ИСТИНА

It was shown that for a collection of curve valuations on (C^2 , 0) and also on (E_8 , 0) (where E_8 is the surface singularity of the E_8-type) the Poincaré series coincides with the Alexander polynomial of the corresponding algebraic link. Moreover, for the plane case (C^2 , 0) the Poincaré series determines the minimal embedded resolution of the curve (and thus the topology of the link), whereas for the E_8 -singularity one has few explicitely described exceptions. In the discussed cases the Alexander polynomial can be expressed as an integral with respect to the Euler characteristic over the space of divisors on the singularity. The coincidence of the Poincaré series with the Alexander polynomial is related with the fact that on (C^2 , 0) and (E_8 , 0) all divisors are Cartier. We define a natural generalization of the Alexander polynomial of an algebraic link on other surface singularity (the Weil-Poincaré series) as the integral over the space of Weil divisors. The Weil-Poincaré series is a power series with rational exponents. We discuss to which extent the Weil-Poincaré series determines the topology of the curve for rational double point surface singularities. We give analogous statements for collections of divisorial valuations. The talk is based on a joint work with A. Campillo and F. Delgado.