On vanishing sums of $\,m\,$th roots of unity in finite fields

Abstract

In an earlier work, the authors have determined all possible weights $n$ for which there exists a vanishing sum $ζ_1+\cdots +ζ_n=0$ of $m$th 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 $n\geq n_0$ 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 $x_1^d+\cdots+x_n^d=0$ with all coordinates not equal to zero over a finite field.