Lattice points, Dedekind sums, and Ehrhart polynomials of lattice polyhedra
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 triangulation one can apply either the inclusion-exclusion property for closed simplices, or simple addition for relative interiors of simplices; the latter is remarked by the author. Finally, it should be noted that interesting applications of the enumeration formulae involving Dedekind sums are given for special simplices in dimensions two and three.
scientific article; zbMATH DE number 1883406