## h-Vectors of Gorenstein polytopes

Autor(en): | Bruns, Winfried Roemer, Tim |

Stichwörter: | affine monoid; Ehrhart function; EHRHART POLYNOMIALS; Gorenstein ring; h-Vector; initial ideal; lattice polytope; Mathematics; triangulation; unimodality |

Erscheinungsdatum: | 2007 |

Herausgeber: | ACADEMIC PRESS INC ELSEVIER SCIENCE |

Journal: | JOURNAL OF COMBINATORIAL THEORY SERIES A |

Volumen: | 114 |

Ausgabe: | 1 |

Startseite: | 65 |

Seitenende: | 76 |

Zusammenfassung: | We show that the Ehrhart h-vector of an integer Gorenstein polytope with a regular unimodular triangulation satisfies McMullen's g-theorem; in particular, it is unimodal. This result generalizes a recent theorem of Athanasiadis (conjectured by Stanley) for compressed polytopes. It is derived from a more general theorem on Gorenstein affine normal monoids M: one can factor K [M] (K a field) by a ``long'' regular sequence in such a way that the quotient is still a normal affine monoid algebra. This technique reduces all questions about the Ehrhart h-vector of P to the Ehrhart h-vector of a Gorenstein polytope Q with exactly one interior lattice point, provided each lattice point in a multiple cP, C is an element of N, can be written as the sum of c lattice points in P. (Up to a translation, the polytope Q belongs to the class of reflexive polytopes considered in connection with mirror symmetry.) If P has a regular unimodular triangulation, then it follows readily that the Ehrhart h-vector of P coincides with the combinatorial h-vector of the boundary complex of a simplicial polytope, and the g-theorem applies. (c) 2006 Elsevier Inc. All rights reserved. |

ISSN: | 00973165 |

DOI: | 10.1016/j.jcta.2006.03.003 |

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