A Recursive Formula and an Estimation for a Specific Exponential Sum (Communications in Mathematical Research, 2022)
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 recursiveformula for evaluating the exponential sum $\sum_{c\in \mathbb{F}_{q^s}} χ^{(s)}(f(x))$. Let $a$ and $b$ betwo elements in $\mathbb{F}_q$ with a $a≠0$, $u$ be a positive integer. We obtain an estimatefor the exponential sum $\sum_{c\in \mathbb{F}^∗_{q^s}}χ^{(s)} (ac^u+bc^{−1})$, where $χ^{(s)}$ is the lifting of anadditive character $χ$ of $\mathbb{F}_q$. Some properties of the sequences constructed fromthese exponential sums are provided too.
Keywords: Exponential sums; finite fields; Dickson polynomials; sequences.