On a conjecture on permutation polynomials over finite fields

TL;DR

This paper confirms a conjecture regarding specific permutation polynomials over finite fields, establishing precise conditions under which these polynomials permute the elements of the field.

Contribution

The paper proves the conjecture that characterizes when certain sums of monomials form permutation polynomials over finite fields.

Findings

Confirmed the conjecture for all cases.

Identified conditions for permutation behavior.

Provided a complete characterization of the polynomials.

Abstract

Let $\Bbb F_q$ be the finite field with $q$ elements and let $p=\text{char}\,\Bbb F_q$. It was conjectured that for integers $e\ge 2$ and $1\le a\le pe-2$, the polynomial $X^{q-2}+X^{q^2-2}+\cdots+X^{q^a-2}$ is a permutation polynomial of $\Bbb F_{q^e}$ if and only if (i) $a=2$ and $q=2$, or (ii) $a=1$ and $\text{gcd}(q-2,q^e-1)=1$. In the present paper we confirm this conjecture.