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Acta Mathematica Vietnamica

, Volume 42, Issue 2, pp 279–288 | Cite as

Computing μ -Sequences of Hypersurface Isolated Singularities via Parametric Local Cohomology Systems

  • Katsusuke NabeshimaEmail author
  • Shinichi Tajima
Article

Abstract

Complex analytic invariants of hypersurface isolated singularities are considered in the context of symbolic computation. The motivations for this paper are computer calculations of μ -sequences that introduced by B. Teissier to study the Whitney equisingularity of deformations of complex hypersurfaces. A new algorithm that utilizes parametric local cohomology systems is proposed to compute μ -sequences. Lists of μ -sequences of some typical cases are also given.

Keywords

μ-sequence Milnor number Local cohomology 

Mathematics Subject Classification (2010)

13D45 32C37 13J05 32A27 

Notes

Acknowledgments

This work has been partly supported by JSPS Grant-in-Aid for Young Scientists (B) (No. 15K17513) and Grant-in-Aid for Scientific Research (C) (No. 15K04891).

References

  1. 1.
    Biviá-Ausina, C.: Generic linear sections of complex hypersurfaces and monomial ideals. Topology Appl. 159, 414–419 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Briançon, J., Speder, J.P.: La trivialité topologique n’implique pas les conditions de Whitney. C. R. Acad. Sci. Paris 280, 365–367 (1975)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Briançon, J., Henry, J.P., Speder, J.P.: Les conditions de Whitney en un point sont analytiques. C. R. Acad. Sci. Paris 282, 279–282 (1976)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Briançon, J., Speder, J.P.: Les conditions de Whitney impliques μ constant. Ann. Inst. Fourier Grenoble 26, 153–163 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Briançon, J., Henry, J.P.G.: Equisingularité générique des familles de surfaces a singularité isolée. Bull. Soc. Math. France 108, 259–281 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gaffney, T.: Polar multiplicities and equisingularity of map germs. Topology 32, 185–223 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Grothendieck, A.: Théorèmes de dualité pour les faisceaux algébriques cohérents. Séminaire Bourbaki 149, 25 pp (1957)Google Scholar
  8. 8.
    Grothendieck, A.: Local Cohomology, Notes by R. Hartshorne. Lecture Notes in Math., vol. 41. Springer (1967)Google Scholar
  9. 9.
    Kouchnirenko, A.G.: Polyèdres de Newton et nombres de Milnor. Invent. Math. 32, 1–31 (1976)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Mima, S.: On the Milnor number of a generic hyperplane section. J. Math. Soc. Jpn. 41–4, 709–724 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Nabeshima, K., Tajima, S.: On efficient algorithm for computing parametric local cohomology classes associated with semi-quasihomogeneous singularities and standard bases. In: Proc. International Symposium on Symbolic and Algebraic Computation (ISSAC 2014), pp. 351–358. ACM (2014)Google Scholar
  12. 12.
    Nabeshima, K., Tajima, S.: Algebraic local cohomology with parameters and parametric standard bases for zero-dimensional ideals. To appear in Journal of Symbolic Computation (2016)Google Scholar
  13. 13.
    Navarro Azunar, V.: Conditions de Whitney et sections planes. Invent. Math. 61, 199–225 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Noro, M., Takeshima, T.: Risa/Asir - A computer algebra system. In: Proc. International Symposium on Symbolic and Algebraic Computation(ISSAC1992), pp. 387–396. ACM. http://www.math.kobe-u.ac.jp/Asir/asir.html (1992)
  15. 15.
    O’Shea, D., Teleman, C.: Limiting tangent spaces and a criterion for μ-constancy. Travaux en Cours 55 Hermann, 79–85 (1997)Google Scholar
  16. 16.
    Tajima, S., Nakamura, Y., Nabeshima, K.: Standard bases and algebraic local cohomology for zero dimensional ideals. Adv. Stud. Pure Math. 56, 341–361 (2009)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Teissier, B.: Cycles évanescents et conditions de Whitney. C. R. Acad. Sc. Paris 276, 1051–1054 (1973)zbMATHGoogle Scholar
  18. 18.
    Teissier, B.: Cycles evanescents, sections planes et conditions de Whitney. Astérisques 7-8, Soc. Math. France, 285–362 (1973)Google Scholar
  19. 19.
    Teissier, B.: Variétés polaires I. Invent. Math. 40, 267–292 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Teissier, B.: Variétés polaires II. Lect. Notes Math. 961, 314–491 (1982)CrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2016

Authors and Affiliations

  1. 1.Graduate School of Science and TechnologyTokushima UniversityTokushimaJapan
  2. 2.Graduate School of Pure and Applied ScienceUniversity of TsukubaTsukubaJapan

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