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+ ··· + ζn= 0 ofmth roots of unity ζiin 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”Wp(m). The main result in this paper guarantees, under suitable conditions, the existence of solutions ofxd1 + ··· +xdn= 0with all coordinates not equal to zeroover a finite field.