Abstract:
The problem of describing the Hilbert functions of
homogeneous ideals of a commutative polynomial ring
containing a fixed monomial ideal $I$ 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 $I$ is a piecewise lexsegment ideal, then it is possible to describe the Hilbert functions
of homogeneous ideals containing $I$ in a way similar to that
suggested by Macaulay for the situation $I=0$.
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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