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Published online by Cambridge University Press:  14 August 2015

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;
Université Savoie Mont Blanc, LAMA, CNRS UMR 5127, F-73000 Chambéry, France;
Sorbonne Universités, UPMC Univ Paris 06, UMR 7586 CNRS, Institut Mathématique de Jussieu, F-75005 Paris, France;


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.

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