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

Authors:
Benjamin Nill
Affiliations:
Research Group Lattice Polytopes, FU Berlin, Arnimallee 3, 14195 Berlin, Germany
Venue:
European Journal of Combinatorics
Year:
2008

Citing 2
Cited 2

Enumerative combinatorics

Enumerative combinatorics
A monotonicity property of h-vectors and h*-vectors

European Journal of Combinatorics - Special issue dedicated to Bernt Lindstro¨m

Lower bounds on the coefficients of Ehrhart polynomials

European Journal of Combinatorics
Note: Lattice polytopes of degree 2

Journal of Combinatorial Theory Series A

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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 than or equal to h"1^*(2d+1)+4d-1. This result generalizes a recent theorem of Batyrev. As an application we deduce from an inequality due to Stanley that the volume of a lattice polytope is bounded by a function depending only on the degree and the two highest non-zero coefficients of the h^*-polynomial.