On vanishing sums of m th roots of unity in finite fields

1996/05/02 by Lam, T. Y., Leung, K. H.
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.math/9605216

Abstract

In an earlier work, the authors have determined all possible weights n for which there exists a vanishing sum ζ₁+⋯ +ζ_n=0 of mth 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 n₀ such that all integers n≥ n₀ 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 x₁^d+⋯+x_n^d=0 with all coordinates not equal to zero over a finite field.

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