Hilbert functions and the theorems of Macaulay and Kruskal–Katona

  • Jürgen HerzogEmail author
  • Takayuki Hibi


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.


Simplicial Complex Polynomial Ring Hilbert Series Hilbert Function Monomial Ideal 
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Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.Fachbereich MathematikUniversität Duisburg-EssenEssenGermany
  2. 2.Department of Pure and Applied Mathematics, Graduate School of Information Science and TechnologyOsaka UniversityToyonakaJapan

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