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


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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  1. Bivia-Ausina, C.: Mixed Newton numbers and isolated complete intersection singularities. Proc. Lond. Math. Soc. 94(3), 749–771 (2007)

    MathSciNet  Article  Google Scholar 

  2. J. Briançon. Le théorème de Kouchnirenko, unpublished lecture note

  3. Eyral, C.: Uniform stable radius, Le numbers and topological triviality for line singularities. Pacific J. Math. 291(2), 359–367 (2017)

    MathSciNet  Article  Google Scholar 

  4. Eyral, C., Oka, M.: Non-compact Newton boundary and Whitney equisingularity for non-isolated singularities. Adv. Math. 316, 94–113 (2017)

    MathSciNet  Article  Google Scholar 

  5. Hamm, H.A.: Lokale topologische Eigenschaften komplexer Raume. Math. Ann. 191, 235–252 (1971)

    MathSciNet  Article  Google Scholar 

  6. Hà, H.V., Phạm, T.S.: Invariance of the global monodromies in families of polynomials of two complex variables. Acta Math. Vietnam. 22(2), 515–526 (1997)

    MathSciNet  MATH  Google Scholar 

  7. Hà, H.V., Zaharia, A.: Families of polynomials with total Milnor number constant. Math. Ann. 313, 481–488 (1996)

    MathSciNet  MATH  Google Scholar 

  8. Huybrechts, D.: Complex Geometry: An Introduction. Springer, Berlin (2005)

    MATH  Google Scholar 

  9. K. Kaveh and A.G. Khovanskii.: On mixed multiplicities of ideals. (2013) arXiv:1310.7979

  10. Kerner, D., Nemethi, A.: Durfee-type bound for some non-degenerate complete intersection singularities. Math. Z. 285, 159–175 (2017)

    MathSciNet  Article  Google Scholar 

  11. Kouchnirenko, A.G.: Polyhedres de Newton et nombre de Milnor. Invent. Math. 32, 1–31 (1976)

    MathSciNet  Article  Google Scholar 

  12. Le, D.T., Ramanujam, C.P.: Invariance of Milnors number implies the invariance of topological type. Amer. J. Math. 98, 67–78 (1976)

    MathSciNet  Article  Google Scholar 

  13. Looijenga., E. J. N.:Isolated singular points on complete intersections, London Mathematical Society lecture note series 77. Cambridge University Press London-New York (1984)

  14. Milnor, J.: Singular points of complex hypersurfaces, Annals of Mathematics Studies 61. Princeton University Press (1968)

  15. Nguyen, T.T., Phạm, P.P., Phạm, T.S.: Bifurcation sets and global monodromies of Newton nondegenerate polynomials on algebraic sets. Publ. RIMS Kyoto Univ. 55, 1–24 (2019)

    MathSciNet  Article  Google Scholar 

  16. Nuno-Ballesteros, J.J., Orefice-Okamoto, B., Tomazella, J.N.: Equisingularity of families of isolated determinantal singularities. Math. Z. 289(3–4), 1409–1425 (2018)

    MathSciNet  Article  Google Scholar 

  17. Mutsuo, O. K. A.: Deformation of Milnor fiberings. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 20 (1973), 397–400. Correction in 27:2 (1980), 463–464

  18. Mutsuo, O.K.A.: On the bifurcation of the multiplicity and topology of the Newton boundary. J. Math. Soc. Japan 31, 435–450 (1979)

    MathSciNet  MATH  Google Scholar 

  19. Mutsuo, O.K.A.: Principal zeta-function of non-degenerate complete intersection singularity. J. Fac. Sci. Univ. Tokyo 37, 11–32 (1990)

    MathSciNet  MATH  Google Scholar 

  20. Mutsuo, O.K.A.: Non-Degenerate Complete Intersection Singularity. Actualit’es Math’ematiques Hermann, Paris (1997)

    MATH  Google Scholar 

  21. Phạm, T.S.: On the topology of the Newton boundary at infinity. J. Math. Soc. Japan 60(4), 1065–1081 (2008)

    MathSciNet  Article  Google Scholar 

  22. Phạm, T.S.: Invariance of the global monodromies in families of nondegenerate polynomials in two variables. Kodai Math. J. 33(2), 294–309 (2010)

    MathSciNet  MATH  Google Scholar 

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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).

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

Mathematics Subject Classification

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