Some generalizations of Macaulay's combinatorial theorem for residue rings

Abstract

The problem of the characterization of the Hilbert functions of homogeneous ideals of a polynomial ring containing a fixed monomial ideal I is considered. Macaulay's result for the polynomial ring is generalized to the case of residue rings modulo some monomial ideals. In particular, necessary and sufficient conditions on an ideal I for Macaulay's theorem to hold are presented in two cases: when I is an ideal of the polynomial ring in two variables and when I is generated by a lexsegment. Macaulay's theorem is also proved for a wide variety of cases when I is generated by monomials in the two largest variables in the lexicographic ordering. In addition, an equivalent formulation of Macaulay's theorem and conditions on the ideal I required for a generalization of this theorem are given.