Jell, Philipp and Scheiderer, Claus and Yu, Josephine (2022) Real Tropicalization and Analytification of Semialgebraic Sets. INTERNATIONAL MATHEMATICS RESEARCH NOTICES, 2022 (2). pp. 928-958. ISSN 1073-7928, 1687-0247
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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 of K-n we define a space S-r(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(an) can be canonically embedded into the real spectrum X-r of X, and we study its relation with the Berkovich analytification of X.
| Item Type: | Article |
|---|---|
| Subjects: | 500 Science > 510 Mathematics |
| Divisions: | Mathematics |
| Depositing User: | Dr. Gernot Deinzer |
| Date Deposited: | 15 Feb 2024 13:01 |
| Last Modified: | 15 Feb 2024 13:01 |
| URI: | https://pred.uni-regensburg.de/id/eprint/57668 |
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