Article

Ehrhart polynomials of integral simplices with prime volumes

07/2011;
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: The characterization of lattice polytopes based upon information about their Ehrhart $h^*$-polynomials is a difficult open problem. In this paper, we finish the classification of lattice polytopes whose $h^*$-polynomials satisfy two properties: they are palindromic (so the polytope is Gorenstein) and they consist of precisely three terms. This extends the classification of Gorenstein polytopes of degree two due to Batyrev and Juny. The proof relies on the recent characterization of Batyrev and Hofscheier of empty lattice simplices whose $h^*$-polynomials have precisely two terms. Putting our theorem in perspective, we give a summary of these and other existing results in this area.

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