ON VANISHING SUMS OF m TH ROOTS OF UNITY IN FINITE FIELDS

T. Y. LAM K. H. LEUNG

Abstract. In an earlier work, the authors have determined all possible weights n for which there exists a vanishing sum ζ_{1} + ⋯ + ζ_{n} = 0 of mth roots of unity ζ_{i} 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 n_{0} such that all integers n ≥ n_{0} are in the “weight set” W_{p}(m). The main result (1.3) in this paper guarantees, under suitable conditions, the existence of solutions of x_{1}^{d} +⋯+x_{n}^{d} = 0 with all coordinates not equal to zero over a finite field.