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

TL;DR

This paper develops recursive formulas and estimates for exponential sums over finite fields involving polynomials and Dickson polynomials, providing new tools for analyzing their properties and related sequences.

Contribution

It introduces recursive evaluation methods for exponential sums and estimates for sums involving polynomial and Dickson polynomial expressions over finite fields.

Findings

Recursive formulas for exponential sums over finite fields.

Estimates for sums involving polynomials and their inverses.

Properties of sequences derived from these exponential sums.

Abstract

Let $\mathbb{F}_{q}$ be a finite field, $\mathbb{F}_{q^s}$ be an extension of $\mathbb{F}_q$, let $f(x)\in \mathbb{F}_q[x]$ be a polynomial of degree $n$ with $\gcd(n,q)=1$. 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 $a\neq 0$, $u$ 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 also.