Real Tropicalization and Analytification of Semialgebraic Sets

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JELL, Philipp, Claus SCHEIDERER, Josephine YU, 2022. Real Tropicalization and Analytification of Semialgebraic Sets. In: International Mathematics Research Notices (IMRN). Oxford University Press (OUP). 2022(2), pp. 928-958. ISSN 1073-7928. eISSN 1687-0247. Available under: doi: 10.1093/imrn/rnaa112

@article{Jell2022Tropi-53167, title={Real Tropicalization and Analytification of Semialgebraic Sets}, year={2022}, doi={10.1093/imrn/rnaa112}, number={2}, volume={2022}, issn={1073-7928}, journal={International Mathematics Research Notices (IMRN)}, pages={928--958}, author={Jell, Philipp and Scheiderer, Claus and Yu, Josephine} }

eng Scheiderer, Claus terms-of-use Yu, Josephine Jell, Philipp Real Tropicalization and Analytification of Semialgebraic Sets 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<sup>n</sup> under this map from a general point of view. For a semialgebraic set S⊆K<sup>n</sup> we define a space S<sup>an</sup><sub>r</sub> 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<sup>an</sup><sub>r</sub> can be canonically embedded into the real spectrum Xr of X⁠, and we study its relation with the Berkovich analytification of X⁠. Yu, Josephine Scheiderer, Claus Jell, Philipp 2021-03-17T12:28:59Z 2022 2021-03-17T12:28:59Z

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