Permutation polynomials of \(\mathbb F_{q^2}\) of the form \(aX+X^{r(q-1)+1}\) (Q2829800)

Let \(q\) be a prime power and \(2\leq r\leq q\). Consider \(f=ax+x^{r(q-1)+1}\in\mathbb F_{q^2}[x]\) with \(a\neq 0\). If \(a^{q+1}=1\) then necessary and sufficient conditions on \(r, q\) and \(a\) for \(f\) to be a permutation polynomial of \(\mathbb F_{q^2}\) have been given by Zieve. It is conjectured that, if \(a^{q+1}\neq 1\) and \(r> 2\) is prime, there are only finitely many \((q,a)\) for which \(f\) is a permutation polynomial. Here the author proves the conjecture by showing that if some \(f\) is a permutation polynomial and \(r>2\) is prime then \(r|q+1\). Further, for fixed \(r>2\) (not necessarily prime), if \(a^{q+1}\neq 1\) and \(r|q+1\) then there are only finitely many \((q,a)\) for which \(f\) is a permutation polynomial.NEWLINENEWLINEThe proof is elementary but complex. It begins by computing power sums \(\sum_{x\in F_{q^2}} f(x)^s\). It is shown that if the result fails then \(p=2,3\) or \(5\). Each of these cases is handled separately.NEWLINENEWLINEFor the entire collection see [Zbl 1345.11003].