Real Tropicalization and Analytification of Semialgebraic Sets

Author:

Jell Philipp1,Scheiderer Claus2,Yu Josephine3

Affiliation:

1. Universität Regensburg, 93040 Regensburg, Germany

2. Fachbereich Mathematik und Statistik, Universität Konstanz, 78457 Konstanz, Germany

3. School of Mathematics, Georgia Tech, Atlanta, GA 30332, USA

Abstract

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 $S \subseteq K^n$ we define a space $S_r^{{\operatorname{an}}}$ 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 $X_r^{{\operatorname{an}}}$ can be canonically embedded into the real spectrum $X_r$ of $X$, and we study its relation with the Berkovich analytification of $X$.

Funder

DFG

NSF

Publisher

Oxford University Press (OUP)

Subject

General Mathematics

Reference19 articles.

1. Tropical spectrahedra;Allamigeon,2020

2. Logarithmic limit sets of real semi-algebraic sets;Alessandrini;Adv. Geom.,2013

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