Blow-analytic equivalence of two variable real analytic function germs

Koike, Satoshi; Parusiński, Adam

doi:10.1090/S1056-3911-09-00527-X

Authors: Satoshi Koike and Adam Parusiński
Journal: J. Algebraic Geom. 19 (2010), 439-472
DOI: https://doi.org/10.1090/S1056-3911-09-00527-X
Published electronically: December 1, 2009
MathSciNet review: 2629597
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Abstract | References | Additional Information

Abstract:

Blow-analytic equivalence is a notion for real analytic function germs, introduced by Tzee-Char Kuo in order to develop real analytic equisingularity theory. In this paper we give complete characterisations of blow-analytic equivalence in the two dimensional case, in terms of the real tree model for the arrangement of real parts of Newton-Puiseux roots and their Puiseux pairs, and in terms of minimal resolutions. These characterisations show that in the two dimensional case the blow-analytic equivalence is a natural analogue of topological equivalence of complex analytic function germs. Moreover, we show that in the two-dimensional case the blow-analytic equivalence can be made cascade, and hence satisfies several geometric properties. It preserves, for instance, the contact order of real analytic arcs.

In the general $n$-dimensional case, we show that a singular real modification satisfies the arc-lifting property.

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Additional Information

Satoshi Koike
Affiliation: Department of Mathematics, Hyogo University of Teacher Education, 942-1 Shimokume, Kato, Hyogo 673-1494, Japan
Email: koike@hyogo-u.ac.jp

Adam Parusiński
Affiliation: Laboratoire Angevin de Recherche en Mathématiques, UMR 6093 du CNRS, Université d’Angers, 2, bd Lavoisier, 49045 Angers cedex, France
Address at time of publication: Laboratoire J. A. Dieudonné U.M.R. C.N.R.S. N 6621, Université de Nice Sophia-Antipolis, Parc Valrose 06108 Nice Cedex 02, France
Email: adam.parusinski@unice.fr

Received by editor(s): April 1, 2008
Received by editor(s) in revised form: December 19, 2008
Published electronically: December 1, 2009
Additional Notes: This research was partially supported by the Grant-in-Aid for Scientific Research (No. 18540084) of the Ministry of Education, Science and Culture of Japan.