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

Abstract: Let $\mathbb{F}<em>{q}$ be a finite field, $\mathbb{F}</em>{q<sup>s}$ be an extension of $\mathbb{F}<em>q$, let $f(x)\in \mathbb{F}_q[x]$ be a polynomial of degree $n$ with $\gcd(n,q)=1$. We present a recursive formula for evaluating the exponential sum $$\sum</em>{c\in \mathbb{F}<em>{q<sup>s}}\chi<sup>{(s)}(f(x))$$. Let $a$ and $b$ be two elements in $\mathbb{F}_q$ with $a\neq 0$, $u$ be a positive integer. We obtain an estimate for the exponential sum $$\sum</em>{c\in \mathbb{F}<sup>*_{q<sup>s}}\chi<sup>{(s)}(ac<sup>u+bc<sup>{-1})$$, where $\chi<sup>{(s)}$ is the lifting of an additive character $\chi$ of $\mathbb{F}_q$. Some properties of the sequences constructed from these exponential sums are provided also.