On a generalized Poincar\'e series of plane valuations

TL;DR

This paper extends the concept of Poincaré series for plane valuations to a generalized motivic version, providing equations for these series in a broader algebraic setting.

Contribution

It introduces equations for the generalized motivic Poincaré series of plane valuations, expanding previous definitions to a new algebraic context.

Findings

Derived equations for the generalized Poincaré series.

Extended the series to valuations on a subfield of complex numbers.

Connected the series to earlier computed cases for specific valuations.

Abstract

Earlier, there were defined two generalized (``motivic'') versions of the Poincar\'e series of a collection of plane valuations on the algebra ${\mathcal O}_{{\mathbb C}^2,0}$ of germs of holomorphic functions in two variables. One of them was defined as an integral with respect to the generalized Euler characteristic over the projectivization of the extended semigroup of the collection. One has a natural version of it for valuations on the algebra ${\mathcal E}_{{\mathbb K}^2,0}$ of germs of holomorphic functions in two variables whose Taylor coefficients are from a fixed subfield ${\mathbb K}$ of the field ${\mathbb C}$ of complex numbers. In this setting the usual Poincar\'e series were computed for one plane curve or divisorial valuation on ${\mathcal E}_{{\mathbb K}^2,0}$. We give equations for the corresponding generalized Poincar\'e series.