Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Acta Arithmetica / All issues

Abstract

We determine all degree-$4$ rational functions $f(X)\in \mathbb {F}_q(X)$ which permute $\mathbb {P}^1(\mathbb {F}_q)$, 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 $f(X)\in \mathbb {F}_q(X)$ which permute $\mathbb {P}^1(\mathbb {F}_q)$ 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 thousands of other positive integers $n$, for each sufficiently large $q$ we determine all degree-$n$ rational functions $f(X)\in \mathbb {F}_q(X)$ which permute $\mathbb {P}^1(\mathbb {F}_q)$ but which are not compositions of lower-degree rational functions in $\mathbb {F}_q(X)$. 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.