A Recursive Formula and an Estimation for a Specific Exponential Sum

Abstract

Let Fq be a finite field and Fqs be an extension of Fq. Let f (x) ∈ Fq[x] be a polynomial of degree n with gcd(n,q) = 1. We present a recursive formula for evaluating the exponential sum ∑c∈Fqs χ (s)( f (x)). Let a and b be two elements in Fq with a 6= 0, u be a positive integer. We obtain an estimate for the exponential sum ∑c∈F∗ qs χ(s)(acu+bc−1), where χ(s) is the lifting of an additive character χ of Fq. Some properties of the sequences constructed from these exponential sums are provided too. AMS subject classifications: 11T23