Poincar\'e series of valuations on functions over a field

TL;DR

This paper computes the Poincaré series associated with valuations on the algebra of formal power series in two variables over a subfield of complex numbers, focusing on curve and divisorial valuations.

Contribution

It provides explicit calculations of the semigroup and classical Poincaré series for valuations on formal power series in two variables.

Findings

Computed the semigroup Poincaré series for valuations.

Derived the classical Poincaré series for valuations.

Enhanced understanding of valuation invariants in algebraic geometry.

Abstract

For a subfield $\K$ of the field $\C$ of complex numbers, we consider curve and divisorial valuations on the algebra $\K[[x,y]]$ of formal power series in two variables with the coeficients in $\K$. We compute the semigroup Poincar\'e series and the classical Poincar\'e series of a valuation.