A New Family of Exceptional Rational Functions. (2nd December 2021) - Ding, Zhiguo; Zieve, Michael E
Abstract: For each odd prime power q, we construct an infinite sequence of rational functions f(X) F_q(X), each of which is exceptional in the sense 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 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.