Cyclotomic Polytopes and Growth Series of Cyclotomic Lattices

The coordination sequence of a lattice L encodes the word-length function with respect toM , a set that generates L as a monoid. We investigate the coordination sequence of the cyclotomic lattice L = Z[ζm], where ζm is a primitive m th root of unity and where M is the set of all m 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 m cyclotomic polytope, the convex hull of the m roots of unity in R.