Permutation polynomials, fractional polynomials, and algebraic curves

In this note we prove a conjecture by Li, Qu, Li, and Fu on permutation trinomials over F32k. In addition, new examples and generalizations of some families of permutation polynomials of F3k and F5k are given. We also study permutation quadrinomials of type Axq(q−1)+1+Bx2(q−1)+1+Cxq+x. Our method is based on the investigation of an algebraic curve associated with a fractional polynomial over a finite field.