A New Approach to Permutation Polynomials over Finite Fields, II

Let $p$ be a prime and $q$ a power of $p$. For $n\ge 0$, let $g_{n,q}\in\Bbb F_p[{\tt x}]$bethepolynomialdefinedbythefunctionalequation$\sum_{a\in\Bbb F_q}({\tt x}+a)^n=g_{n,q}({\tt x}^q-{\tt x})$\mathrm{.}Whenis$g_{n,q}$apermutationpolynomial(PP)of$\Bbb F_{q^e}$? This turns out to be a challenging question with remarkable breath and depth, as shown in the predecessor of the present paper. We call a triple of positive integers $(n,e;q)$ {\em desirable} if $g_{n,q}$ is a PP of ${\mathbb{F}}_{q^e}$. In the present paper, we find many new classes of desirable triples whose corresponding PPs were previously unknown. Several new techniques are introduced for proving a given polynomial is a PP.