Vanishing Sums ofmth Roots of Unity in Finite Fields

In an earlier work, the authors determined all possible weightsnfor which there exists a vanishing sum ζ₁+ · · · + ζ_n= 0 ofmth roots of unity ζ_iin characteristic 0. In this paper, the same problem is studied in finite fields of characteristicp. For givenmandp, results are obtained on integersn₀such that all integersn≥n₀are in the “weight set”W_p(m). The main result in this paper guarantees, under suitable conditions, the existence of solutions ofx^d1 + · · · +x^d_n= 0with all coordinates not equal to zeroover a finite field.