Hilbert functions and the theorems of Macaulay and Kruskal–Katona
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Chapter 6 offers basic material on combinatorics of monomial ideals. First we recall the concepts of Hilbert functions and Hilbert polynomials, and their relationship to the f-vector of a simplicial complex is explained. We study in detail the combinatorial characterization of Hilbert functions of graded ideals due to Macaulay together with its squarefree analogue, the Kruskal–Katona theorem, which describes the possible face numbers of simplicial complexes. Lexsegment ideals as well as squarefree lexsegment ideals play the key role in the discussion.
KeywordsSimplicial Complex Polynomial Ring Hilbert Series Hilbert Function Monomial Ideal
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