Hilbert function of local Artinian level rings in codimension two

In this paper we characterize the possible Hilbert functions of a local Artinian level ring of given type and socle degree in the codimension two case. In particular, given an “admissible” numerical function h, an effective method is given to produce a zero-dimensional ideal I of R=k〚x,y〛 such that R/I is a local Artinian level ring having h as Hilbert function. As a consequence we recover the well-known result proved by Briançon and Iarrobino concerning the Hilbert functions of a complete intersection of height two.We extend notions of standard or Gröbner bases from the graded to the local algebra context. The principal tools concern the concepts of enhanced standard basis and generic initial ideals which are defined for an ideal in the formal power series ring R=k〚x1,…,xn〛.