In an earlier work, the authors have 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 (1.3) in this paper guarantees,
under suitable conditions, the existence of solutions of x1d+⋯+xnd=0
with all coordinates not equal to zero over a finite field