  Ailon and Rudnick have shown that if $a,b \in C[T]$ are multiplicatively
independent polynomials, then $\deg(\gcd(a^n-1,b^n-1))$ is bounded for all
$n\ge1$. We show that if instead $a,b \in F[T]$ for a finite field $F$ of
characteristic $p$, then $\deg(\gcd(a^n-1,b^n-1))$ 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., $gec(n,p)=1$).
