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, advancing understanding of their properties and applications in finite field theory.

Contribution

It introduces new recursive formulas and bounds for exponential sums related to polynomials and Dickson polynomials over finite fields.

Findings

Recursive formula for exponential sums of polynomial evaluations.

Estimate for exponential sums involving linear combinations and 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.