A New Family of Exceptional Rational Functions

Abstract

Abstract For each odd prime power $q$, we construct an infinite sequence of rational functions $f(X) \in{\mathbb{F}}_q(X)$, each of which is exceptional in the sense that for infinitely many $n$ the map $c \mapsto f(c)$ induces a bijection of ${\mathbb{P}}^1({\mathbb{F}}_{q^n})$. Moreover, each of our functions $f(X)$ is indecomposable in the sense that it cannot be written as the composition of lower-degree rational functions in ${\mathbb{F}}_q(X)$. These are the first known examples of wildly ramified indecomposable exceptional rational functions $f(X)$, other than linear changes of polynomials. In case $q$ is not a power of $3$, these are also the first known examples of indecomposable exceptional rational functions $f(X)$ over ${\mathbb{F}}_q$ which have non-solvable monodromy groups and have arbitrarily large degree.