Some generalizations of Macaulay's combinatorial theorem for residue rings
Sbornik. Mathematics, Tome 192 (2001) no. 9, pp. 1399-1416
Voir la notice de l'article provenant de la source Math-Net.Ru
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.
@article{SM_2001_192_9_a6,
author = {D. A. Shakin},
title = {Some generalizations of {Macaulay's} combinatorial theorem for residue rings},
journal = {Sbornik. Mathematics},
pages = {1399--1416},
publisher = {mathdoc},
volume = {192},
number = {9},
year = {2001},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SM_2001_192_9_a6/}
}
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/