Lattice points, Dedekind sums, and Ehrhart polynomials of lattice polyhedra
type: Item
instance of: scholarly article
title: Lattice points, Dedekind sums, and Ehrhart polynomials of lattice polyhedra
zbMATH Open document ID: 1021.52010
author: Beifang Chen
published in: Discrete \& Computational Geometry
publication date: 2003-03-17T00:00:00Z
review text: If \(P\) is a convex polytope with vertices in the integer lattice \(\mathbb{Z}^N\) in euclidean space \(\mathbb{R}^N\), and \(L(P,m)\) and \(L(P^0,m)\) denote the number of lattice points in \(mP\) and \(mP^0\), respectively (with \(P^0\) the relative interior of \(P\)), then it is well known that \(L(P,m)\) and \(L(P^0,m)\) are polynomials in the positive integer variable \(m\), which satisfy the reciprocity law \[ L(P^0,m) = (-1)^{\dim P}L(P,-m). \] Few of the coefficients \(c_s(P)\) of the polynomial \(L(P,m)\) are known in general terms; if \(\dim P = n\), then \(c_n(P)\) is lattice volume, \(c_{n-1}(P)\) is half the lattice surface area, and \(c_0(P) = \chi(P)\) is the Euler characteristic. In this paper, the author gives several descriptions of all the coefficients when \(P =\sigma\), a simplex. Let me just describe the simplest formula. If \(\sigma\) has vertices \(\alpha_0,\dots,\alpha_n\) in \(\mathbb{Z}^N\), the determined set of \(\sigma\) at its vertex \(\alpha_0\) is \[ D(\sigma,\alpha_0)=\left\{\sum_{j=1}^n u_j(\alpha_j-\alpha_0)\in\mathbb{Z}^n: 0\leq u_j < 1\right\}. \] If \(\gamma=\sum_{j=1}^n u_j(\alpha_j-\alpha_0)\in D(\sigma,\alpha_0)\), write \(|\gamma|=\sum_{j=1}^nu_j\). Then the \(j\)th coefficients of \(L(\sigma^0,m)\) and \(L(\sigma,m)\) are \[ c_j(\sigma^0) = \frac{1}{n!}\sum_{\gamma\in D(\sigma,\alpha_0)}s_{n-j}(\lceil|\gamma |\rceil-1,\dots,\lceil|\gamma |\rceil -n), \] \[ c_j(\sigma) = \frac{1}{n!}\sum_{\gamma\in D(\sigma,\alpha_0)}s_{n-j}(1-\lceil|\gamma |\rceil,\dots,n-\lceil|\gamma |\rceil), \] where \(s_{n-j}\) is the \((n-j)\)th symmetric function. (The two formulae are equivalent, since \(c_j(\sigma^0) = (-1)^{n-j}c_j(\sigma)\) by the reciprocity law). The author also produces generalizations of his formulae, for instance, for simplices \(\sigma\) such that \(\lambda\sigma\) is a lattice polytope for some \(\lambda > 0\). In principle, knowing the lattice point enumerator for simplices gives it for all polytopes, since from a trian…
Mathematics Subject Classification ID: 52B20
zbMATH DE Number: 1883406
zbMATH Keywords: Ehrhart polynomials
zbMATH Keywords: lattice polyhedra
zbMATH Keywords: convex polytope
zbMATH Keywords: lattice points
zbMATH Keywords: reciprocity law
zbMATH Keywords: enumeration formulae
zbMATH Keywords: Dedekind sums
reviewed by: Peter McMullen
MaRDI profile type: Publication
full work available at URL: s00454-002-2759-7
OpenAlex ID: W2077818624
DOI: 10.1007/S00454-002-2759-7
description: scientific article; zbMATH DE number 1883406