Milnor and Tjurina numbers for a hypersurface germ with isolated singularity

Yongqiang Liu

Milnor and Tjurina numbers for a hypersurface germ with isolated singularity

Algebraic Topology 2018-07-04 v2

Authors: Yongqiang Liu

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 $\dfrac{\mu}{\tau} \leq n$ . As an application, we give a lower bound for the Tjurina number in terms of $n$ and the multiplicity of $f$ at the origin.

Keywords

hypersurface germ isolated singularity Milnor number Tjurina number analytic function germ multiplicity

Cite

@article{arxiv.1708.09716,
  title  = {Milnor and Tjurina numbers for a hypersurface germ with isolated singularity},
  author = {Yongqiang Liu},
  journal= {arXiv preprint arXiv:1708.09716},
  year   = {2018}
}

Comments

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

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