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Abstract: 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.
B I B L I O G R A F I A[1] S. S. Abhyankar, Irreducibility criterion for germs of analytic functions of two complex variables, Adv. in Math. 74 (1989), 190-257.
[2] S. S. Abhyankar, T. Moh, Embeddings of line in the plane, J. Reine Angew. Math. 276 (1975), 148-166.
[3] H. Bresinsky, Semigroup corresponding to algebroid branches in the plane, Proc. of the AMS 32 (1972), 381-384.
[4] E. Brieskorn, H. Knörrer, Plane Algebraic Curves, Birkhäuser, Boston 1986.
[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.