On the non-analyticity locus of an arc-analytic function
Authors:
Krzysztof Kurdyka and Adam Parusiński
Journal:
J. Algebraic Geom. 21 (2012), 61-75
DOI:
https://doi.org/10.1090/S1056-3911-2011-00553-5
Published electronically:
March 1, 2011
MathSciNet review:
2846679
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Abstract |
References |
Additional Information
Abstract:
Let $X$ be a real analytic manifold. A function $f:X\to \mathbb {R}$ is called arc-analytic if it is real analytic on each real analytic arc. In real analytic geometry there are many examples of arc-analytic functions that are not real analytic. They appear while studying the arc-symmetric sets and the blow-analytic equivalence.
In this paper we show that the non-analyticity locus of an arc-analytic function is arc-symmetric. We also discuss the behavior of the non-analyticity locus under blowings-up. By a result of Bierstone and Milman, an arc-analytic function $f:X\to \mathbb {R}$ that satisfies a polynomial equation with real analytic coefficients, can be made analytic, over any relatively compact subset of $X$, by a sequence of blowings-up with smooth centers. We show that these centers can be chosen, at each stage of the resolution, inside the non-analyticity loci.
References
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References
- E. Bierstone and P. D. Milman, Semianalytic and Subanalytic sets, Publ. I.H.E.S. 67 (1988), 5–42. MR 972342 (89k:32011)
- E. Bierstone and P. D. Milman, Arc-analytic functions, Invent. Math. 101 (1990), 411–424. MR 1062969 (92a:32011)
- E. Bierstone and P. D. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 (1997), 207–302. MR 1440306 (98e:14010)
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- J. Bochnak and J. Siciak, Analytic functions in topological vector spaces, Studia Math., XXXIX (1971), 77–112. MR 0313811 (47:2365)
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- H. Hironaka, Subanalytic sets in Number Theory, Algebraic Geometry and Commutative Algebra (Kinokuniya, Tokyo), 1973, volume in honor of Yasuo Akizuki, 453–493. MR 0377101 (51:13275)
- T.-C. Kuo, On classification of real singularities, Invent. math. 82 (1985), 257–262. MR 809714 (87d:58025)
- K. Kurdyka, Points réguliers d’un ensemble sous-analytique, Ann. Inst. Fourier, Grenoble 38 (1988), 133–156. MR 949002 (89g:32010)
- K. Kurdyka, Ensembles semi-algébriques symétriques par arcs, Math. Ann. 281, no. 3 (1988), 445–462. MR 967023 (89j:14015)
- K. Kurdyka, S. Łojasiewicz, and M. Zurro, Stratifications distinguées comme un util en géométrie semi-analytique, Manuscripta Math. 86, 81–102 (1995). MR 1314150 (96a:32013)
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- L. Paunescu, Implicit function theorem for locally blow-analytic functions, Ann. Inst. Fourier, Grenoble 51 (2001), 1089–1010. MR 1849216 (2003d:58062)
- M. Tamm, Subanalytic sets in the calculus of variation, Acta Math. 146 (1981), no. 3-4, 167–199. MR 611382 (82h:32012)
- J. Włodarczyk, Simple Hironaka resolution in characteristic zero, J. Amer. Math. Soc. 18 (2005), no. 4, 779–822. MR 2163383 (2006f:14014)
Additional Information
Krzysztof Kurdyka
Affiliation:
Laboratoire de Mathématiques, UMR 5175 du CNRS, Université de Savoie, Campus Scientifique, 73 376 Le Bourget–du–Lac Cedex, France
Email:
kurdyka@univ-savoie.fr
Adam Parusiński
Affiliation:
Laboratoire J.-A. Dieudonné UMR 6621 du CNRS, Université de Nice - Sophia Antipolis Parc Valrose 06108 Nice Cedex 02 France
Email:
Adam.PARUSINSKI@unice.fr
Received by editor(s):
April 1, 2009
Received by editor(s) in revised form:
November 6, 2009
Published electronically:
March 1, 2011
Additional Notes:
This research was done at the Mathematisches Forschungsinstitut Oberwolfach during a stay within the Research in Pairs Programme from January 20 to February 2, 2008. We would like to thank the MFO for excellent working conditions.