Real Poincaré series of a plane divisorial valuation

Earlier, there was computed the Poincaré 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 and therefore determines the embedded topology of the the collection of the curves. Recently, the authors defined two versions of the Poincaré series of a valuation or of a collection of valuations in the real setting. These two Poincaré series (and also the so called semigroup Poincaré series, which essentially makes sense only for one valuation) were computed for one plane curve valuation. Here we compute them for a plane divisorial valuation.