On the value-semigroup of a simple complete ideal in a two-dimensional regular local ring

Let R be a two-dimensional regular local ring with maximal ideal m, and let ℘ be a simple complete m-primary ideal which is residually rational. Let R0 := R $ R1 $ · · · $ Rr be the quadratic sequence associated to ℘, let Γ℘ be the value-semigroup associated to ℘, and let (ej(℘))0≤j≤r be the multiplicity sequence of ℘. We associate to ℘ a sequence (γi(℘))0≤i≤g of natural integers which we call the formal characteristic sequence of ℘. We show that the value-semigroup of ℘, the multiplicity sequence of ℘ and the formal characteristic sequence of ℘ are equivalent data; furthermore, we give a new proof of the fact that Γ℘ is symmetric. In order to prove these results, we use the Hamburger-Noether tableau of ℘ [ cf. [3] ]; we also give a formula for c℘, the conductor of Γ℘, in terms of the Hamburger-Noether tableau of ℘