The Poincaré series of a simple complete ideal of a two-dimensional regular local ring

The aim of this work is to introduce both a classical and a motivic Poincaré series associated with a residually rational simple complete mm-primary ideal ℘℘ of a two-dimensional regular local ring (R,m)(R,m). We describe them in terms of the generators of the value semigroup of ℘℘, and compare them with the Poincaré series arising from a general element ff for ℘℘.