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Smooth parametrizations in dynamics, analysis, diophantine and computational geometry

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Abstract

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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Acknowledgments

The author would like to thank D. Burguet, G. Comte, O. Friedland, Y. Ishii, G. Jones, G. Liao, P. Milman, B. Mourrain, J. Pila, R. Pierzchava, M. Thomas, A. Wilkie for useful discussions, and for explaining him some topics presented below. Special thanks belong to RIMS Institute, Kyoto, and to the organizers of the conference there in July 2013 on semi-algebraic techniques in Dynamics, which inspired a good part of this paper.

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  1. Department of Mathematics, The Weizmann Institute of Science, 76100, Rehovot, Israel

    Y. Yomdin

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Correspondence to Y. Yomdin.

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This research was supported by the ISF, Grant No. 779/13, and by the Yeda-Sela Foundation.

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Yomdin, Y. Smooth parametrizations in dynamics, analysis, diophantine and computational geometry. Japan J. Indust. Appl. Math. 32, 411–435 (2015). https://doi.org/10.1007/s13160-015-0176-6

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