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Preprint Number 705

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705. R. Cluckers, G. Comte, F. Loeser
Non-archimedean Yomdin-Gromov parametrizations and points of bounded height

Submission date: 7 April 2014.


We prove an analogue of the Yomdin-Gromov Lemma for p-adic definable sets and more broadly in a non-archimedean, definable context. This analogue 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 C((t)), in analogy to results by Pila and Wilkie, resp. by Bombieri and Pila. 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.

Mathematics Subject Classification: 03C98, 11D88 (Primary) 03C65, 11G50, 14G05, 14G20 (Secondary)

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Full text arXiv 1404.1952: pdf, ps.

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