This article is about certain properties of higher order polars of a germ of complex analytic curve , say defined near the origin of by a convergent power series . Recall that the first -polar of is the germ defined by the analytic equation . If the -axis is not an irreducible component of then is not identically zero, so is defined. Then it is not hard to see that is not an irreducible component of the first polar and the process may be iterated, getting the second polar . Repeating we may define -polars of any order . The main result obtained in this paper is of the following type. Let be an irreducible germ of plane complex analytic curve (a branch) defined by , having a Puiseux series , with characteristic exponents , so that if then . If , let be an integer such that . Then the germ admits a decomposition , where the intersection multiplicities , can be expressed in terms of (explicit formulas are given). Moreover, for all , each branch of has a Puiseux series which coincides with up to terms of degree (but not for higher degree terms).
The author shows some interesting consequences of this rather technical result. For instance:
(a) an explicit formula for the intersection number , where is any branch of , in terms of , ;
(b) with notation as above, the numbers coincide with the first polar invariants of (introduced by Teissier);
(c) a new proof of a formula expressing in terms of , the ’s and the ’s, obtained before by A. Dickenstein and C. Sessa [Manuscr. Math. 37, 1-9, (1982; Zbl 0496.14018)]. A special case of (a) gives us a new proof of a result of M. Merle [Invent. Math. 41, 103-111 (1997; Zbl 0371.14003)], expressing the polar invariants of in terms of its characteristic exponents. By means of examples the author shows that similar results for reducible germs are no longer valid.