VC density of definable families over valued fields

نویسندگان

چکیده

We prove a tight bound on the number of realized 0=1 patterns (or equivalently Vapnik–Chervonenkis codensity) definable families in models theory algebraically closed valued fields with non-archimedean valuation. Our result improves best known this direction proved by Aschenbrenner, Dolich, Haskell, Macpherson and Starchenko, who weaker restricted case where characteristics field $K$ its residue are both assumed to be 0. The obtained here is optimal without any restriction characteristics. obtain aforementioned as consequence another bounding Betti numbers semi-algebraic subsets certain Berkovich analytic spaces, mirroring similar results already o-minimal structure for real well fields. latter first possibly independent interest. Its proof relies heavily recent Hrushovski Loeser topology spaces.

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ژورنال

عنوان ژورنال: Journal of the European Mathematical Society

سال: 2021

ISSN: ['1435-9855', '1435-9863']

DOI: https://doi.org/10.4171/jems/1056