# A factorization theorem for the polar of a curve with two branches

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A factorization theorem for the polar of a curve with two branches COMPOSITIO MATHEMATICA FÉLIXDELGADO DE LAMATA A factorization theorem for the polar of a curve with two branches Compositio Mathematica, tome 92, no 3 (1994), p. 327-375. <http://www.numdam.org/item?id=CM_1994__92_3_327_0> © Foundation Compositio Mathematica, 1994, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/legal.php). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 327 A factorization theorem for the polar of a curve with two branches FÉLIX DELGADO DE LA MATA Universidad Valladolid, Depto. de Algebra, Fac. Ciencias, 47005 Valladolid, Spain Received 12 July 1990; accepted in final form 14 June 1993 Compositio Mathematica 92: 327-375, 1994. © 1994 Kluwer Academic Publishers. Printed in the Netherlands. Introduction Let f E C[[X, Y]] be a reduced formal series over the complex field C (i.e. f = II i f where the fis are irreducible and fi *fj if i -# j), and let h E C[[X, Y]] be a regular parameter (i.e. h defines a nonsingular plane algebroid curve). The polar of f with respect to h, P( f, h), is the algebroid curve defined by: Examples by Pham show that the topological type of P( f , h) depends on the analytic type of C, the curve defined by f = 0, and not only on its topological type, even for h transversal to f However we may wonder what information of P( f, h) depends on the topological type of C. Roughly speaking: Assuming the topological type of C fixed, What can we say about the topological type of P(f, h) The best results about this question have been obtained by Lê,

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