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

TL;DR

This paper introduces blow-spherical equivalence for real analytic sets, generalizing existing notions, and explores its implications for regularity and multiplicity invariance, including classifications of real analytic curve germs.

Contribution

It defines blow-spherical equivalence, extends multiplicity invariance results, and classifies germs of real analytic curves, advancing understanding of real analytic set regularity and invariants.

Findings

Blow-spherical regularity implies $C^1$ smoothness for real analytic curves.

Multiplicity mod 2 is invariant under blow-spherical homeomorphisms for curves and surfaces.

Complete classification of germs of real analytic curves achieved.

Abstract

This article is devoted to studying multiplicity and regularity of real analytic sets. We present an equivalence for real 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 regularity, we show that blow-spherical regularity of real analytic implies $C^1$ smoothness only in the case of real analytic curves. On multiplicity, we present a generalization for Gau-Lipman's Theorem about differential invariance of the multiplicity in the complex and real cases, we show that the multiplicity ${\rm mod}\,2$ is invariant by 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. We present also a complete classification of the germs of real analytic curves.