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Uniform stable radius and Milnor number for non-degenerate isolated complete intersection singularities

Abstract

We prove that for two germs of analytic mappings \(f,g:({\mathbb {C}}^n,0) \rightarrow ({\mathbb {C}}^p,0)\) with the same Newton polyhedra which are (Khovanskii) non-degenerate and their zero sets are complete intersections with isolated singularity at the origin, there is a piecewise analytic family \(\{f_t\}\) of analytic maps with \(f_0=f, f_1=g\) which has a so-called uniform stable radius for the Milnor fibration. As a corollary, we show that their Milnor numbers are equal. Also, a formula for the Milnor number is given in terms of the Newton polyhedra of the component functions. This is a generalization of the result by C. Bivia-Ausina. Consequently, we obtain that the Milnor number of a non-degenerate isolated complete intersection singularity is an invariant of Newton boundaries.

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Acknowledgements

The author is supported by the International Centre for Research and Postgraduate Training in Mathematics – Institute of Mathematics – Vietnam Academy of Science and Technology under the Grant ICRTM01\(\_\)2020.06 and by the bilateral joint research project between Vietnam Academy of Science and Technology (VAST) and the Japan Society for the Promotion of Science (JSPS) under the Grant QTJP01.02/21-23. He also would like to thank anonymous referee(s) for the detail corrections and valuable comments that have helped him improve the quality and presentation of the paper.

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Correspondence to Tat Thang Nguyen.

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Nguyen, T.T. Uniform stable radius and Milnor number for non-degenerate isolated complete intersection singularities. manuscripta math. 168, 571–589 (2022). https://doi.org/10.1007/s00229-021-01323-5

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  • DOI: https://doi.org/10.1007/s00229-021-01323-5

Keywords

  • Isolated complete intersection singularity
  • Milnor fibration
  • Milnor number
  • Mixed newton number
  • Nondegenerate
  • Uniform stable radius

Mathematics Subject Classification

  • 14D05
  • 14D06
  • 14B05
  • 14B07