if $f,g\in\C[T]$ are multiplicatively independent polynomials, then there exists $h \in \C[T]$ such that $$\label{eq:AR}
\gcd(f^n-1,g^n-1) \mid h$$ for all $n\ge 1$. Examining their argument one can easily see that the same statement holds in a larger generality; namely there exists $\widetilde h \in \C[T]$ such that $$\label{eq:AR Gen}
\gcd(f^n-1,g^m-1) \mid \widetilde h$$ for all $n,m\ge 1$.

In the case of finite fields $\F_q$ of characteristic $p$, Silverman [@Si] proves that even more restrictions on the polynomials $f,g\in\F_q[T]$ does not allow a similar conclusion as the result of [@AR]. In particular, Silverman proves that the analogue of  is false in a very strong sense: there exists a constant $c(f,g;q)$, depending only on $f$, $g$ and $q$, such that $$\deg \gcd(f^n-1,g^n-1)\ge c(f,g;q) n$$ for infinitely many $n$.

More results in positive characteristic are obtained in [@CZ13; @Lau], as well as variants for elliptic divisibility sequences [@Si04; @Si05].

In this paper we present some extensions of the Ailon-Rudnick Theorem [@AR Theorem 1] over $\C$, both in the univariate and multivariate cases. Although the method of proof in the univariate case is similar to, or reduces to using, [@AR], we find these extensions exciting and we hope they will be of independent interest. 