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

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...

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Bibliographic Details
Published in:Mathematische Zeitschrift 2022-05, Vol.301 (1), p.385-410
Main Author: Sampaio, José Edson
Format: Article
Language:English
Subjects:
Equivalence
Hyperspaces
Invariants
Mathematical analysis
Mathematics
Mathematics and Statistics
Regularity
Smoothness
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Summary: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 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.
ISSN:0025-5874
1432-1823
DOI:10.1007/s00209-021-02928-y