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Computations and Stability of the Fukui Invariant

Published online by Cambridge University Press:  04 December 2007

Shuzo Izumi ,
Satoshi Koike  and
Tzee-Char Kuo
Shuzo Izumi
Affiliation:
Department of Mathematics and Physics, Faculty of Science and Engeneering, Kinki University, Higashi-Osaka 577-8502, Japan. E-mail: izumi@math.kindai.ac.jp
Satoshi Koike
Affiliation:
Department of Mathematics, Hyogo University of Teacher Education, Yashiro, Kato, Hyogo 673-1405, Japan. E-mail: koike@sci.hyogo-u.ac.jp
Tzee-Char Kuo
Affiliation:
School of Mathematics and Statistics, University of Sydney, Sydney, NSW, 2006, Australia. E-mail: tck@maths.usyd.edu.au
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Abstract

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T. Fukui introduced an invariant for the blow-analytic equivalence of real singularities. For a nondegenerate analytic function (germ) f, he discovered a formula for computing the one-dimensional invariant, denoted by A(f) := A1(f). We find a formula for A(f) for any f (real or complex, degenerate or not). We then define, and characterise, various notions of stability of A(f), using the formula. For real analytic f, the Fukui invariant with sign is defined, and computed by a similar formula. In the case where f is an analytic function of two complex variables, A(f) can also be computed using the tree-model of f.

Keywords

Fukui invariantblow-analyticitysimplificationtree-model

Information

Type
Research Article
Information
Compositio Mathematica , Volume 130 , Issue 1 , January 2002 , pp. 49 - 74
Copyright
© 2002 Kluwer Academic Publishers