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Oberseminar SoSe 2009 Zusammenfassung des Vortrags von S. M. Gusein-Zade (Moskau)

Poincaré series of collections of plane valuations

In earlier papers with A. Campillo and F. Delgado, there were given formulae for the Poincaré series of multi-index filtrations on the ring of germs of functions of two variables defined by collections of valuations corresponding to (reducible) plane curve singularities and by collections of divisorial ones. It was shown that the Poincaré series of a collection of divisorial valuations determines the topology of the collection of divisors. Here we give a formula for the Poincaré series of a general collection of valuations on the ring of germs of functions of two variables centred at the origin and prove a generalization of the statement that the Poincaré series determines the topology of the collection.