The coordinator polynomial of some cyclotomic lattices
Let \(L\) be a lattice of rank \(r\), and let \(M\) be a subset that generates \(L\) as a monoid. Let \(S(n)\) denote the number of elements in \(L\) with \textit{word length} \(n\) with respect to \(M\), that is, the number of lattice elements that are expressed as a sum from \(M\) with a minimal number of \(n\) terms. The \textit{growth series} \(G\) of \((L, M)\) is the generating function \( G (x) := \sum_{ n \geq 0 } S (n) \, x^n\). It was proved by Benson and Billington that \(G (x) = {{ h(x) } \over { (1-x)^r }}\) where \(h(x)\), the \textit{coordinator polynomial} of \(L\), is a polynomial of degree \(\leq r\). The paper under review considers the case \(L = {\mathbb Z}[e^{ 2\pi i/m }]\), the ring of integers in the cyclotomic field of order \(m\), where \(M\) consists of all \(m\)th roots of unity. The study of growth series of these cyclotomic lattices was initiated in [``Conjectures on the size of constellations constructed from direct sums of PSK kernels'', Lect. Notes Comput. Sci. 1719, 420--429 (1999; Zbl 1016.94510)], where \textit{M. G. Parker} offered a series of conjectures about the coordinator polynomial of \({\mathbb Z}[e^{ 2\pi i/m }]\). The paper under review studies the problem of computing the coordinator polynomial \(h_m(x)\) for \({\mathbb Z}[e^{ 2\pi i/m }]\) from the viewpoint of unimodular triangulations and Ehrhart polynomials (i.e., lattice-point enumeration functions) of lattice polytopes. The authors illustrate how one of Parker's theorems (namely, an explicit formula for \(h_m(x)\) when \(m\) is prime) can be reproved from this point of view and prove one of Parker's conjecture, namely an explicit formula for \(h_m(x)\) in the case \(m = 2p\), where \(p\) is an odd prime. All but one of the remaining conjectures of Parker were subsequently proved in [\textit{M. Beck} and \textit{S. Hoşten}, ``Cyclotomic polytopes and growth series of cyclotomic lattices'', Math. Res. Lett. 13, No. 4, 607--622 (2006; Zbl 1175.11030)].
scientific article; zbMATH DE number 5077682