A new family of exceptional rational functions

TL;DR

This paper constructs an infinite family of indecomposable, exceptional rational functions over finite fields with non-solvable monodromy groups, expanding the known examples in algebraic function theory.

Contribution

It introduces the first known examples of wildly ramified, indecomposable exceptional rational functions with non-solvable monodromy groups over finite fields.

Findings

Constructed infinite sequences of exceptional rational functions for each odd prime power q.

These functions are indecomposable and have non-solvable monodromy groups.

First examples with arbitrarily large degree and wild ramification, other than linear polynomial changes.

Abstract

For each odd prime power q, we construct an infinite sequence of rational functions f(X) in F_q(X), each of which is exceptional, which means that for infinitely many n the map c-->f(c) induces a bijection of P^1(F_{q^n}). Moreover, each of our functions f(X) is indecomposable, which means that it cannot be written as the composition of lower-degree rational functions in F_q(X). In case q is not a power of 3, these are 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. These are also the first known examples of wildly ramified indecomposable exceptional rational functions f(X), other than linear changes of polynomials.