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A characterization of semi-quasihomogeneous functions in terms of the Milnor number

Authors: Masako Furuya and Masataka Tomari
Translated by:
Journal: Proc. Amer. Math. Soc. 132 (2004), 1885-1890
MSC (2000): Primary 14B05; Secondary 13H15, 32S05
Published electronically: January 7, 2004
MathSciNet review: 2053957
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Abstract: We give an inequality on the Milnor number $\mu (f)$ of a hypersurface isolated singularity in terms of the weighted Taylor expansion $ f = f_{\rho }+ f_{\rho +1} + \cdots $ for a given weight on the coordinates. Here the equality holds if and only if the leading term $f_{\rho }$ defines an isolated singularity. This gives a characterization of the semi-quasihomogeneous condition in terms of $\mu $. Our proof uses a result on multiplicity of filtered rings and is given by purely algebraic arguments.

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Additional Information

Masako Furuya
Affiliation: 2-155, Makinohara, Matsudo-city, Chiba, 270-2267, Japan

Masataka Tomari
Affiliation: Department of Mathematics, College of Humanities and Sciences, Nihon University, Setagaya, Tokyo, 156-0045, Japan

Keywords: Milnor number, semi-quasihomogeneous function, multiplicity of local rings, filtered rings
Received by editor(s): June 28, 2001
Received by editor(s) in revised form: March 15, 2003
Published electronically: January 7, 2004
Communicated by: Michael Stillman
Article copyright: © Copyright 2004 American Mathematical Society

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