Permutation polynomials, fractional polynomials, and algebraic curves

In this note we prove a conjecture by Li, Qu, Li, and Fu on permutation trinomials over ${\mathbb{F}}_3^{2k}$. In addition, new examples and generalizations of some families of permutation polynomials of ${\mathbb{F}}_{3^k}$ and ${\mathbb{F}}_{5^k}$ are given. We also study permutation quadrinomials of type $A{x}^{q(q-1)+1}+B{x}^{2(q-1)+1}+C{x}^q+x$. Our method is based on the investigation of an algebraic curve associated with a {fractional polynomial} over a finite field.