Abstract
The problem of describing the Hilbert functions of homogeneous ideals of a commutative polynomial ring containing a fixed monomial ideal is considered. For this purpose the notion of a piecewise lexsegment ideal is introduced generalizing the notion of a lexsegment ideal. It is proved that if is a piecewise lexsegment ideal, then it is possible to describe the Hilbert functions of homogeneous ideals containing in a way similar to that suggested by Macaulay for the situation . Moreover, a generalization of extremal properties of lexsegment ideals is obtained (the inequality for the Betti numbers, behaviour under factorization by homogeneous generic forms).
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