Milnor and Tjurina numbers for a hypersurface germ with isolated singularity

Abstract

Assume that $f:({\mathbb{C}}^n,0)\to (\mathbb{C},0)$ is an analytic function germ at the origin with only isolated singularity. Let $\mu$ and $\tau$ be the corresponding Milnor and Tjurina numbers. We show that $\frac{\mu }{\tau }\le n$. As an application, we give a lower bound for the Tjurina number in terms of $n$ and the multiplicity of $f$ at the origin.

5 pages. A remark is added to explain the recent result for isolated plane curve case due to A. Dimca and G.-M. Greuel. Some typos fixed

References in corpus (1)

• Milnor numbers of projective hypersurfaces with isolated singularities

Cited by in corpus (1)

• A Note on 1-Forms for Reduced Curve Singularities