4 problems

Finiteness conjecture. There are only finitely many pairs $(q,a)$ for which $f$ is a permutation polynomial of $\mathbb{F}_{q^2}$.

Let $a,b\in\mathbb{Z}$. Let $p$ and $q$ be odd primes with $p\equiv -1 \\(\operatorname{mod}~p)$. Define … Suppose that $\mathcal K(|ap|)=\mathcal K(|b^q+a|)$, where…

Let $m$ be a nonnegative integer, set $q=2^{2m+1}$, and assume $e=2$, $d=3$, and $a^{q+1}\ne 1$. Define … A polynomial is a permutation binomial of $\mathbb F_{q^2}$ if it permutes…

Let $q$ be a prime power, $e\ge 2$, $d=1$, and $n<q^e-q$. Write … A polynomial is a permutation binomial of $\mathbb F_{q^e}$ if it permutes $\mathbb F_{q^e}$. In the cases…