Poincare series for several plane divisorial valuations

We compute the (generalized) Poincare series of the multi-index filtration defined by a finite collection of divisorial valuations on the ring O-C2,(0) of germs of functions of two variables. We use the method initially elaborated by the authors and Campillo for computing the similar Poincare series for the valuations defined by the irreducible components of a plane curve singularity. The method is essentially based on the notions of the so-called extended semigroup and of the integral with respect to the Euler characteristic over the projectivization of the space of germs of functions of two variables. The last notion is similar to (and inspired by) the notion of the motivic integration.