On the Hilbert function of one-dimensional local complete intersections

The Hilbert function of standard graded algebras are well understood by Macaulay's theorem and very little is known in the local case, even if we assume that the local ring is a complete intersection. An extension to the power series ring $R$ of the theory of Gr\"{o}bner bases (w.r.t. local degree orderings) enable us to characterize the Hilbert function of one dimensional quadratic complete intersections $A=R/I$, and we give a structure theorem of the minimal system of generators of $I$ in terms of the Hilbert function. We find several restrictions for the Hilbert function of $A$ in the case that $I$ is a complete intersection of type $(2,b). $ Conditions for the Cohen-Macaulyness of the associated graded ring of $A$ are given.