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

摘要

Ailon and Rudnick have shown that if a,b∈C[T] aremultiplicatively independent polynomials, thendeg(gcd(an-1,bn-1)) is bounded for all n≧1. Weshow that if instead a,b∈F[T] for a finite field F ofcharacteristic p, then deg(gcd(an-1,bn-1)) is largerthan 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.,potdivide n).