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INTERMEDIATE SUMS ON POLYHEDRA: COMPUTATION AND REAL EHRHART THEORY

Part of: Combinatorics Discrete mathematics in relation to computer science Polytopes and polyhedra Discrete geometry

Published online by Cambridge University Press:  05 September 2012

V. Baldoni ,
N. Berline ,
M. Köppe  and
M. Vergne
V. Baldoni
Affiliation:
Dipartimento di Matematica, Università degli Studi di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Roma, Italy (email: baldoni@mat.uniroma2.it)
N. Berline
Affiliation:
Centre de Mathématiques Laurent Schwartz, École Polytechnique, 91128 Palaiseau Cedex, France (email: nicole.berline@math.polytechnique.fr)
M. Köppe
Affiliation:
Department of Mathematics, University of California, Davis, One Shields Avenue, Davis, CA 95616, U.S.A. (email: mkoeppe@math.ucdavis.edu)
M. Vergne
Affiliation:
Université Paris 7 Diderot, Institut Mathématique de Jussieu, 16 rue Clisson, 75013 Paris, France (email: vergne@math.jussieu.fr)
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Abstract

We study intermediate sums, interpolating between integrals and discrete sums, which were introduced by A. Barvinok in [Computing the Ehrhart quasi-polynomial of a rational simplex. Math. Comp. 75 (2006), 1449–1466]. For a given polytope 𝔭 with facets parallel to rational hyperplanes and a rational subspace L, we integrate a given polynomial function h over all lattice slices of the polytope 𝔭 parallel to the subspace L and sum up the integrals. We first develop an algorithmic theory of parametric intermediate generating functions. Then we study the Ehrhart theory of these intermediate sums, that is, the dependence of the result as a function of a dilation of the polytope. We provide an algorithm to compute the resulting Ehrhart quasi-polynomials in the form of explicit step-polynomials. These formulas are naturally valid for real (not just integer) dilations and thus provide a direct approach to real Ehrhart theory.

MSC classification

Secondary: 05A15: Exact enumeration problems, generating functions 52C07: Lattices and convex bodies in $n$ dimensions 68R05: Combinatorics 68U05: Computer graphics; computational geometry 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry)
Type
Research Article
Information
Mathematika , Volume 59 , Issue 1 , January 2013 , pp. 1 - 22
Copyright
Copyright © University College London 2012

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