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 ?1+ ··· + ?n= 0 ofmth roots of unity ?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”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.