Low-degree permutation rational functions over finite fields

TL;DR

This paper classifies low-degree permutation rational functions over finite fields, providing explicit results for degrees 4, 8, and 32, and introduces a new Galois-theoretic approach for understanding such functions.

Contribution

It offers a complete classification of degree-4 permutation rational functions and partial classifications for degrees 8 and 32, along with a novel Galois-theoretic characterization of additive polynomials.

Findings

Classified all degree-4 permutation rational functions over finite fields.

Determined permutation functions of degrees 8 and 32 under certain conditions.

Introduced a new Galois-theoretic method for analyzing permutation rational functions.

Abstract

We determine all degree-4 rational functions f(X) in F_q(X) which permute P^1(F_q), and answer two questions of Ferraguti and Micheli about the number of such functions and the number of equivalence classes of such functions up to composing with degree-one rational functions. We also determine all degree-8 rational functions f(X) in F_q(X) which permute P^1(F_q) in case q is sufficiently large, and do the same for degree 32 in case either q is odd or f(X) is a nonsquare. Further, for most other positive integers n<4096, for each sufficiently large q we determine all degree-n rational functions f(X) in F_q(X) which permute P^1(F_q) but which are not compositions of lower-degree rational functions in F_q(X). Some of these results are proved by using a new Galois-theoretic characterization of additive (linearized) polynomials among all rational functions, which is of independent interest.