A Recursive Formula and an Estimation for a Specific Exponential Sum

Abstract

Let $\mathbb{F}_q$ be a finite field and $\mathbb{F}_{q^s}$ be an extension of $\mathbb{F}_q$. Let $f(x)\in \mathbb{F}_q[x]$ be a polynomial of degree $n$ with ${\rm gcd}(n,q) = 1$. We present a recursive formula for evaluating the exponential sum $\sum_{c\in \mathbb{F}_{q^s}} \chi^{(s)}(f(x))$. Let $a$ and $b$ be two elements in $\mathbb{F}_q$ with a $a≠0$, $u$ be a positive integer. We obtain an estimate for the exponential sum $\sum_{c\in \mathbb{F}^∗_{q^s}}\chi^{(s)} (ac^u+bc^{−1})$, where $\chi^{(s)}$ is the lifting of an additive character $\chi$ of $\mathbb{F}_q$. Some properties of the sequences constructed from these exponential sums are provided too.