Let f(X) in Q(X) be a rational function. For almost all primes p we can reduce the coefficients of f and consider f_p:=f mod p as a function on the projective line P^1(F_p) over F_p. Here we continue the arithmetic aspects of joint work with Guralnick and Saxl, and classify the functions f such that f_p is a bijection for infinitely many primes p. This is the rational function analog of the classical conjecture of Schur (1923), solved by Fried (1970), which considered the case that f is a polynomial. Thereby we also answer a question of J. G. Thompson about the minimal field of definition of a certain rational function of degree 25.
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