Permutation Polynomials of $\Bbb F_{q^2}$ of the form $a{\tt X}+{\tt X}^{r(q-1)+1}$

TL;DR

This paper investigates permutation polynomials of a specific form over finite fields, establishing necessary conditions and finiteness results that support a recent conjecture in the field.

Contribution

It provides new necessary conditions for such polynomials to be permutations and proves finiteness of solutions under certain parameters, advancing understanding of permutation binomials.

Findings

If the polynomial is a permutation, then gcd(r, q+1) > 1.

Under certain conditions, only finitely many (q, a) make the polynomial a permutation.

The results confirm a recent conjecture about the structure of these permutation binomials.

Abstract

Let $q$ be a prime power, $2\le r\le q$, and $f=a{\tt X}+{\tt X}^{r(q-1)+1}\in\Bbb F_{q^2}[{\tt X}]$, where $a\ne 0$. The conditions on $r,q,a$ that are necessary and sufficient for $f$ to be a permutation polynomial (PP) of ${\Bbb F}_{q^2}$ are not known. (Such conditions are known under an additional assumption that $a^{q+1}=1$.) In this paper, we prove the following: (i) If $f$ is a PP of ${\Bbb F}_{q^2}$, then $\text{gcd}(r,q+1)>1$ and $(-a)^{(q+1)/\text{gcd}(r,q+1)}\ne 1$. (ii) For a fixed $r>2$ and subject to the conditions that $q+1\equiv 0\pmod r$ and $a^{q+1}\ne 1$, there are only finitely many $(q,a)$ for which $f$ is a PP of ${\Bbb F}_{q^2}$. Combining (i) and (ii) confirms a recent conjecture regarding the type of permutation binomial considered here.