Residue sums of Dickson polynomials over finite fields

TL;DR

This paper derives a closed-form expression for the sum of distinct residues of Dickson polynomials over finite fields of odd characteristic and characterizes the size of their image sets, advancing understanding of polynomial residue behavior.

Contribution

It provides the first non-trivial classification of residue sums for Dickson polynomials of arbitrary degree over finite fields.

Findings

Closed-form sum over residues of Dickson polynomials derived

Complete characterization of the size of the polynomial's image set

First classification of residue sums for unbounded degree polynomials

Abstract

Given a polynomial with integral coefficients, one can inquire about the possible residues it can take in its image modulo a prime $p$. The sum over the distinct residues can sometimes be computed independent of the prime $p$; for example, Gauss showed that the sum over quadratic residues vanishes modulo a prime. In this paper we provide a closed form for the sum over distinct residues in the image of Dickson polynomials of arbitrary degree over finite fields of odd characteristic, and prove a complete characterization of the size of the image set. Our result provides the first non-trivial classification of such a sum for a family of polynomials of unbounded degree.