Permutation Polynomials of ${\mathbb{F}}_{q^2}$ of the form $a\mathtt{X}+{\mathtt{X}}^{r(q-1)+1}$

Abstract

Let $q$ be a prime power, $2\le r\le q$, and $f=a\mathtt{X}+{\mathtt{X}}^{r(q-1)+1}\in {\mathbb{F}}_{q^2}[\mathtt{X}]$, where $a\ne 0$. The conditions on $r,q,a$ that are necessary and sufficient for $f$ to be a permutation polynomial (PP) of ${\mathbb{F}}_{q^2}$ 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 ${\mathbb{F}}_{q^2}$, 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 $q+1\equiv 0(\mathrm{mod}r)$ and $a^{q+1}\ne 1$, there are only finitely many $(q,a)$ for which $f$ is a PP of ${\mathbb{F}}_{q^2}$. Combining (i) and (ii) confirms a recent conjecture regarding the type of permutation binomial considered here.

24 pages

References in corpus (2)

• Permutation polynomials on F_q induced from bijective Redei functions on subgroups of the multiplicative group of F_q
• Determination of a Type of Permutation Trinomials over Finite Fields