On a variant of the Ailon–Rudnick theorem in finite characteristic

Let L be a field of characteristic p, and let a, b, c, d ε L(T). Assume that a and b are algebraically independent over F_p. Then for each fixed positive integer n, we prove that there exist at most finitely many λ ε L satisfying f(a(λ)) = c(λ) and g(b(λ)) = d(λ) for some polynomials f, g ε F_pn[Z] such that f(a) ≠ c and g(b) ≠ d. Our result is a characteristic p variant of a related statement proven by Ailon and Rudnick.