Poincare series of collections of plane valuations

TL;DR

This paper extends the understanding of Poincare series for collections of valuations on plane curve singularities, providing a general formula and showing it encodes the topology of the valuation collection.

Contribution

It introduces a formula for the Poincare series of general valuations and proves it determines the topology of the valuation collection.

Findings

Derived a general formula for Poincare series of valuations

Proved Poincare series encodes the topology of valuations

Extended previous results to more general collections

Abstract

In earlier papers there were given formulae for the Poincare 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 Poincare series of a collection of divisorial valuations determines the topology of the collection of divisors. Here we give a formula for the Poincare 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 Poincare series determines the topology of the collection.