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

Furuya, Masako; Tomari, Masataka

doi:10.1090/S0002-9939-04-07383-6

A characterization of semi-quasihomogeneous functions in terms of the Milnor number
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by Masako Furuya and Masataka Tomari PDF

Proc. Amer. Math. Soc. 132 (2004), 1885-1890 Request permission

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.

References

V. I. Arnol′d, Normal forms of functions in the neighborhood of degenerate critical points, Uspehi Mat. Nauk 29 (1974), no. 2(176), 11–49 (Russian). Collection of articles dedicated to the memory of Ivan Georgievič Petrovskiĭ (1901–1973), I. MR 0516034
Furuya, M.: Lower bound of Milnor number, preprint. (An early version of this is found in AG/9901107v2.)
A. G. Kouchnirenko, Polyèdres de Newton et nombres de Milnor, Invent. Math. 32 (1976), no. 1, 1–31 (French). MR 419433, DOI 10.1007/BF01389769
Lê Dũng Tráng and C. P. Ramanujam, The invariance of Milnor’s number implies the invariance of the topological type, Amer. J. Math. 98 (1976), no. 1, 67–78. MR 399088, DOI 10.2307/2373614
Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
John Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. MR 0239612
John Milnor and Peter Orlik, Isolated singularities defined by weighted homogeneous polynomials, Topology 9 (1970), 385–393. MR 293680, DOI 10.1016/0040-9383(70)90061-3
D. Rees, ${\mathfrak {a}}$-transforms of local rings and a theorem on multiplicities of ideals, Proc. Cambridge Philos. Soc. 57 (1961), 8–17. MR 118750, DOI 10.1017/s0305004100034800
Masataka Tomari, Multiplicity of filtered rings and simple $K3$ singularities of multiplicity two, Publ. Res. Inst. Math. Sci. 38 (2002), no. 4, 693–724. MR 1917161, DOI 10.2977/prims/1145476194
Masataka Tomari and Keiichi Watanabe, Filtered rings, filtered blowing-ups and normal two-dimensional singularities with “star-shaped” resolution, Publ. Res. Inst. Math. Sci. 25 (1989), no. 5, 681–740. MR 1031224, DOI 10.2977/prims/1195172704