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

Let $f={\tt X}^r(a+{\tt X}^{2(q-1)})\in{\Bbb F}_{q^2}[{\tt X}]$, where $a\in{\Bbb F}_{q^2}^*$ and $r\ge 1$. The parameters $(q,r,a)$ for which $f$ is a permutation polynomial (PP) of ${\Bbb F}_{q^2}$ have been determined in the following cases: (i) $a^{q+1}=1$; (ii) $r=1$; (iii) $r=3$. These parameters together form three infinite families. For $r>3$ (there is a good reason not to consider $r=2$) and $a^{q+1}\ne 1$, computer search suggested that $f$ is not a PP of ${\Bbb F}_{q^2}$ when $q$ is not too small relative to $r$. In the present paper, we prove that this claim is true. In particular, for each $r>3$, there are only finitely many $(q,a)$, where $a^{q+1}\ne 1$, for which $f$ is a PP of ${\Bbb F}_{q^2}$.