### 2006-30

#### Notes on the roots of Ehrhart polynomials

Series: 2006-30, Preprints

MSC:
52C07 Lattices and convex bodies in $n$ dimensions
11H06 Lattices and convex bodies

Abstract:
We determine lattice polytopes of smallest volume with a given
number of interior lattice points. We show that the Ehrhart
polynomials of those with one interior lattice point have
largest roots with norm of order $n^2$, where $n$ is the
dimension. This improves on the previously best known bound
$n$ and complements a recent result of Braun [8] where it is
shown that the norm of a root of a Ehrhart polynomial is at
most of order $n^2$.
For the class of 0-symmetric lattice polytopes we present
a conjecture on the smallest volume for a given number of
interior lattice points and prove the conjecture for
crosspolytopes.
We further give a characaterisation of the roots of Ehrhart
polynomials in the 3-dimensional case and we classify for
$n \leq 4$ all lattice polytopes whose roots of their
Ehrhart polynomials have all real part -1/2. These polytopes
belong to the class of reflexive polytopes.

Keywords:
Lattice polytopes, Ehrhart polynomial, reflexive polytopes