Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
Published electronically: January 27, 2004
MathSciNet review: 2051130
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 32B20, 14P20

Retrieve articles in all journals with MSC (2000): 32B20, 14P20

Additional Information

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

Laurentiu Paunescu
Affiliation: School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia

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