On the polar curves of a reduced plane curve (Q2766356)

Let \(C\) be a reduced germ of an analytic plane curve with an isolated singularity at the origin and let \(P(\tau)\) be the generic polar curve of \(C\). Thinking of \(C\) in terms of Puiseux expansions of its branches, the author states the following decomposition theorem:NEWLINENEWLINEThe branches of \(P(\tau)\) are divided into bunches, all the branches of the same bunch having the same contact with each branch of \(C\); as a consequence, the first several terms of the Puiseux expansion of a branch of \(P(\tau)\) are independent of \(\tau\), they depend only on the bunch to which the branch belongs.NEWLINENEWLINEThese facts produce numerical invariants of the topological type of \(C\) [see \textit{M. Merle}, Invent. Math. 41, 103--111 (1977; Zbl 0371.14003)] for the irreducible case; in particular they imply a theorem of \textit{Lê Dung Trang, F. Michel} and \textit{C. Weber} [Compos. Math. 72, No.1, 87--113 (1989; Zbl 0705.32021)] about the behaviour of polar curves in an embedded resolution of the singularities of \(C\). The author studies how the topological type of \(C\) depends on the contact between branches of \(P(\tau)\) and branches of \(C\); she introduces the so-called matrix of partial polar invariants, which depends only on the topological type of \(C\) and which determines it. Some results of this paper were announced by the author earlier [\textit{E. García Barroso}, C. R. Acad. Sci., Paris, Sér. I, Math. 326, No. 1, 59--62 (1998; Zbl 0954.14022)].