Vanishing sums of mth roots of unity in finite fields

Abstract

In an earlier work, the authors determined all possible weights n for which there exists a vanishing sum ζ1 + · · · + ζn = 0 of mth roots of unity ζi in characteristic 0. In this paper, the same problem is studied in finite fields of characteristic p. For given m and p, results are obtained on integers n0 such that all integers n ≥ n0 are in the "weight set" Wp(m). The main result in this paper guarantees, under suitable conditions, the existence of solutions of xd 1 + · · · + xd n = 0 with all coordinates not equal to zero over a finite field. © 1996 Academic Press, Inc.