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Non-degenerate jumps of Milnor numbers of quasihomogeneous singularities

Volume 123 / 2019

Tadeusz Krasiński, Justyna Walewska Annales Polonici Mathematici 123 (2019), 369-386 MSC: Primary 32S05; Secondary 32S30. DOI: 10.4064/ap180914-15-4 Published online: 23 September 2019

Abstract

Let $f_0: (\mathbb {C}^n,0)\rightarrow (\mathbb {C},0)$ be a holomorphic function germ having an isolated critical point at $0\in \mathbb {C}^n$ and let $[f_0]$ be the singularity generated by $f_0$, i.e. the equivalence class of $f_0$ with respect to right-left holomorphic equivalence. The non-degenerate jump of Milnor number $\lambda ^{ {\rm nd}}(f_0)$ of $f_0$ is the minimal non-zero difference between the Milnor number of $f_0$ and the Milnor number of a generic element of $(f_t)$ among all holomorphic non-degenerate deformations $(f_t)$ of $f_0$. For the class $[f_0]$ we define $\lambda ^{ {\rm nd}}([f_0])$ as the minimum of $\lambda ^{ {\rm nd}}(g_0)$ over $g_0\in [f_0]$. We give a formula for $\lambda ^{ {\rm nd}}([f_0])$ when $f_0$ is quasihomogeneous in two variables.

Authors

  • Tadeusz KrasińskiFaculty of Mathematics and Computer Science
    University of Łódź
    Banacha 22
    90-238 Łódź, Poland
    e-mail
  • Justyna WalewskaFaculty of Mathematics and Computer Science
    University of Łódź
    Banacha 22
    90-238 Łódź, Poland
    e-mail

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