Hilbert-Samuel sequences of homogeneous finite type

Commutative local Artinian algebras over a field can be thought of as algebras of functions on a non-reduced scheme that has only one point. This paper deals with the problem of classification of such algebras. They have a natural invariant: Hilbert-Samuel sequence. We say that a Hilbert-Samuel sequence is of homogeneous finite type, if it corresponds to a finite number of isomorphism classes of graded local algebras. We find all the Hilbert-Samuel sequences of homogeneous finite type in the case of algebras that can be generated by two elements. For all other sequences we prove the existence of infinite families of graded algebras that admit two generators.