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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
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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  • 1. D. Alekseevsky, A. Kriegl, P. W. Michor, and M. Losil, Choosing roots of polynomials smoothly, Israel Journal of Mathematics, 105 (1998), 203-233. MR 2000c:58017
  • 2. E. Bierstone and P. D. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math., 67 (1988), 5-42. MR 89k:32011
  • 3. E. Bierstone and P. D. Milman, Arc-analytic functions, Invent. Math., 101 (1990), 411-424. MR 92a:32011
  • 4. E. Bierstone, P. D. Milman, and A. Parusinski, A function which is arc-analytic but not continuous, Proc. Amer. Math. Soc., 113 (1991), 419-423. MR 91m:32008
  • 5. J. Dieudonné, Sur un théorème de Glaeser, J. Analyse Math., Vol. 23 (1970), pp. 85-88. MR 42:4678
  • 6. T. Fukui, Seeking invariants for blow-analytic equivalence, Compositio Math., 105 (1997), 95-108. MR 98c:32009
  • 7. T. Fukui, S. Koike and T.-C. Kuo, Blow-analytic equisingularities, properties, problems and progress, in ``Real analytic and algebraic singularities'', Pitman Research Notes in Mathematics Series, 381, Longman, Harlow, 1998, pp. 8-29. MR 99a:32051
  • 8. G. Glaeser, Racine carrée d'une fonction différentiable, Annales de l'Institut Fourier (Grenoble), Vol. 13 (1963), fasc. 2, pp. 203-210. MR 29:1294
  • 9. S. Izumi, S. Koike and T.-C. Kuo, Computations and Stability of the Fukui Invariant, Compositio Mathematica 130(1) (2002), 49-73. MR 2003a:32050
  • 10. T.-C. Kuo, On classification of real singularities, Invent. Math., 82 (1985), 257-262. MR 87d:58025
  • 11. T.-C. Kuo and Y.C. Lu, On analytic function germs of two complex variables, Topology, 16 (1977), 299-310. MR 57:704
  • 12. T.-C. Kuo and A. Parusinski, Newton Polygon Relative to an Arc, in Real and Complex Singularities (São Carlos, 1998), Chapman & Hall Res. Notes Math., 412, Boca Raton, FL, 2000, pp. 76-93. MR 2000j:32043
  • 13. K. Kurdyka, Ensembles semi-algébriques symétriques par arcs, Math. Ann., 282 (1988), 445-462. MR 89j:14015
  • 14. K. Kurdyka, A counterexample to subanalyticty of an arc-analytic function, Ann. Polon. Math. 55 (1991), 241-243. MR 92j:32019
  • 15. K. Kurdyka, An arc-analytic function with nondiscrete singular set, Ann. Polon. Math. 59, 1 (1994), 251-254. MR 95g:32009
  • 16. K. Kurdyka and L. Paunescu, Arc-analyticity is an open property for subanalytic functions, preprint, Univ.-Savoie, 2002.
  • 17. S. \Lojasiewicz, Ensembles semi-analytiques, preprint, Inst. Hautes Études Sci., 1965.
  • 18. A. Parusinski, Subanalytic functions, Trans. Amer. Math. Soc. 344, 2 (1994), 583-595. MR 94k:32006
  • 19. L. Paunescu, An example of blow-analytic homeomorphism in ``Real analytic and algebraic singularities'', Pitman Research Notes in Mathematics Series, 381, Longman, Harlow, 1998, pp. 62-63. MR 98i:32001

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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

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

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