The coordination sequence of a lattice L encodes the word-length function with respect to M, 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-th 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-th cyclotomic polytope, the convex hull of the m roots of unity in Rφ(m), and combine results from commutative algebra, number theory, and discrete geometry.
This is joint work with Serkan Hosten (SF State).