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Polynomial approximation of Berkovich spaces and definable types

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Abstract

We investigate affine Berkovich spaces over maximally complete fields and prove that they may be approximated by simpler spaces when the only functions we need to evaluate are polynomials with bounded degrees. We derive applications to semi-algebraic sets and recover a result of E. Hrushovski and F. Loeser claiming that points of Berkovich spaces give rise to definable types (a model-theoretic notion of tameness).

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Notes

  1. It is unfortunate that two notions of type appear in this paper. The model-theoretic notion is not to be confused with the one used by V. Berkovich to classify the points of the line.

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Acknowledgments

In 2010, I was given the opportunity to attend the summer conference of the MRC program on model theory of fields in Snowbird Resort, Utah. I would like to thank all the participants who explained the basics of model theory to me, with much patience and insight. I would also like to express my sincere gratitude to the referee for his precise and numerous comments and suggestions, which contributed to improve the paper.

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  1. Institut de Recherche Mathématique Avancée, 7, rue René Descartes, 67084 , Strasbourg, France

    Jérôme Poineau

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  1. Jérôme Poineau
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Correspondence to Jérôme Poineau.

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The research for this article was partially supported by the ANR projects BERKO: ANR 07-JCJC-0004-CSD5 and GLOBES: ANR-12-JS01-0007-01.

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Poineau, J. Polynomial approximation of Berkovich spaces and definable types. Math. Ann. 358, 949–970 (2014). https://doi.org/10.1007/s00208-013-0979-1

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  • DOI: https://doi.org/10.1007/s00208-013-0979-1

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