# Roots of certain polynomials over finite fields

## TL;DR

This paper explicitly determines the roots of specific polynomials over finite fields, resolving an open problem and a conjecture, and introduces a new approach for analyzing such polynomials.

## Contribution

It provides explicit root characterizations for a class of polynomials over finite fields and introduces a novel method for this type of problem.

## Key findings

- Explicit root formulas for polynomials over finite fields.
- Resolution of an open problem and a conjecture.
- Introduction of a new analytical approach.

## Abstract

We determine the roots in F_{q^3} of the polynomial X^{2q^k+1} + X + c for each positive integer k and each c in F_q, where q is a power of 2. We introduce a new approach for this type of question, and we obtain results which are more explicit than the previous results in this area. Our results resolve an open problem and a conjecture of Zheng, Kan, Zhang, Peng, and Li.