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

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.