Characterization of jacobian Newton polygons of plane branches and new criteria of irreducibility
[Caractérisation des polygones de Newton jacobiens des branches planes et de nouveaux critères d’irréductibilité]
Annales de l'Institut Fourier, Tome 60 (2010) no. 2, pp. 683-709.

Nous caractérisons de deux manières différentes les polygones de Newton jacobiens des branches planes. Ces caractérisations donnent, en particulier, des critères combinatoires d’irréductibilité des séries complexes en deux variables et des conditions nécessaires pour qu’une courbe dans le plan complexe soit le discriminant d’une branche plane.

In this paper we characterize, in two different ways, the Newton polygons which are jacobian Newton polygons of a plane branch. These characterizations give in particular combinatorial criteria of irreducibility for complex series in two variables and necessary conditions which a complex curve has to satisfy in order to be the discriminant of a complex plane branch.

DOI : 10.5802/aif.2536
Classification : 32S55, 14H20
Keywords: Irreducible plane curve, jacobian Newton polygon, polar invariant, approximate root
Mot clés : courbe plane irréductible, polygone de Newton jacobien, invariant polaire, racine approchée
Barroso, Evelia R. García 1 ; Gwoździewicz, Janusz 2

1 Universidad de La Laguna Facultad de Matemáticas Departamento de Matemática Fundamental 38271 La Laguna, Tenerife (Espagne)
2 Technical University Department of Mathematics 25-314 Kielce (Pologne)
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Barroso, Evelia R. García; Gwoździewicz, Janusz. Characterization of jacobian Newton polygons of plane branches  and new criteria of irreducibility. Annales de l'Institut Fourier, Tome 60 (2010) no. 2, pp. 683-709. doi : 10.5802/aif.2536. http://archive.numdam.org/articles/10.5802/aif.2536/

[1] Abhyankar, S. S. Irreducibility Criterion for Germs of Analytic Functions of Two Complex Variables, Advances in Mathematics, Volume 74 (1989), pp. 190-257 | DOI | MR | Zbl

[2] Abhyankar, S. S.; Moh, T. Newton-Puiseux Expansions and Generalized Tschirnhausen Transformation, J. Reine Angew. Math., Volume 260 (1973), pp. 47-83 261 (1973), 29–54 | DOI | MR | Zbl

[3] Bresinsky, H. Semigroups corresponding to algebroid branches in the plane, Proc. of the AMS, Volume 32 (1972) no. 2, pp. 381-384 | DOI | MR | Zbl

[4] Casas-Alvero, E. Singularities of plane curves, Lecture Note Series, 276, London Mathematical Society, 2000 | MR | Zbl

[5] Eggers, H. Polarinvarianten und die Topologie von Kurvensingularitaten, 147, Bonner Mathematische Schriften, 1983 | MR | Zbl

[6] García Barroso, E. R. Sur les courbes polaires d’une courbe plane réduite, London Math. Soc., Volume 81, Part 1 (2000), pp. 1-28 | Zbl

[7] García Barroso, E. R.; Teissier, B. Concentration multi-échelles de courbure dans des fibres de Milnor, Comment. Math. Helv., Volume 74 (1999), pp. 398-418 | DOI | MR | Zbl

[8] Gwoździewicz, J.; Płoski, A. On the Merle formula for polar invariants, Bull. Soc. Sci. Lett. Łódź, Volume 41 (1991), pp. 61-67 | MR | Zbl

[9] Gwoździewicz, J.; Płoski, A. On the Approximate Roots of polynomials, Annales Polonici Mathematici LX3, 1995 (199–210) | Zbl

[10] Gwoździewicz, J.; Płoski, A. On the polar quotients of an analytic plane curve, Kodai Math. J., Volume 25 (2002), pp. 43-53 | DOI | MR | Zbl

[11] Kouchnirenko, A. G. Polyèdres de Newton et nombres de Milnor, Invent. Math., Volume 32 (1976), pp. 1-31 | DOI | MR | Zbl

[12] Kuo, T. C. Generalized Newton-Puiseux Theory and Hensel’s Lemma in C[[x,y]], Canadian Journal of Mathematics, Volume 41 (1989) no. 6, pp. 1101-1116 | DOI | MR | Zbl

[13] Kuo, T. C.; Lu, Y. C. On analytic function germs of two complex variables, Topology, Volume 16 (1977), pp. 299-310 | DOI | MR | Zbl

[14] Kuo, T. C.; Parusiński, A. Newton-Puiseux roots of Jacobian determinants, J. Algebraic Geometry, Volume 13 (2004), pp. 579-601 | MR | Zbl

[15] Lenarcik, A. On the Jacobian Newton polygon of plane curve singularities, Manuscripta Math., Volume 125 (2008), pp. 309-324 | DOI | MR | Zbl

[16] Maisonobe, P. Lieu discriminant d’un germe analytique de corang 1 de (C 2 ,0) vers (C 2 ,0), Ann. Inst. Fourier, Volume 32 (1982) no. 4, pp. 91-118 | DOI | Numdam | MR | Zbl

[17] Maugendre, H. Discriminant d’un germe (g,f):( 2 ,0)( 2 ,0) et quotients de contact dans la résolution of f.g, Annales de la Faculté de Sciences de Toulouse, 6e série, tome 3 (1998) no. 3, pp. 497-525 | Numdam | MR | Zbl

[18] Maugendre, H. Discriminant of a germ φ:( 2 ,0)( 2 ,0) and Seifert fibred manifolds, Journal London Mathematical Society, Volume 59 (1999) no. 1, pp. 207-226 | DOI | MR | Zbl

[19] Merle, M. Invariants polaires des courbes planes, Invent. Math., Volume 41 (1977), pp. 103-111 | DOI | MR | Zbl

[20] Płoski, A. Newton polygons and the Łojasiewicz exponent of a holomorphic mapping of C 2 , Ann. Polon. Math. LI (1990), pp. 275-281 | MR | Zbl

[21] Popescu-Pampu, P. Approximate Roots, Fields Institute Communications, Volume 33 (2003), pp. 285-321 | MR | Zbl

[22] Teissier, B. The hunting of invariants in the geometry of discriminants, Proc. Nordic summer school (1976), pp. 565-677 (Per Holm, editor, Sijthoff and Noordhoff 1978) | MR | Zbl

[23] Teissier, B. Variétés polaires.I. Invariants polaires des singularités des hypersurfaces, Invent. Math., Volume 40 (1977), pp. 267-292 | DOI | MR | Zbl

[24] Teissier, B. Polyèdre de Newton Jacobien et équisingularité, Séminaire sur les Singularités, Publications Math. Université Paris VII, 1980, pp. 193-221 | MR

[25] Walker, R. Algebraic Curves, Princeton University Press, 1950 | MR | Zbl

[26] Wall, C. T. C. Chains on the Eggers tree and polar curves, Rev. Mat. Iberoamericana, Volume 19 (2003) no. 2, pp. 745-754 | MR | Zbl

[27] Wall, C. T. C. Singular points of plane curves, Student Texts, Volume 63, London Mathematical Society, 2004 | MR | Zbl

[28] Zariski, O. Le problème des modules pour les branches planes, Centre de Maths, École Polytechnique, 1975 (Reprinted by Hermann, Paris 1986) | Zbl

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