Poincare series of collections of plane valuations
A. Campillo, F. Delgado, S.M. Gusein-Zade, F. Hernando

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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAlgebraic Geometry and Number Theory · Commutative Algebra and Its Applications · Meromorphic and Entire Functions
