A new approach to permutation polynomials over finite fields, II

Let p be a prime and q a power of p. For n⩾0, let gn,q∈Fp[x] be the polynomial defined by the functional equation ∑a∈Fq(x+a)n=gn,q(xq−x). When is gn,q a permutation polynomial (PP) of Fqe? 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) desirable if gn,q is a PP of Fqe. 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.