Some generalizations of Macaulay's combinatorial theorem for residue rings

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.