need.

Our results: multivariate case
------------------------------

For our first result in the multivariate case, we reduce the problem to the univariate case using Hilbert’s irreducibility theorem (see Section \[sec:Hilb\]), and to control the degree for such specialisation we also couple this approach with a transformation involving the Kronecker substitution. We obtain:

\[thm:multivAR\] Let $h_1,h_2\in\C[T]$ and $F,G\in\C[X_1,\ldots,X_{\ell}]$. We denote by $D=\max_{i=1\ldots,\ell}\(\deg_{X_i}F,\deg_{X_i}G\)$. If $F, G$ are multiplicatively independent in $\C(X_1,\ldots,X_{\ell})^*/\C^*$, then for all $n,m\ge 1$ we have $$\deg\gcd\(h_1\(F^n\),h_2\(G^m\)\)\le  d_{h_1} d_{h_2} \(44(D+1)^{2{\ell}}\)^{(D+1)^{\ell}}.$$

We note that if $h_1=h_2=T-1$ as in [@AR Theorem 1], then in Theorem \[thm:multivAR\] we need $F,G$ to be just multiplicatively independent.

Theorem \[thm:multivAR\] is proved in Section \[sec:multivar1\].

Another natural extension of [@AR Theorem 1] to the multivariate case is related to the fact that the greatest common divisor of two univariate polynomials is given by their common zeros. Thus [@AR Theorem 1] says that the number of common zeros of $f^n-1$ and $g^m-1$, for two polynomials $f,g\in\C[T]$, is bounded by a constant depending only on $f$ and $g$ for all $n,m\ge 1$, and Lemma \[lem:univAR\] gives a uniform bound.

For positive integers $\ell,D\ge 1$, we denote $$\label{eq:gamma}
 \gamma_{\ell}(D)=\binom{\ell+1+D^\ell}{\ell+1}.$$

We now obtain the following result proved in Section \[sec:multivar2\]. This 