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A note on evaluations of some exponential sums

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Moisio M.

Acta Arithmetica

2000

tom 93

nr 2

117-119

1. Introduction. The recent article [1] gives explicit evaluations for exponential sums of the form $S(a,p^{α}+1) := ∑_{x∈𝔽_q} χ(ax^{p^{α}+1})$ where χ is a non-trivial additive character of the finite field $𝔽_q$, $q = p^e$ odd, and $a ∈ 𝔽*_q$. In my dissertation [5], in particular in [4], I considered more generally the sums S(a,N) for all factors N of $p^{α}+1$. The aim of the present note is to evaluate S(a,N) in a short way, following [4]. We note that our result is also valid for even q, and the technique used in our proof can also be used to evaluate certain sums of the form $∑_{x∈𝔽_q} χ(ax^{p^{α}+1} + bx)$.

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1 Acta Arithmetica

1 Moisio M.

1 2000

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A note on evaluations of some exponential sums