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Abstract

Let $q$ be a prime power, $\mathbb F_q$ be the finite field of order $q$ and
$\mathbb F_q(x)$ be the field of rational functions over $\mathbb F_q$. In this
paper we classify all rational functions $\varphi\in \mathbb F_q(x)$ of degree
3 that induce a permutation of $\mathbb P^1(\mathbb F_q)$. Our methods are
constructive and the classification is explicit: we provide equations for the
coefficients of the rational functions using Galois theoretical methods and
Chebotarev Density Theorem for global function fields. As a corollary, we
obtain that a permutation rational function of degree 3 permutes $\mathbb F_q$
if and only if it permutes infinitely many of its extension fields. As another
corollary, we derive the well-known classification of permutation polynomials
of degree 3. As a consequence of our classification, we can also show that
there is no complete permutation rational function of degree $3$ unless $3\mid
q$ and $\varphi$ is a polynomial.