Lattice polytopes having h∗-polynomials with given degree and linear coefficient

2007/05/31 by Benjamin Nill · 1 citation
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #math.CO #msc:52B20

paper · pdf · doi:10.1016/j.ejc.2007.11.002

published as Eur. J. Comb. 29 (2008), 1596-1602 · AMS-LaTeX, 9 pages; introduction improved

arxiv created 2007/11/29 · openalex publication_date 2008/02/21 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

The h^*-polynomial of a lattice polytope is the numerator of the generating function of the Ehrhart polynomial. Let P be a lattice polytope with h^*-polynomial of degree d and with linear coefficient h^*₁. We show that P has to be a lattice pyramid over a lower-dimensional lattice polytope, if the dimension of P is greater or equal to h^*₁ (2d+1) + 4d-1. This result has a purely combinatorial proof and generalizes a recent theorem of Batyrev.

Citations

• Lower bounds on the coefficients of Ehrhart polynomials

Cited by

• Lower bounds on the coefficients of Ehrhart polynomials