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Polar invariants of plane curve singularities: intersection theoretical approach

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This article, based on the talk given by one of the authors at the Pierret-tefest in Castro Urdiales in June 2008, is an overview of a number of recent results on the polar invariants of plane curve singularities.
Wydawca
Rocznik
Strony
303--323
Opis fizyczny
Bibliogr. 46 poz.
Twórcy
autor
autor
  • Department of Mathematics Technical University Al. 1000 L PP 7 25-314 Kielce, Poland, matjg@tu.kielce.pl
Bibliografia
  • [1] S. S. Abhyankar, Irreducibility criterion for germs of analytic functions of two complex variables, Adv. in Math. 74 (1989), 190–257.
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  • [3] H. Bresinsky, Semigroup corresponding to algebroid branches in the plane, Proc. of the AMS 32 (1972), 381–384.
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  • [5] E. Casas-Alvero, Singularities of Plane Curves, London Math. Soc. Lecture Note Ser. 276, Cambridge Univ. Press, Cambridge, 2000.
  • [6] F. Delgado de la Mata, An arithmetical factorization for the critical point set of some map germs from C2 to C2, Singularities (Lille 1991), 61–100. London Math. Soc. Lecture Note Ser. 201, 1994.
  • [7] F. Delgado de la Mata, A factorization theorem for the polar of a curve with two branches, Compositio Math. 92 (1994), 327–375.
  • [8] H. Eggers, Polarinvarianten und die Topologie von Kurvensingularitäten, Bonner Math. Schriften 147, Universität Bonn, Bonn 1982.
  • [9] R. Ephraim, Special polars and curves with one place at infinity (P. Orlik ed.), Proc. of Symp. in Pure Math., Vol 40 Part 1, AMS, Providence, 1983, 353–359.
  • [10] E. García Barroso, Sur les courbes polaires d’une courbe plane réduite, Proc. London Math. Soc. (3), 81 (2000), 1–28.
  • [11] E. García Barroso, J. Gwoździewicz, Characterization of jacobian Newton polygons of plane branches and new criteria of irreducibility, arXiv:085.4257 (to appear in Ann. Inst. Fourier (Grenoble) vol. 60 no. 2).
  • [12] E. R. García Barroso, T. Krasiński, A. Płoski, The Łojasiewicz numbers and plane curve singularities, Ann. Pol. Math. 87 (2005), 127–150.
  • [13] E. R. García Barroso, A. Płoski, Pinceaux de courbes planes et invariants polaires, Ann. Pol. Math. 82 (2004), 113–128.
  • [14] E. R. García Barroso, A. Lenarcik, A. Płoski, Characterization of nondegenerate plane curve singularities, Univ. Iagel. Acta Math. 45 (2007), 27–36.
  • [15] J. Gwoździewicz, A. Lenarcik, A. Płoski, The jacobian Newton polygon and equisingularity of plane curve singularities (in preparation).
  • [16] J. Gwoździewicz, A. Płoski, On the Merle formula for polar invariants, Bull. Soc. Sci. Lett. Łódź 41 (7) (1991), 61–67.
  • [17] J. Gwoździewicz, A. Płoski, On the approximate roots of polynomials, Ann. Polon. Math. 3 (1995), 199–210.
  • [18] J. Gwoździewicz, A. Płoski, On the polar quotients of an analytic plane curve, Kodai Math. J. 25 (2002), 43–53.
  • [19] J. Gwoździewicz, A. Płoski, Łojasiewicz exponents and singularities at infinity of polynomials in two complex variables, Coll. Math. 103 (2005), 47–60.
  • [20] S. Izumi, S. Koike, T-Ch. Kuo, Computation and stability of the Fukui Invariant , Compositio Math. 130 (2002), 49–73.
  • [21] A. G. Kouchnirenko, Polyèdres de Newton et nombre de Milnor, Invent. Math. 32 (1976), 1–31.
  • [22] T-C. Kuo, Generalized Newton-Puiseux Theory and Hensel’s lemma in C[[x, y]], Canad. J. Math. 41 (1989), 1101–1116.
  • [23] T-C. Kuo, Y. C. Lu, On analytic function germ of two complex variables, Topology 16 (1977), 299–310.
  • [24] D. T. Lê, Topological use of polar curves, Algebraic geometry, Arcata 1974, Proc. Sym. Pure Math., vol 29 (AMS Providence) RI (1975), 507–512.
  • [25] D. T. Lê, F. Michel, C. Weber, Sur le comportement des polaires associées aux germes de courbes planes, Compositio Math. 72 (1989), 87–113.
  • [26] D. T. Lê, F. Michel, C. Weber, Courbes polaires et topologie des courbes planes, Ann. Sci. École Norm. Sup. 24 (1991), 141–169.
  • [27] D. T. Lê, C. P. Ramanujam, The invariance of Milnor’s number implies the invariance of the topological type, Amer. J. Math. 98 (1976), 67–78.
  • [28] A. Lenarcik, A. Płoski, Polar invariants of plane curves and the Newton polygon, Kodai Math. J. 23 (2000), 309–319.
  • [29] A. Lenarcik, M. Masternak, A. Płoski, Factorization of the polar curve and the Newton polygon, Kodai Math. J. 26 (2003), 288–303.
  • [30] A. Lenarcik, Polar quotients of a plane curve and the Newton algorithm, Kodai Math. J. 27 (2004), 336–353.
  • [31] A. Lenarcik, On the jacobian Newton polygon of plane curve singularities, Manuscripta Math. 125 (2008), 309–324.
  • [32] M. Merle, Invariants polaires des courbes planes, Invent. Math. 41 (1977), 103–111.
  • [33] A. Płoski, The Milnor number of a plane algebroid curve, in Materiały XVI Konferencji Szkoleniowej z Analizy i Geometrii Zespolonej, Łódź (1995), 73–82.
  • [34] A. Płoski, On the maximal polar quotient of an analytic plane curve, Kodai Math. J. 24 (2001), 120–133.
  • [35] A. Płoski, Polar quotients and singularities at infinity of polynomials in two complex variables, Ann. Polon. Math. 78 (2002), 49–58.
  • [36] A. Płoski, On the special values for pencils of plane curve singularities, Univ. Iagel. Acta. Math. 42 (2004), 7–13.
  • [37] H. J. S. Smith, On the higher singularities of plane curves, Proc. London Math. Soc. 6 (1875), 153–182.
  • [38] B. Teissier, Cycles évanescents, sections planes et conditions de Whitney, Astérisque (Société Mathématique de France), No 7-8, 1973.
  • [39] B. Teissier, Introduction to equisingularity problems, Proc. Sym. Pure Math., vol 29 (AMS Providence) RI (1975), 593–632.
  • [40] B. Teissier, The hunting of invariants in the geometry of discriminants, Nordic Summer School/NAVF Symposium in Mathematics, Oslo, August 5–25, 1976.
  • [41] B. Teissier, Varietés polaires I. Invariants polaires des singularités des hypersurfaces, Invent. Math. 40 (1977), 267–292.
  • [42] B. Teissier, Polyèdre de Newton Jacobien et équisingularité, Séminaire sur les Singularités, Publications Math., Université Paris VII, 7 (1980), 193–221, http://pepole.math.jussieu.fr/˜teissier/articles-Teissier.html.
  • [43] B. Teissier, Introduction to Curve Singularities, Singularity Theory, Editors D. T. Lê, K. Saito, B. Teissier, Word Scientific 1991.
  • [44] C. T. C. Wall, Chains on the Eggers tree and polar curves, Rev. Mat. Ibera 19 (2003), 745–754.
  • [45] C. T. C. Wall, Singular Points of Plane Curves, Cambridge University Press, 2004.
  • [46] O. Zariski, Le problème de modules pour les branches des courbes planes, Lecture Notes (ed. F. Kmety and M. Merle), École Polytechnique, 1973.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-PWA5-0027-0044
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