Characterization of jacobian Newton polygons of plane branches and new criteria of irreducibility

Evelia R. García Barroso[1]; Janusz Gwoździewicz[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)

Annales de l’institut Fourier (2010)

  • Volume: 60, Issue: 2, page 683-709
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Barroso, Evelia R. García, and Gwoździewicz, Janusz. "Characterization of jacobian Newton polygons of plane branches and new criteria of irreducibility." Annales de l’institut Fourier 60.2 (2010): 683-709. <http://eudml.org/doc/116285>.

@article{Barroso2010,
abstract = {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.},
affiliation = {Universidad de La Laguna Facultad de Matemáticas Departamento de Matemática Fundamental 38271 La Laguna, Tenerife (Espagne); Technical University Department of Mathematics 25-314 Kielce (Pologne)},
author = {Barroso, Evelia R. García, Gwoździewicz, Janusz},
journal = {Annales de l’institut Fourier},
keywords = {Irreducible plane curve; jacobian Newton polygon; polar invariant; approximate root; irreducible plane curve; Jacobian Newton polygon},
language = {eng},
number = {2},
pages = {683-709},
publisher = {Association des Annales de l’institut Fourier},
title = {Characterization of jacobian Newton polygons of plane branches and new criteria of irreducibility},
url = {http://eudml.org/doc/116285},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Barroso, Evelia R. García
AU - Gwoździewicz, Janusz
TI - Characterization of jacobian Newton polygons of plane branches and new criteria of irreducibility
JO - Annales de l’institut Fourier
PY - 2010
PB - Association des Annales de l’institut Fourier
VL - 60
IS - 2
SP - 683
EP - 709
AB - 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.
LA - eng
KW - Irreducible plane curve; jacobian Newton polygon; polar invariant; approximate root; irreducible plane curve; Jacobian Newton polygon
UR - http://eudml.org/doc/116285
ER -

References

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  1. S. S. Abhyankar, Irreducibility Criterion for Germs of Analytic Functions of Two Complex Variables, Advances in Mathematics 74 (1989), 190-257 Zbl0683.14001MR997097
  2. S. S. Abhyankar, T. Moh, Newton-Puiseux Expansions and Generalized Tschirnhausen Transformation, J. Reine Angew. Math. 260 (1973), 47-83 Zbl0272.12102MR337955
  3. H. Bresinsky, Semigroups corresponding to algebroid branches in the plane, Proc. of the AMS 32 (1972), 381-384 Zbl0218.14003MR291171
  4. E. Casas-Alvero, Singularities of plane curves, 276 (2000), London Mathematical Society Zbl0967.14018MR1782072
  5. H. Eggers, Polarinvarianten und die Topologie von Kurvensingularitaten, 147 (1983), Bonner Mathematische Schriften Zbl0559.14018MR701391
  6. E. R. García Barroso, Sur les courbes polaires d’une courbe plane réduite, London Math. Soc. 81, Part 1 (2000), 1-28 Zbl1041.14008
  7. E. R. García Barroso, B. Teissier, Concentration multi-échelles de courbure dans des fibres de Milnor, Comment. Math. Helv. 74 (1999), 398-418 Zbl0956.32028MR1710694
  8. J. Gwoździewicz, A. Płoski, On the Merle formula for polar invariants, Bull. Soc. Sci. Lett. Łódź 41 (1991), 61-67 Zbl0893.32006MR1187123
  9. J. Gwoździewicz, A. Płoski, On the Approximate Roots of polynomials, (1995), Annales Polonici Mathematici LX3 Zbl0826.13012
  10. J. Gwoździewicz, A. Płoski, On the polar quotients of an analytic plane curve, Kodai Math. J. 25 (2002), 43-53 Zbl1007.32018MR1891798
  11. A. G. Kouchnirenko, Polyèdres de Newton et nombres de Milnor, Invent. Math. 32 (1976), 1-31 Zbl0328.32007MR419433
  12. T. C. Kuo, Generalized Newton-Puiseux Theory and Hensel’s Lemma in C [ [ x , y ] ] , Canadian Journal of Mathematics 41 (1989), 1101-1116 Zbl0716.13015MR1018453
  13. T. C. Kuo, Y. C. Lu, On analytic function germs of two complex variables, Topology 16 (1977), 299-310 Zbl0378.32001MR460711
  14. T. C. Kuo, A. Parusiński, Newton-Puiseux roots of Jacobian determinants, J. Algebraic Geometry 13 (2004), 579-601 Zbl1061.32002MR2047682
  15. A. Lenarcik, On the Jacobian Newton polygon of plane curve singularities, Manuscripta Math. 125 (2008), 309-324 Zbl1139.32014MR2373063
  16. P. Maisonobe, Lieu discriminant d’un germe analytique de corang 1 de ( C 2 , 0 ) vers ( C 2 , 0 ) , Ann. Inst. Fourier 32 (1982), 91-118 Zbl0487.32008MR694129
  17. H. Maugendre, Discriminant d’un germe ( g , f ) : ( ℂ 2 , 0 ) → ( ℂ 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), 497-525 Zbl0936.32012MR1677138
  18. H. Maugendre, Discriminant of a germ φ : ( ℂ 2 , 0 ) → ( ℂ 2 , 0 ) and Seifert fibred manifolds, Journal London Mathematical Society 59 (1999), 207-226 Zbl0941.58027MR1688499
  19. M. Merle, Invariants polaires des courbes planes, Invent. Math. 41 (1977), 103-111 Zbl0371.14003MR460336
  20. A. Płoski, Newton polygons and the Łojasiewicz exponent of a holomorphic mapping of C 2 , Ann. Polon. Math. LI (1990), 275-281 Zbl0764.32012MR1093999
  21. P. Popescu-Pampu, Approximate Roots, Fields Institute Communications 33 (2003), 285-321 Zbl1036.13017MR2018562
  22. B. Teissier, The hunting of invariants in the geometry of discriminants, Proc. Nordic summer school (1976), 565-677 Zbl0388.32010MR568901
  23. B. Teissier, Variétés polaires.I. Invariants polaires des singularités des hypersurfaces, Invent. Math. 40 (1977), 267-292 Zbl0446.32002MR470246
  24. B. Teissier, Polyèdre de Newton Jacobien et équisingularité, Séminaire sur les Singularités (1980), 193-221, Publications Math. Université Paris VII MR683624
  25. R. Walker, Algebraic Curves, (1950), Princeton University Press Zbl0039.37701MR33083
  26. C. T. C. Wall, Chains on the Eggers tree and polar curves, Rev. Mat. Iberoamericana 19 (2003), 745-754 Zbl1057.14032MR2023205
  27. C. T. C. Wall, Singular points of plane curves, Student Texts 63 (2004), London Mathematical Society Zbl1057.14001MR2107253
  28. O. Zariski, Le problème des modules pour les branches planes, Centre de Maths, École Polytechnique (1975) Zbl0317.14004

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