Determination of a class of permutation quadrinomials

We determine all permutation polynomials over Fq2$\mathbb {F}_{q^2}$ of the form XrA(Xq−1)$X^r A(X^{q-1})$ where, for some Q$Q$ that is a power of the characteristic of Fq$\mathbb {F}_q$ , we have r≡Q+1(modq+1)$r\equiv Q+1\pmod {q+1}$ and all terms of A(X)$A(X)$ have degrees in {0,1,Q,Q+1}$\lbrace 0,1,Q,Q+1\rbrace$ . We use this classification to resolve eight conjectures and open problems from the literature, and we list 77 recent results from the literature that follow immediately from the simplest special cases of our result. Our proof makes a novel use of geometric techniques in a situation where they previously did not seem applicable, namely to understand the arithmetic of high‐degree rational functions over small finite fields, despite the fact that in this situation the Weil bounds do not provide useful information.