Ehrhart polynomials of integral simplices with prime volumes

Source: arXiv

ABSTRACT For an integral convex polytope $\Pc \subset \RR^N$ of dimension $d$, we call
$\delta(\Pc)=(\delta_0, \delta_1,..., \delta_d)$ the $\delta$-vector of $\Pc$
and $\vol(\Pc)=\sum_{i=0}^d\delta_i$ its normalized volume. In this paper, we
will establish the new equalities and inequalities on $\delta$-vectors for
integral simplices whose normalized volumes are prime. Moreover, by using
those, we will classify all the possible $\delta$-vectors of integral simplices
with normalized volume 5 and 7.

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    ABSTRACT: For any lattice polytope $P$, we consider an associated polynomial $\bar{\delta}_{P}(t)$ and describe its decomposition into a sum of two polynomials satisfying certain symmetry conditions. As a consequence, we improve upon known inequalities satisfied by the coefficients of the Ehrhart $\delta$-vector of a lattice polytope. We also provide combinatorial proofs of two results of Stanley that were previously established using techniques from commutative algebra. Finally, we give a necessary numerical criterion for the existence of a regular unimodular lattice triangulation of the boundary of a lattice polytope. Comment: 11 pages. v2: minor changes, more detailed proof of Lemma 2.12. To appear in Trans. Amer. Math. Soc
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