Hilbert function, generalized Poincaré series and topology of plane valuations

Abstract

To a multi-index filtration (say, on the ring of germs of functions on a germ of a complex analytic variety) one associates several invariants: the Hilbert function, the Poincare series, the generalized Poincare series, and the generalized semigroup Poincare series. The Hilbert function and the generalized Poincare series are equivalent in the sense that each of them determines the other one. We show that for a filtration on the ring of germs of holomorphic functions in two variables defined by a collection of plane valuations both of them are equivalent to the generalized semigroup Poincare series and determine the topology of the collection of valuations, i.e. the topology of its minimal resolution.