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A New Deterministic Method for Computing Milnor Number of an ICIS

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Computer Algebra in Scientific Computing (CASC 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12865))

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Abstract

The Milnor number of an isolated complete intersection singularity (ICIS) is considered in the context of symbolic computation. Based on the classical Lê-Greuel formula, a new method for computing Milnor numbers is introduced. Key ideas of our approach are the use of auxiliary indeterminates and the concept of local cohomology with coefficients in the field of rational functions of auxiliary indeterminates. The resulting algorithm is described and some examples are given for illustration.

This work has been partly supported by JSPS Grant-in-Aid for Scientific Research (C) (18K03320 and 18K03214).

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References

  1. Afzal, D., Afzal, F., Mulback, M., Pfister, G., Yaqub, A.: Unimodal ICIS, a classifier. Studia Scientiarum Math. Hungarica 54, 374–403 (2017)

    MATH  Google Scholar 

  2. Altman, A., Kleiman, S.: Introduction to Grothendieck Duality Theory. LNM, vol. 146. Springer, Heidelberg (1970). https://doi.org/10.1007/BFb0060932

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. Brieskorn, E.: Vue d’ensembre sur les problèmes de monodromie. Astérisque 7(8), 393–413 (1973)

    MathSciNet  MATH  Google Scholar 

  5. Callejas-Brdregal, R., Morgado, M.F.Z., Saia, M., Seade, J.: The Lê-Greuel formula for functions on analytic spaces. Tohoku Math. J. 68, 439–456 (2016)

    MathSciNet  MATH  Google Scholar 

  6. Carvalho, R.S., Oréfice-Okamoto, B., Tomazzela, J.N.: \( \mu \)-constant deformations of functions on an ICIS. J. Singul. 19, 163–176 (2019)

    MathSciNet  MATH  Google Scholar 

  7. Damon, J.N.: Topological invariants of \(\mu \)-constant deformations of complete intersection. Quart. J. Math. Oxford 40, 139–159 (1989)

    Article  MathSciNet  Google Scholar 

  8. Gaffney, T.: Polar multiplicities and equisingularity of map germs. Topology 32, 185–223 (1993)

    Article  MathSciNet  Google Scholar 

  9. Gaffney, T.: Multiplicities and equisingularity of ICIS germs. Invent. Math. 123, 209–220 (1996)

    Article  MathSciNet  Google Scholar 

  10. Giusti, M.: Classification des singularités isolées simples d’intersections complètes. In: Singularities Part I, Proceedings of Symposia in Pure Mathematics, vol. 40, pp. 457–494. AMS (1983)

    Google Scholar 

  11. Greuel, G.-M.: Der Gauss-Manin-Zusammenhang isolierter Singularitäten von vollständigen Durchschnitten. Math. Ann. 214, 235–266 (1973)

    Article  Google Scholar 

  12. Greuel, G.-M., Hamm, H.A.: Invarianten quasihomogener vollständiger Durchschnitte. Invent. Math. 49, 67–86 (1978)

    Article  MathSciNet  Google Scholar 

  13. Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley, Hoboken (1976)

    MATH  Google Scholar 

  14. Grothendieck, A.: Théorèmes de dualité pour les faisceaux algébriques cohérents. Séminaire Bourbaki 149, 169–193 (1957)

    Google Scholar 

  15. Hartshorne, R.: Local Cohomology. LNM, vol. 41. Springer, Heidelberg (1967). https://doi.org/10.1007/BFb0073971

    Book  Google Scholar 

  16. Hamm, H.: Lokale topologische Eigenschaften komplexer Räume. Math. Ann. 191, 235–252 (1971)

    Article  MathSciNet  Google Scholar 

  17. Kunz, E.: Residues and Duality for Projective Algebraic Varieties. American Mathematical Society, Providence (2009)

    Google Scholar 

  18. Lê Dũng Tráng: Calcule du nobmre de cycles évanouissants d’une hypersurface complexe. Ann. Inst. Fourier 23, 261–270 (1973)

    Article  Google Scholar 

  19. Lê Dũng Tráng: Calculation of Milnor number of isolated singularity of complete intersection (in Russian). Funktsional. Analiz i ego Prilozhen. 8, 45–49 (1974)

    MATH  Google Scholar 

  20. Looijenga, E.J.N.: Isolated Singular Points on Complete Intersections. London Mathematical Society Lecture Note Series, Cambridge, vol. 77 (1984)

    Google Scholar 

  21. Martin, B., Pfister, G.: Milnor number of complete intersections and Newton polygons. Math. Nachr. 110, 159–177 (1983)

    Article  MathSciNet  Google Scholar 

  22. Milnor, J.: Singular points of complex hypersurfaces. Ann. Math. Stud. 61, 591–648 (1968)

    MathSciNet  MATH  Google Scholar 

  23. Nabeshima, K., Tajima, S.: Algebraic local cohomology with parameters and parametric standard bases for zero-dimensional ideals. J. Symb. Comput. 82, 91–122 (2017)

    Article  MathSciNet  Google Scholar 

  24. Nabeshima, K., Tajima, S.: A new method for computing limiting tangent spaces of isolated hypersurface singularity via algebraic local sohomology. In: Advanced Studies in Pure Mathematics, vol. 78, pp. 331–344 (2018)

    Google Scholar 

  25. Nabeshima, K., Tajima, S.: Computing logarithmic vector fields and Bruce-Roberts Milnor numbers via local cohomology classes. Rev. Roumaine Math. Pures Appl. 64, 521–538 (2019)

    MathSciNet  MATH  Google Scholar 

  26. Nabeshima, K., Tajima, S.: Alternative algorithms for computing generic \(\mu ^{\ast }\)-sequences and local Euler obstructions of isolated hypersurface singularities. J. Algebra Appl. 18(8) (2019). 1959156 (13pages)

    Google Scholar 

  27. Nabeshima, K., Tajima, S.: Testing zero-dimensionality of varieties at a point. Math. Comput. Sci. 15, 317–331 (2021)

    Article  MathSciNet  Google Scholar 

  28. Nguyen, T.T.: Uniform stable radius and Milnor number for non-degenerate isolated complete intersection singularities. arXiv:1912.10655v2 (2019)

  29. Noro, M., Takeshima, T.: Risa/Asir- a computer algebra system. In: Proceedings of International Symposium on Symbolic and Algebraic Computation (ISSAC), pp. 387–396. ACM (1992)

    Google Scholar 

  30. Oka, M.: Principal zeta-function of non-degenerate complete intersection singularity. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 37, 11–32 (1990)

    MathSciNet  MATH  Google Scholar 

  31. Palamodov, V.P.: Multiplicity of holomorphic mappings. Funktsional. Analiz i ego Prilozhen. 1, 54–65 (1967)

    Google Scholar 

  32. Saito, K.: Calcule algébrique de la monodromie. Astérisque 7(8), 195–211 (1973)

    MATH  Google Scholar 

  33. Tajima, S.: On polar varieties, logarithmic vector fields and holonomic D-modules. RIMS Kôkyûroku Bessatsu 40, 41–51 (2013)

    MathSciNet  MATH  Google Scholar 

  34. Tajima, S., Nakamura, Y., Nabeshima, K.: Standard bases and algebraic local cohomology for zero dimensional ideals. In: Advanced Studies in Pure Mathematics, vol. 56, pp. 341–361 (2009)

    Google Scholar 

  35. Tajima, S., Shibuta, T., Nabeshima, K.: Computing logarithmic vector fields along an ICIS germ via Matlis duality. In: Boulier, F., England, M., Sadykov, T.M., Vorozhtsov, E.V. (eds.) CASC 2020. LNCS, vol. 12291, pp. 543–562. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-60026-6_32

    Chapter  Google Scholar 

  36. Teissier, B.: Cycles évanescents, sections planes et conditions de Whitney, Singularités, à Cargèse. Astérisque 7(8), 285–362 (1973)

    MATH  Google Scholar 

  37. Teissier, B.: Variétés polaires. I. Inventiones Mathematicae, vol. 40, pp. 267–292 (1977)

    Google Scholar 

  38. Teissier, B.: Varietes polaires II Multiplicites polaires, sections planes, et conditions de whitney. In: Aroca, J.M., Buchweitz, R., Giusti, M., Merle, M. (eds.) Algebraic Geometry. LNM, vol. 961, pp. 314–491. Springer, Heidelberg (1982). https://doi.org/10.1007/BFb0071291

    Chapter  Google Scholar 

  39. Wall, C. T. C.: Classification of unimodal isolated singularities complete intersections. In: Proceedings of Symposia in Pure Mathematics, vol. 40, Part II, pp. 625–640 (1983)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Graduate School of Science and Technology, Niigata University, Ikarashi 2-no-cho, 8050, Nishi-ku Niigata, Japan

    Shinichi Tajima

  2. Department of Applied Mathematics, Tokyo University of Science, 1-3, Kagurazaka, Tokyo, Japan

    Katsusuke Nabeshima

Authors
  1. Shinichi Tajima
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  2. Katsusuke Nabeshima
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Correspondence to Katsusuke Nabeshima .

Editor information

Editors and Affiliations

  1. Université de Lille, Villeneuve d’Ascq, France

    François Boulier

  2. Coventry University, Coventry, UK

    Matthew England

  3. Plekhanov Russian University of Economics, Moscow, Russia

    Timur M. Sadykov

  4. Institute of Theoretical and Applied Mechanics, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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Tajima, S., Nabeshima, K. (2021). A New Deterministic Method for Computing Milnor Number of an ICIS. In: Boulier, F., England, M., Sadykov, T.M., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2021. Lecture Notes in Computer Science(), vol 12865. Springer, Cham. https://doi.org/10.1007/978-3-030-85165-1_22

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  • DOI: https://doi.org/10.1007/978-3-030-85165-1_22

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  • Publisher Name: Springer, Cham

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  • Online ISBN: 978-3-030-85165-1

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