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Arc-analytic roots of analytic functions are Lipschitz


Authors: Krzysztof Kurdyka and Laurentiu Paunescu
Translated by:
Journal: Proc. Amer. Math. Soc. 132 (2004), 1693-1702
MSC (2000): Primary 32B20, 14P20
DOI: https://doi.org/10.1090/S0002-9939-04-07323-X
Published electronically: January 27, 2004
MathSciNet review: 2051130
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Abstract: Let $g$ be an arc-analytic function (i.e., analytic on every analytic arc) and assume that for some integer $r$ the function $g^r$ is real analytic. We prove that $g$ is locally Lipschitz; even $C^1$if $r$ is less than the multiplicity of $g^r$. We show that the result fails if $g^r$ is only a $C^k$, arc-analytic function (even blow-analytic), $k\in {\mathbb N}$. We also give an example of a non-Lipschitz arc-analytic solution of a polynomial equation $P(x,y)= y^d +\sum_{i=1}^{d}a_i(x)y^{d-i}$, where $a_i$ are real analytic functions.


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Additional Information

Krzysztof Kurdyka
Affiliation: Laboratoire de Mathématiques (LAMA), Université de Savoie, UMR 5127 CNRS, 73-376 Le Bourget-du-Lac cedex, France
Email: Krzysztof.Kurdyka@univ-savoie.fr

Laurentiu Paunescu
Affiliation: School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
Email: laurent@maths.usyd.edu.au

DOI: https://doi.org/10.1090/S0002-9939-04-07323-X
Keywords: Real analytic, subanalytic, arc-analytic, Lipschitz
Received by editor(s): November 15, 2002
Published electronically: January 27, 2004
Additional Notes: The second author thanks Université de Savoie and CNRS for support.
Communicated by: Jozef Dodziuk
Article copyright: © Copyright 2004 American Mathematical Society

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