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 $K^n$ 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 $X$.

Pro-definability of spaces of definable types

“…Similarly, our aim is to view V as a model-theoretic analogue of the Huber analytification of V . When T is RCVF, V is a good candidate to be the model-theoretic counterpart of the analytification of semi-algebraic sets defined by Jell, Scheiderer and Yu in [14]. The set V is also tightly related to the set of residue field dominated types as defined by Ealy, Haskell and Maříková in [9].…”

“…We hope this can serve as a basis towards a model theory of adic spaces. In the same spirit, working in RCVF, there are spaces of definable types which can be seen as the model-theoretic counterpart of the analytification of semi-algebraic sets as recently defined by P. Jell, C. Scheiderer and J. Yu in [14]. This article aims to lay down a foundation for a model-theoretic study of such spaces.…”

Beautiful pairs

“…(1) When T Γ is the theory of trivial pairs of DOAG, F corresponds to the class of definable types which are orthogonal to Γ. The set F V (K), also denoted by " V (K), can be viewed as the model theoretic analogue of the Berkovich analytification of semi-algebraic sets as defined in [25].…”

“…On the one hand, various geometric spaces such as schemes of finite type, Berkovich (resp. Huber) analytifications of algebraic varieties or analytifications of semi-algebraic sets as defined in [25], can be (loosely speaking) represented by spaces of definable types within a particular theory T . This idea was remarkably exploited by E. Hrushovski and F. Loeser [23], who developed a modeltheoretic approach to non-archimedean semi-algebraic geometry, having striking consequences on the topology of the Berkovich analytification of algebraic varieties.…”

Tropical Bisectors and Voronoi Diagrams

“…More recently, however, tropical semi-algebraic sets and their intrinsic geometry came into the picture; see [1,15]. For instance, their metric properties appear in [2] as a tool to show that standard versions of the interior point method of linear programming exhibit an exponential complexity in the unit cost model.…”