  In this paper we present some extensions of the Ailon-Rudnick Theorem, which
says that if $f,g\in{\mathbb C}[T]$, then $\gcd(f^n-1,g^m-1)$ is bounded for
all $n,m\ge 1$. More precisely, using a uniform bound for the number of torsion
points on curves and results on the intersection of curves with algebraic
subgroups of codimension at least $2$, we present two such extensions in the
univariate case. We also give two multivariate analogues of the Ailon-Rudnick
Theorem based on Hilbert's irreducibility theorem and a result of Granville and
Rudnick about torsion points on hypersurfaces.
