= 0$, with $x^n , y^m \in S$ for some nonzero integers $n$ and $m$, where $S$ is the group of $S$-units of some fixed number field. This leads again to further generalisations.

It is certainly interesting to obtain a similar result as Theorem \[thm:multivAR\] for $$\gcd\(H_1\(F_1^{n_1},\ldots,F_{s}^{n_{s}}\),H_2\(G_1^{m_1},\ldots,G_{r}^{m_{r}}\)\),$$ with polynomials $H_1\in\C[Y_1,\ldots,Y_s]$, $H_2\in\C[Z_1,\ldots,Z_r]$ and also $F_1,\ldots,F_s,G_1,\ldots,G_r\in\C[X_1,\ldots,X_{\ell}]$.

If one chooses $$H_1=Y_1\cdots Y_s-1,\quad H_2=Z_1\cdots Z_r-1,$$ then following the same proof as for Theorem \[thm:multivAR\], we reduce (via specialisations) the problem to Theorem \[thm:genAR2\], and thus get that $$\label{eq:degmultiv}
\deg \gcd\(H_1\(F_1^{n_1},\ldots,F_{s}^{n_{s}}\),H_2\(G_1^{m_1},\ldots,G_{r}^{m_{r}}\)\)$$ is bounded by a constant depending only on $F_1,\ldots,F_s,G_1,\ldots,G_r$.

However, the approach of Theorem \[thm:multivAR\] does not seem to work for more general multivariate polynomials $H_1,H_2$.

Dynamical analogues
-------------------

Another interesting direction of research is obtaining dynamical analogues of the results of Ailon-Rudnick [@AR] and Silverman [@Si]. That is, investigating the greatest common divisors of polynomials iterates.

More precisely, let $\K$ be a field and $f,g\in\K[X]$. We define $$f^{(0)}=T,\quad f^{(n)}=f(f^{(n-1)}),\quad  n\ge 1,$$ and similarly for $g$.

Give, under some natural conditions, an upper bound for $$\deg\gcd\(f^{(n)},g^{(m)}\).$$

\[prob:gcd 1\] Show, under some natural conditions, that the iterates of $f$ and $g$ are coprime for infinitely many $n,m$.

We note that some conditions on $f,g$ are 