A New Family of Exceptional Rational Functions

For each odd prime power q, we construct an infinite sequence of rational functions f(X) is an element of F-q(X) , each of which is exceptional in the sense that for infinitely many n the map c bar right arrow f (c) induces a bijection of P-1 (F-qn). 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 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 F-q which have non-solvable monodromy groups and have arbitrarily large degree.

Automated summary

This article presents a construction of an infinite sequence of exceptional rational functions f(X) in F-q(X), where q is an odd prime power. These functions induce bijections of P-1 (F-qn) for infinitely many n and cannot be decomposed as compositions of lower-degree rational functions in F-q(X). These are the first known examples of wildly ramified indecomposable exceptional rational functions other than linear changes of polynomials.