On Sums of Roots of Unity

Abstract

We make two remarks on linear forms over ${\cal Z}$ in complex roots of unity. First we show that a Liouville type lower bound on the absolute value of a nonvanishing form can be derived from the time complexity upper bound on Tarski algbera. Second we exhibit an efficient randomized algorithm for deciding whether a given form vanishes. In the special case where the periods of the roots of unity are mutually coprime, we can eliminate randomization. This efficiency is surprising given the doubly exponential smallness of the Liouville bound.