Primitive values of rational functions at primitive elements of a finite field

Given a prime power q and an integer n >= 2, we establish a sufficient condition for the existence of a primitive pair (alpha, f(alpha)) where alpha is an element of F-q and f (x) is an element of F-q(x) is a rational function of degree sum n. (Here f = f(1)/f(2), where f(1), f(2) are coprime polynomials of degree n(1), n(2), respectively, and the sum of their degrees n(1) + n(2) = n.) For any n, such a pair is guaranteed to exist for sufficiently large q. Indeed, when n = 2, such a pair definitely does not exist only for 28 values of q and possibly (but unlikely) only for at most 3911 other values of q. (c) 2020 Elsevier Inc. All rights reserved.