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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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)
Eggers, H.: Polarinvarianten und die Topologie von Kurvensingularitäten. Bonner Math. Schriften 147, Universität Bonn, Bonn (1982)
García Barroso E.R.: Sur les courbes polaires d’une courbe plane réduite. Proc. London Math. Soc. III 81(1), 1–28 (2000)
García Barroso, E.R., Płoski, A.: Sur l’exposant de contact des courbes analytiques planes. Octobre 2002, Travail en cours.
García Barroso, E.R., Płoski, A.: An approach to plane algebroid branches. (2011, in press)
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)
Gwoździewicz J., Lenarcik A., Płoski A.: Polar invariants of plane curve singularities: intersection theoretical approach. Demonstratio Math. 43(2), 303–323 (2010)
Kouchnirenko A.G.: Polyèdres de Newton et nombres de Milnor. Inv. Math. 32, 1–31 (1976)
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.
Lenarcik A.: On the jacobian Newton polygon of plane curve singularities. Manuscripta Math 125, 309–324 (2008). doi:10.1007/s00229-007-0150-y
Lenarcik, A.: On the łojasiewicz exponent, special direction and maximal polar quotient. arXiv:1112.5578v1 (in press)
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)
McNeal J.D., Nemethi A.: The order of contact of a holomorphic ideal in C 2. Math. Z. 250, 873–883 (2005)
Merle M.: Invariants polaires des courbes planes. Invent. Math. 41, 103–111 (1977)
Płoski , Płoski : Remarque sur la multiplicité d’intersection des branches planes. Bull. Pol. Acad. Sci. Math. 33(11–12), 601–605 (1985)
Teissier, B.: Variétés polaires. Invent. Math. 40, 267–292 (1977)
Teissier, B.: Polyèdre de Newton Jacobien et équisingularité. In: Séminaire sur les Singularités, vol.7, pp. 193–221. Math. Univ., Paris VII (1980)
Wall C.T.C.: Chains on the Eggers tree and polar curves. Rev. Mat. Ibero 19, 745–754 (2003)
Wall, C.T.C.: Singular Points of Plane Curves. Cambridge University Press (2004)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Professor Evelia Rosa García Barroso
Rights and permissions
About this article
Cite this article
Lenarcik, A. Eggers tree and jacobian Newton polygon. manuscripta math. 142, 233–244 (2013). https://doi.org/10.1007/s00229-012-0600-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-012-0600-z