06-30

Notes on the roots of Ehrhart polynomials

by Bey, C.; Henk, M.; Wills, J. M.

 

Preprint series: 06-30, Preprints

MSC:
52C07 Lattices and convex bodies in $n$ dimensions, See Also {11H06, 11H31, 11P21}
11H06 Lattices and convex bodies, See also {11P21, 52C05, 52C07}

 

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


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