Let w denote a primitive nth root of unity. The ring Z[w] is dense in the complex plane for n = 5 and n >= 7, so we assume n lies in the set {5} u {7,8,9,...}. Every element of Z[w] is of the form k_1*w + . . . + k_(n-1)*w^(n-1) for integers k_j, 1 <= j <= n-1. For convenience let K denote (k_1,...,k_(n-1)), and call this element of Z[w] by the name w[K]. Given real M > 0, what can be said about sm(M,n) := min {|w[K]| : ||K|| <= M, and w[K] is not 0} in terms of M (where ||K|| is the Euclidean norm)* ? Is there a reasonably sharp asymptotic upper bound for sm(M,n) as M -> oo ? --Dan _______________ * or else for the taxicab norm ||K|| = |k_1|+...+|k_(n-1)| _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele