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Manuscripta Mathematica

, Volume 142, Issue 1–2, pp 233–244 | Cite as

Eggers tree and jacobian Newton polygon

  • Andrzej LenarcikEmail author
Article
  • 126 Downloads

Abstract

For a germ f = 0 of an isolated plane curve singularity defined by \({f \in \mathbb{C}\{X, Y\}}\) we consider the jacobian Newton polygon \({\nu_\mathbf{J}(f)}\) introduced by Bernard Teissier. For two such germs f = 0, g = 0 we study the case \({\nu_\mathbf{J}(f) = \nu_\mathbf{J}(g)}\) . When f and g are irreducible then the germs f = 0, g = 0 are equisingular (Merle’s result). The same is true for f,g unitangent and nondegenerate in the Kouchnirenko sense (author’s result). We generalize these theorems. We formulate our result in terms of the Eggers tree.

Mathematics Subject Classifications (2000)

Primary 32S55 Secondary 14H20 

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References

  1. 1.
    Ch¸dzyński J., Płoski A.: An inequality for the intersection multiplicity of analytic curves. Bull. Polish Acad. Sci. Math. 36(3–4), 113–117 (1988)MathSciNetGoogle Scholar
  2. 2.
    Eggers, H.: Polarinvarianten und die Topologie von Kurvensingularitäten. Bonner Math. Schriften 147, Universität Bonn, Bonn (1982)Google Scholar
  3. 3.
    García Barroso E.R.: Sur les courbes polaires d’une courbe plane réduite. Proc. London Math. Soc. III 81(1), 1–28 (2000)zbMATHCrossRefGoogle Scholar
  4. 4.
    García Barroso, E.R., Płoski, A.: Sur l’exposant de contact des courbes analytiques planes. Octobre 2002, Travail en cours.Google Scholar
  5. 5.
    García Barroso, E.R., Płoski, A.: An approach to plane algebroid branches. (2011, in press)Google Scholar
  6. 6.
    García Barroso, E.R., Lenarcik, A., Płoski, A.: Characterization of non-degenerate plane curve singularities, Univ. Iagel. Acta Math. 45, 27–36 (2007), with Erratum: Univ. Iagel. Acta Math. 47, 321–322 (2009)Google Scholar
  7. 7.
    Gwoździewicz J., Lenarcik A., Płoski A.: Polar invariants of plane curve singularities: intersection theoretical approach. Demonstratio Math. 43(2), 303–323 (2010)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Kouchnirenko A.G.: Polyèdres de Newton et nombres de Milnor. Inv. Math. 32, 1–31 (1976)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Lejeune-Jalabert M.: Sur l’équivalence des courbes algebroïdes planes. Coefficients de Newton. Contribution à à l’etude des singularités du poit du vue du polygone de Newton. Paris VII, Thèse d’Etat., Janvier 1973. See also In: Lê Dũng Trãng (ed.): Travaux en Cours, 36, Introduction à à la théorie des singularités I, 1988.Google Scholar
  10. 10.
    Lenarcik A.: On the jacobian Newton polygon of plane curve singularities. Manuscripta Math 125, 309–324 (2008). doi: 10.1007/s00229-007-0150-y MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Lenarcik, A.: On the łojasiewicz exponent, special direction and maximal polar quotient. arXiv:1112.5578v1 (in press)Google Scholar
  12. 12.
    Maugendre H.: Discriminant d’un germe \({\Phi:(\mathbb{C}^2,0)\rightarrow(\mathbb{C}^2,0)}\) et résolution minimale de f · g. Ann. Fac. Sci. Toulouse. VI Sér. Math 7(3), 497– (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    McNeal J.D., Nemethi A.: The order of contact of a holomorphic ideal in C 2. Math. Z. 250, 873–883 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Merle M.: Invariants polaires des courbes planes. Invent. Math. 41, 103–111 (1977)Google Scholar
  15. 15.
    Płoski , Płoski : Remarque sur la multiplicité d’intersection des branches planes. Bull. Pol. Acad. Sci. Math. 33(11–12), 601–605 (1985)zbMATHGoogle Scholar
  16. 16.
    Teissier, B.: Variétés polaires. Invent. Math. 40, 267–292 (1977)Google Scholar
  17. 17.
    Teissier, B.: Polyèdre de Newton Jacobien et équisingularité. In: Séminaire sur les Singularités, vol.7, pp. 193–221. Math. Univ., Paris VII (1980)Google Scholar
  18. 18.
    Wall C.T.C.: Chains on the Eggers tree and polar curves. Rev. Mat. Ibero 19, 745–754 (2003)zbMATHCrossRefGoogle Scholar
  19. 19.
    Wall, C.T.C.: Singular Points of Plane Curves. Cambridge University Press (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of MathematicsKielce University of TechnologyKielcePoland

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