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NON-ARCHIMEDEAN YOMDIN–GROMOV PARAMETRIZATIONS AND POINTS OF BOUNDED HEIGHT

Part of: Model theory Arithmetic algebraic geometry Arithmetic problems. Diophantine geometry Diophantine equations

Published online by Cambridge University Press:  14 August 2015

RAF CLUCKERS ,
GEORGES COMTE  and
FRANÇOIS LOESER
RAF CLUCKERS
Affiliation:
Université Lille 1, Laboratoire Painlevé, CNRS - UMR 8524, Cité Scientifique, 59655 Villeneuve d’Ascq Cedex, France Katholieke Universiteit Leuven, Department of Mathematics, Celestijnenlaan 200B, B-3001 Leuven, Belgium; Raf.Cluckers@math.univ-lille1.fr
GEORGES COMTE
Affiliation:
Université Savoie Mont Blanc, LAMA, CNRS UMR 5127, F-73000 Chambéry, France; Georges.Comte@univ-savoie.fr
FRANÇOIS LOESER
Affiliation:
Sorbonne Universités, UPMC Univ Paris 06, UMR 7586 CNRS, Institut Mathématique de Jussieu, F-75005 Paris, France; Francois.Loeser@upmc.fr

Abstract

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We prove an analog of the Yomdin–Gromov lemma for $p$-adic definable sets and more broadly in a non-Archimedean definable context. This analog keeps track of piecewise approximation by Taylor polynomials, a nontrivial aspect in the totally disconnected case. We apply this result to bound the number of rational points of bounded height on the transcendental part of $p$-adic subanalytic sets, and to bound the dimension of the set of complex polynomials of bounded degree lying on an algebraic variety defined over $\mathbb{C}(\!(t)\!)$, in analogy to results by Pila and Wilkie, and by Bombieri and Pila, respectively. Along the way we prove, for definable functions in a general context of non-Archimedean geometry, that local Lipschitz continuity implies piecewise global Lipschitz continuity.

MSC classification

Primary: 03C98: Applications of model theory 11D88: $p$-adic and power series fields
Secondary: 11G50: Heights 14G05: Rational points
Type
Research Article
Information
Forum of Mathematics, Pi , Volume 3 , 2015 , e5
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2015

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