Characterization of jacobian Newton polygons of plane branches and new criteria of irreducibility
Annales de l'Institut Fourier, Volume 60 (2010) no. 2, p. 683-709
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.
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.
DOI : https://doi.org/10.5802/aif.2536
Classification:  32S55,  14H20
Keywords: Irreducible plane curve, jacobian Newton polygon, polar invariant, approximate root
@article{AIF_2010__60_2_683_0,
     author = {Barroso, Evelia R. Garc\'\i a and Gwo\'zdziewicz, Janusz},
     title = {Characterization of jacobian Newton polygons of plane branches  and new criteria of irreducibility},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {60},
     number = {2},
     year = {2010},
     pages = {683-709},
     doi = {10.5802/aif.2536},
     mrnumber = {2667790},
     zbl = {1197.32012},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2010__60_2_683_0}
}
Barroso, Evelia R. García; Gwoździewicz, Janusz. Characterization of jacobian Newton polygons of plane branches  and new criteria of irreducibility. Annales de l'Institut Fourier, Volume 60 (2010) no. 2, pp. 683-709. doi : 10.5802/aif.2536. http://www.numdam.org/item/AIF_2010__60_2_683_0/

[1] Abhyankar, S. S. Irreducibility Criterion for Germs of Analytic Functions of Two Complex Variables, Advances in Mathematics, Tome 74 (1989), pp. 190-257 | Article | MR 997097 | Zbl 0683.14001

[2] Abhyankar, S. S.; Moh, T. Newton-Puiseux Expansions and Generalized Tschirnhausen Transformation, J. Reine Angew. Math., Tome 260 (1973), pp. 47-83 (261 (1973), 29–54) | Article | MR 337955 | Zbl 0272.12102

[3] Bresinsky, H. Semigroups corresponding to algebroid branches in the plane, Proc. of the AMS, Tome 32 (1972) no. 2, pp. 381-384 | Article | MR 291171 | Zbl 0218.14003

[4] Casas-Alvero, E. Singularities of plane curves, London Mathematical Society, Lecture Note Series, Tome 276 (2000) | MR 1782072 | Zbl 0967.14018

[5] Eggers, H. Polarinvarianten und die Topologie von Kurvensingularitaten, Bonner Mathematische Schriften Tome 147 (1983) | MR 701391 | Zbl 0559.14018

[6] García Barroso, E. R. Sur les courbes polaires d’une courbe plane réduite, London Math. Soc., Tome 81, Part 1 (2000), pp. 1-28 | Zbl 1041.14008

[7] García Barroso, E. R.; Teissier, B. Concentration multi-échelles de courbure dans des fibres de Milnor, Comment. Math. Helv., Tome 74 (1999), pp. 398-418 | Article | MR 1710694 | Zbl 0956.32028

[8] Gwoździewicz, J.; Płoski, A. On the Merle formula for polar invariants, Bull. Soc. Sci. Lett. Łódź, Tome 41 (1991), pp. 61-67 | MR 1187123 | Zbl 0893.32006

[9] Gwoździewicz, J.; Płoski, A. On the Approximate Roots of polynomials, Annales Polonici Mathematici LX3 (1995) (199–210) | Zbl 0826.13012

[10] Gwoździewicz, J.; Płoski, A. On the polar quotients of an analytic plane curve, Kodai Math. J., Tome 25 (2002), pp. 43-53 | Article | MR 1891798 | Zbl 1007.32018

[11] Kouchnirenko, A. G. Polyèdres de Newton et nombres de Milnor, Invent. Math., Tome 32 (1976), pp. 1-31 | Article | MR 419433 | Zbl 0328.32007

[12] Kuo, T. C. Generalized Newton-Puiseux Theory and Hensel’s Lemma in C[[x,y]], Canadian Journal of Mathematics, Tome 41 (1989) no. 6, pp. 1101-1116 | Article | MR 1018453 | Zbl 0716.13015

[13] Kuo, T. C.; Lu, Y. C. On analytic function germs of two complex variables, Topology, Tome 16 (1977), pp. 299-310 | Article | MR 460711 | Zbl 0378.32001

[14] Kuo, T. C.; Parusiński, A. Newton-Puiseux roots of Jacobian determinants, J. Algebraic Geometry, Tome 13 (2004), pp. 579-601 | MR 2047682 | Zbl 1061.32002

[15] Lenarcik, A. On the Jacobian Newton polygon of plane curve singularities, Manuscripta Math., Tome 125 (2008), pp. 309-324 | Article | MR 2373063 | Zbl 1139.32014

[16] Maisonobe, P. Lieu discriminant d’un germe analytique de corang 1 de (C 2 ,0) vers (C 2 ,0), Ann. Inst. Fourier, Tome 32 (1982) no. 4, pp. 91-118 | Article | Numdam | MR 694129 | Zbl 0487.32008

[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 1677138 | Zbl 0936.32012

[18] Maugendre, H. Discriminant of a germ φ:( 2 ,0)( 2 ,0) and Seifert fibred manifolds, Journal London Mathematical Society, Tome 59 (1999) no. 1, pp. 207-226 | Article | MR 1688499 | Zbl 0941.58027

[19] Merle, M. Invariants polaires des courbes planes, Invent. Math., Tome 41 (1977), pp. 103-111 | Article | MR 460336 | Zbl 0371.14003

[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 1093999 | Zbl 0764.32012

[21] Popescu-Pampu, P. Approximate Roots, Fields Institute Communications, Tome 33 (2003), pp. 285-321 | MR 2018562 | Zbl 1036.13017

[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 568901 | Zbl 0388.32010

[23] Teissier, B. Variétés polaires.I. Invariants polaires des singularités des hypersurfaces, Invent. Math., Tome 40 (1977), pp. 267-292 | Article | MR 470246 | Zbl 0446.32002

[24] Teissier, B. Polyèdre de Newton Jacobien et équisingularité, Séminaire sur les Singularités, Publications Math. Université Paris VII (1980), pp. 193-221 | MR 683624

[25] Walker, R. Algebraic Curves, Princeton University Press (1950) | MR 33083 | Zbl 0039.37701

[26] Wall, C. T. C. Chains on the Eggers tree and polar curves, Rev. Mat. Iberoamericana, Tome 19 (2003) no. 2, pp. 745-754 | MR 2023205 | Zbl 1057.14032

[27] Wall, C. T. C. Singular points of plane curves, Student Texts, London Mathematical Society, Tome 63 (2004) | MR 2107253 | Zbl 1057.14001

[28] Zariski, O. Le problème des modules pour les branches planes, Centre de Maths, École Polytechnique (1975) (Reprinted by Hermann, Paris 1986) | Zbl 0317.14004