Full classification of permutation rational functions and complete rational functions of degree three over finite fields

Let q be a prime power, Fq be the finite field of order q and Fq(x) be the field of rational functions over Fq. In this paper we classify and count all rational functions φ∈ Fq(x) of degree 3 that induce a permutation of P1(Fq). As a consequence of our classification, we can show that there is no complete permutation rational function of degree 3 unless 3 ∣ q and φ is a polynomial.

Let q be a prime power, Fq be the finite field of order q and Fq(x) be the field of rational functions over Fq. In this paper we classify all rational functions φ∈Fq(x) of degree 3 that induce a permutation of P1(Fq). 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 Fq 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∣q and φ is a polynomial.