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Multiplicity, regularity and blow-spherical equivalence of real analytic sets

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This article is devoted to studying multiplicity and regularity of analytic sets. We present an equivalence for analytic sets, named blow-spherical equivalence, which generalizes differential equivalence and subanalytic bi-Lipschitz equivalence and, with this approach, we obtain several applications on analytic sets. On multiplicity, we present a generalization for Gau–Lipman’s Theorem about differential invariance of the multiplicity in the complex and real cases, and we show that the multiplicity \(\mathrm{mod}\,2\) is invariant under blow-spherical homeomorphisms in the case of real analytic curves and surfaces and also for a class of real analytic foliations and is invariant by (image) arc-analytic blow-spherical homeomorphisms in the case of real analytic hypersurfaces, generalizing some results proved by G. Valette. On regularity, we show that blow-spherical regularity of real analytic sets implies \(C^1\) smoothness only in the case of real analytic curves. We present also a complete classification of the germs of real analytic curves.

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Acknowledgements

The author wishes to thank Eurípedes C. da Silva for his interest in this research. The author also wishes to thank the anonymous referee for an accurate reading of the paper and essential suggestions and comments, which allowed to improve the exposition and added clarifications at several points.

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  1. Departamento de Matemática, Universidade Federal do Ceará, Rua Campus do Pici, s/n, Bloco 914, Pici, Fortaleza, CE, 60440-900, Brazil

    José Edson Sampaio

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  1. José Edson Sampaio
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Correspondence to José Edson Sampaio.

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The author was partially supported by CNPq-Brazil Grant 303811/2018-8.

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Sampaio, J.E. Multiplicity, regularity and blow-spherical equivalence of real analytic sets. Math. Z. 301, 385–410 (2022). https://doi.org/10.1007/s00209-021-02928-y

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