Non-Archimedean Yomdin–gromov Parametrizations and Points of Bounded Height

Cluckers, Raf; Comte, G.; Loeser, François

doi:10.1017/fmp.2015.4

Cited by 21 publications

(88 citation statements)

References 39 publications

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“…2.5.Some results about Lipschitz continuity. We recall now some results of[3] about reaching global Lipschitz continuity from local Lipschitz continuity.…”

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confidence: 99%

Motivic local density

Forey¹

2016

Math. Z.

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Abstract. We develop a theory of local densities and tangent cones in a motivic framework, extending work by Cluckers-Comte-Loeser about p-adic local density. We prove some results about geometry of definable sets in Henselian valued fields of characteristic zero, both in semi-algebraic and subanalytic languages, and study Lipschitz continuous maps between such sets. We prove existence of regular stratifications satisfying analogous of Verdier condition (w f ). Using Cluckers-Loeser theory of motivic integration, we define a notion of motivic local density with values in the Grothendieck ring of the theory of the residue sorts. We then prove the existence of a distinguished tangent cone and that one can compute the local density on this cone endowed with appropriate motivic multiplicities. As an application, we prove a uniformity theorem for p-adic local density.

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“…2.5.Some results about Lipschitz continuity. We recall now some results of[3] about reaching global Lipschitz continuity from local Lipschitz continuity.…”

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confidence: 99%

Motivic local density

Forey¹

2016

Math. Z.

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“…A related result on o-minimal stratifications was obtained in [29]. Very recently, a version of C k parametrization theorem was obtained in [22] for p-adic definable sets, and more broadly, in a non-archimedean, definable context. In particular, piecewise approximation by Taylor polynomials was extended in [22] to this setting.…”

Section: K -Parametrizationmentioning

confidence: 96%

“…Very recently, a version of C k parametrization theorem was obtained in [22] for p-adic definable sets, and more broadly, in a non-archimedean, definable context. In particular, piecewise approximation by Taylor polynomials was extended in [22] to this setting. This result was applied in [22] to bounding the number of rational points of a given height on the transcendental part of p-adic subanalytic sets.…”

Section: K -Parametrizationmentioning

confidence: 99%

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Smooth parametrizations in dynamics, analysis, diophantine and computational geometry

Yomdin

2015

Japan J. Indust. Appl. Math.

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Smooth parametrization consists in a subdivision of the mathematical objects under consideration into simple pieces, and then parametric representation of each piece, while keeping control of high order derivatives. The main goal of the present paper is to provide a short overview of some results and open problems on smooth parametrization and its applications in several apparently rather separated domains: smooth dynamics, diophantine geometry, approximation theory, and computational geometry. The structure of the results, open problems, and conjectures in each of these domains shows in many cases a remarkable similarity, which we try to stress. Sometimes this similarity can be easily explained, sometimes the reasons remain somewhat obscure, and it motivates some natural questions discussed in the paper. We present also some new results, stressing interconnection between various types and various applications of smooth parametrization.

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“…One may hope that such an approach could allow a more direct generalization to different algebraic contexts where the analytic notion of C rparametrization may be more difficult to recover. In particular we consider it an interesting direction to check whether the method developed in this paper can offer an alternative approach to the work of Cluckers, Comte and Loeser [4] on nonarchimedean analogs of the Pila-Wilkie theorem. We remark in this context that in our primary model-theoretic reference [5] the complex-analytic and p-adic contexts are treated in close analogy.…”

Section: 2mentioning

confidence: 99%