On Exponential Sums, Nowton identities and Dickson Polynomials over Finite Fields

Abstract. Let Fq be a finite field, 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 also.