization and analytification of semialgebraic sets

TL;DR

This paper introduces a real tropicalization map that incorporates signs for semialgebraic sets over real closed fields, establishing a real analogue of tropical fundamental theorems and exploring topological properties of real analytification.

Contribution

It defines the real analytification space and proves its homeomorphism to the inverse limit of real tropicalizations, extending tropical geometry to semialgebraic sets with signs.

Findings

Real tropicalization map accounts for signs in semialgebraic sets.

Real analytification is homeomorphic to inverse limit of tropicalizations.

Tropicalization of semialgebraic sets described by finitely many inequalities.

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 $K^n$ under this map from a general point of view. For a semialgebraic set $S \subset K^n$ we define a space $S_r^{\text{an}}$ 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 $X_r^{\text{an}}$ can be canonically embedded into the real spectrum $X_r$ of $X$, and we study its relation with the Berkovich analytification of $X$.