Title: Permutation polynomials, fractional polynomials, and algebraic curves,
Abstract: 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 $Ax^{q(q-1)+1} + Bx^{2(q-1)+1} + Cx^{q} + x$. Our method
is based on the investigation of an algebraic curve associated with a
{fractional polynomial} over a finite field.