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Existence of Moduli for Bi-Lipschitz Equivalence of Analytic Functions

Published online by Cambridge University Press:  04 December 2007

Jean-Pierre Henry  and
Adam Parusiński
Jean-Pierre Henry
Affiliation:
Centre de Mathématiques, (Unité associé au CNRS No169), Ecole Polytechnique, F-91128 Palaiseau Cedex, France. e-mail: henry@cmat.polytechnique.fr
Adam Parusiński
Affiliation:
Département de Mathématiques, Université d'Angers, 2, bd Lavoisier, 49045 Angers Cedex 1, France. e-mail: parus@tonton.univ-angers.fr
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Abstract

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We show that the bi-Lipschitz equivalence of analytic function germs (${\open C}^{2}$, 0)→(${\open C}$, 0) admits continuous moduli. More precisely, we propose an invariant of the bi-Lipschitz equivalence of such germs that varies continuously in many analytic families ft: (${\open C}^{2}$, 0)→(${\open C}$, 0). For a single germ f the invariant of f is given in terms of the leading coefficients of the asymptotic expansions of f along the branches of generic polar curve of f.

Keywords

analytic function germsbi-Lipschitz equivalencemodulipolar curveNewton polygon
Type
Research Article
Information
Compositio Mathematica , Volume 136 , Issue 2 , April 2003 , pp. 217 - 235
Copyright
© 2003 Kluwer Academic Publishers