Geodesic
Some generalizations of Macaulay's combinatorial theorem for residue rings
Sbornik. Mathematics, Tome 192 (2001) no. 9, pp. 1399-1416 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     author = {D. A. Shakin},
     title = {Some generalizations of {Macaulay's} combinatorial theorem for residue rings},
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     url = {http://geodesic.mathdoc.fr/item/SM_2001_192_9_a6/}
}
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D. A. Shakin. Some generalizations of Macaulay's combinatorial theorem for residue rings. Sbornik. Mathematics, Tome 192 (2001) no. 9, pp. 1399-1416. http://geodesic.mathdoc.fr/item/SM_2001_192_9_a6/

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