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A DEFINABLE $p$-ADIC ANALOGUE OF KIRSZBRAUN’S THEOREM ON EXTENSIONS OF LIPSCHITZ MAPS

Part of: Algebraic number theory: local and $p$-adic fields Model theory Topological fields

Published online by Cambridge University Press:  20 October 2015

Raf Cluckers  and
Florent Martin
Raf Cluckers
Affiliation:
Université de Lille, Laboratoire Painlevé, CNRS - UMR 8524, Cité Scientifique, 59655 Villeneuve d’Ascq Cedex, France KU Leuven, Department of Mathematics, Celestijnenlaan 200B, B-3001 Leuven, Belgium (Raf.Cluckers@math.univ-lille1.fr) URL http://math.univ-lille1.fr/∼cluckers
Florent Martin
Affiliation:
Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany (florent.martin@mathematik.uni-regensburg.de) URL http://homepages.uni-regensburg.de/∼maf55605/
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Abstract

A direct application of Zorn’s lemma gives that every Lipschitz map $f:X\subset \mathbb{Q}_{p}^{n}\rightarrow \mathbb{Q}_{p}^{\ell }$ has an extension to a Lipschitz map $\widetilde{f}:\mathbb{Q}_{p}^{n}\rightarrow \mathbb{Q}_{p}^{\ell }$. This is analogous to, but easier than, Kirszbraun’s theorem about the existence of Lipschitz extensions of Lipschitz maps $S\subset \mathbb{R}^{n}\rightarrow \mathbb{R}^{\ell }$. Recently, Fischer and Aschenbrenner obtained a definable version of Kirszbraun’s theorem. In this paper, we prove in the $p$-adic context that $\widetilde{f}$ can be taken definable when $f$ is definable, where definable means semi-algebraic or subanalytic (or some intermediary notion). We proceed by proving the existence of definable Lipschitz retractions of $\mathbb{Q}_{p}^{n}$ to the topological closure of $X$ when $X$ is definable.

Keywords

p-adic semi-algebraic functionsp-adic subanalytic functionsLipschitz continuous functionsp-adic cell decompositiondefinable retractions

MSC classification

Primary: 03C98: Applications of model theory 12J25: Non-Archimedean valued fields
Secondary: 03C60: Model-theoretic algebra 11S80: Other analytic theory (analogues of beta and gamma functions, $p$-adic integration, etc.)
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 17 , Issue 1 , February 2018 , pp. 39 - 57
Copyright
© Cambridge University Press 2015 

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