Poincare Series of Non-divisorial Valuations on Two-dimensional Function Fields

TL;DR

This paper computes the Poincare series for non-divisorial valuations in two-dimensional function fields using dual graphs and generating sequences, extending previous results on divisorial valuations.

Contribution

It introduces a method to compute Poincare series for non-divisorial valuations, linking them to dual graphs and offering an alternative valuation classification.

Findings

Poincare series reflect dual graph data

Extension of Galindo's results to non-divisorial valuations

Provides an alternative classification of valuations

Abstract

We use dual graphs and generating sequences of valuations to compute the Poincare series of non-divisorial valuations on function fields of dimension two. The Poincare series are shown to reflect data from the dual graphs and hence carry equivalent information. This extends Galindo's earlier results on the Poincare series of divisorial valuations and consequently offers an alternative classification of valuations via Poincare series.