On a generalized Poincar\'e series of plane valuations

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.