On the non-analyticity locus of an arc-analytic function

Kurdyka, Krzysztof; Parusiński, Adam

doi:10.1090/S1056-3911-2011-00553-5

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.

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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.