Real Poincar\'e series of a plane divisorial valuation

TL;DR

This paper extends the concept of Poincaré series to real plane divisorial valuations, providing explicit computations and exploring their properties, which enhances understanding of valuations in complex geometry.

Contribution

The authors compute the Poincaré series for a plane divisorial valuation in the real setting, expanding previous work limited to complex valuations.

Findings

Computed Poincaré series for real plane divisorial valuations.

Established connections with algebraic links and Alexander polynomials.

Extended the framework of valuation invariants to the real case.

Abstract

Earlier, there was computed the Poincar\'e series of a valuation or of a collection of valuations on the ring of germs of holomorphic functions in two variables. For a collection of several plane curve valuations it appeared to coincide with the Alexander polynomial of the corresponding algebraic link. Recently, the authors defined Poincar\'e series of a valuation or of a collection of valuations in the real setting. (Actually, there were defined three versions of them, however, one of them was found to be ``not computable''.) These two Poincar'e series were computed for one plane curve valuation. Here we compute them for a plane divisorial valuation.