2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/69974Ailon and Rudnick have shown that if $a,b \in C[T]$ are multiplicatively independent polynomials, then $°(\gcd(a^n-1,b^n-1))$ 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 $°(\gcd(a^n-1,b^n-1))$ 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., $gec(n,p)=1$).Number TheoryAlgebraic Geometry11T55; 11R58; 11D61text