Common divisors of an − 1 and bn − 1 over function fields

Ailon and Rudnick have shown that if a, b ∈ C[T ] are multiplicatively independent polynomials, then deg ( gcd(an − 1, bn − 1)) is bounded for all n ≥ 1. We show that if instead a, b ∈ F[T ] for a finite field F of characteristic p, then deg ( gcd(an − 1, bn − 1)) is larger than Cn for a constant C = C(a, b) > 0 and for infinitely many n, even if n is restricted in various reasonable ways (e.g., p n).