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Higher order polar germs. (English) Zbl 0985.14012

This article is about certain properties of higher order polars of a germ of complex analytic curve C, say defined near the origin of 2 by a convergent power series f(x,y). Recall that the first x-polar of f is the germ P x (f) defined by the analytic equation f/y=0. If the y-axis Y is not an irreducible component of C then f/y is not identically zero, so P x (f) is defined. Then it is not hard to see that Y is not an irreducible component of the first polar and the process may be iterated, getting the second polar P x (2) (f). Repeating we may define x-polars P x (r) (f) of any order r. The main result obtained in this paper is of the following type. Let C be an irreducible germ of plane complex analytic curve (a branch) defined by f=0, having a Puiseux series s= i1 a i x i/n , with characteristic exponents m 1 /n,,m k /n, so that if n i :=GCD{n,m 1 ,,m i } then n k =1. If 1r<n, let u(r) be an integer such that n u(r)-1 >rn u(r) . Then the germ P x (r) (f) admits a decomposition P x (r) (f)=D 1 (r) ++D u(r) (r) , where the intersection multiplicities [D i (r) ·Y], 1i<u(r) can be expressed in terms of r,n,n 1 ,,n u(r)-1 (explicit formulas are given). Moreover, for all i, each branch of D i (r) has a Puiseux series which coincides with s up to terms of degree m i -1/n (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 [C·B], where B is any branch of P x (r) i (f), in terms of [B,Y], n,n 1 ,,n i-1 ,m 1 ,,m i ;

(b) with notation as above, the numbers [C·B]/[Y·B] coincide with the first u(r) polar invariants of C (introduced by Teissier);

(c) a new proof of a formula expressing [C·P x (r) (f)] in terms of n, the n i ’s and the m j ’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 C in terms of its characteristic exponents. By means of examples the author shows that similar results for reducible germs C are no longer valid.


MSC:
14H20Singularities, local rings
32B10Germs of analytic sets, local parametrization
32S05Local singularities (analytic spaces)
14B05Singularities (algebraic geometry)
32S10Invariants of analytic local rings