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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Arc-analytic roots of analytic functions are Lipschitz
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by Krzysztof Kurdyka and Laurentiu Paunescu PDF
Proc. Amer. Math. Soc. 132 (2004), 1693-1702 Request permission


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:
  • Laurentiu Paunescu
  • Affiliation: School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
  • Email:
  • 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
  • © Copyright 2004 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 132 (2004), 1693-1702
  • MSC (2000): Primary 32B20, 14P20
  • DOI:
  • MathSciNet review: 2051130