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

TL;DR

This paper provides a complete classification of degree 3 rational functions over finite fields that permute the projective line, using Galois theory and density theorems, and explores their properties and limitations.

Contribution

It offers an explicit, constructive classification of permutation rational functions of degree 3 over finite fields, including conditions for their completeness and permutation behavior.

Findings

Permutation rational functions of degree 3 permute infinitely many extension fields.

No complete permutation rational functions of degree 3 exist unless they are polynomials and 3 divides q.

The classification recovers known results for permutation polynomials of degree 3.

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.