Third Power of the Reversed Dickson Polynomial over Finite Fields

TL;DR

This paper evaluates the third power sum of reversed Dickson polynomials over finite fields, providing new conditions for these polynomials to be permutation polynomials.

Contribution

It extends previous work by explicitly calculating the third power sum, offering new criteria for permutation polynomial characterization.

Findings

Derived the value of the third power sum over finite fields.

Established new necessary conditions for permutation polynomial status.

Enhanced understanding of reversed Dickson polynomial properties.

Abstract

Let $D_n(1,x)$ be the $n$th reversed Dickson polynomial. The power sums $\sum_{a\in\Bbb F_q}D_n(1,a)^i$, $i=1,2$, have been determined recently. In this paper we give an evaluation of the sum $\sum_{a\in\Bbb F_q}D_n(1,a)^3$. This result implies new necessary conditions for $D_n(1,x)$ to be a permutation polynomial over $\Bbb F_q$.