2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/130837The coordination sequence of a lattice $Ł$ encodes the word-length function with respect to $M$, a set that generates $Ł$ as a monoid. We investigate the coordination sequence of the cyclotomic lattice $Ł= \Z[ζ_m]$, where $ζ_m$ is a primitive $mþ$ root of unity and where $M$ is the set of all $mþ$ roots of unity. We prove several conjectures by Parker regarding the structure of the rational generating function of the coordination sequence; this structure depends on the prime factorization of $m$. Our methods are based on unimodular triangulations of the $mþ$ cyclotomic polytope, the convex hull of the $m$ roots of unity in $\R^{ϕ(m)}$, with respect to a canonically chosen basis of $Ł$.15 pagesCombinatoricsCommutative AlgebraNumber Theory52C07, 13D40, 11H06; 14M25, 52B20text