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Ehrhart Polynomials and Successive Minima

Part of: Polytopes and polyhedra Discrete geometry

Published online by Cambridge University Press:  21 December 2009

Martin Henk ,
Achill Schürmann  and
Jörg M. Wills
Martin Henk
Affiliation:
Universität Magdeburg, IAG, Universitätsplatz 2, D-39106 Magdeburg, Germany. E-Mail: henk@math.uni-magdeburg.de
Achill Schürmann
Affiliation:
Universität Magdeburg, Institut für Algebra und Geometrie, Universitätsplatz 2, D-39106 Magdeburg, Germany. E-mail: achill@math.uni-magdeburg.de
Jörg M. Wills
Affiliation:
Universität Siegen, Mathematisches Institut, ENC, D-57068 Siegen, Germany. E-mail: wills@mathematik.uni-siegen.de
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Abstract

The Ehrhart polynomials for the class of 0-symmetric convex lattice polytopes in Euclidean n-space ℝn are investigated. It turns out that the roots of the Ehrhart polynomial and Minkowski's successive minima of such polytopes are closely related by their geometric and arithmetic means. It is also shown that the roots of the Ehrhart polynomials of lattice n-polytopes with or without interior lattice points differ essentially. Furthermore, the structure of the roots in the planar case is studied. Here it turns out that their distribution reflects basic properties of lattice polygons.

MSC classification

Secondary: 52C07: Lattices and convex bodies in $n$ dimensions 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry)
Type
Research Article
Information
Mathematika , Volume 52 , Issue 1-2 , December 2005 , pp. 1 - 16
Copyright
Copyright © University College London 2005

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