\[lem:multindep\], to reduce via specialisations to the univariate case and thus use Theorem \[thm:genAR1\].

We denote $d=D+1$, that is $d>\deg_{X_j}F,\deg_{X_j}G$ for any $j=1,\ldots,{\ell}$. We define the polynomials $$\begin{split}
&\widetilde{F}(X_1,\ldots,X_{\ell})=F\(X_1,X_2+X_1^d,\ldots,X_{\ell}+X_1^{d^{{\ell}-1}}\),\\
&\widetilde{G}(X_1,\ldots,X_{\ell})=G\(X_1,X_2+X_1^d,\ldots,X_{\ell}+X_1^{d^{{\ell}-1}}\).
\end{split}$$

The polynomials $\widetilde{F},\widetilde{G}$ have the property that $$\deg \widetilde{F},\deg \widetilde{G}\le D\frac{d^{{\ell}}-1}{d-1}<(D+1)^{\ell}$$ and $$\deg \widetilde{F}(X_1,\alpha_2,\ldots,\alpha_{\ell})=\deg \widetilde{F},\ \deg \widetilde{G}(X_1,\alpha_2,\ldots,\alpha_{\ell})=\deg \widetilde{G}$$ for any specialisation $(\alpha_2,\ldots,\alpha_{\ell})\in\C^{{\ell}-1}$.

Moreover, we note that the polynomials $\widetilde{F}, \widetilde{G}$ are also multiplicatively independent in $\C(X_1,\ldots,X_{\ell})^*/\C^*$. Indeed, if this would not be the case, then there exist $i_1,i_2$ not both zero and $a\in\C$ such that $$\widetilde{F}^{i_1}\widetilde{G}^{i_2}=a.$$ Composing this polynomial identity with the polynomial automorphism $$\label{eq:Kro}
(X_1,\ldots,X_{\ell})\to \(X_1,X_2-X_1^d,\ldots,X_{\ell}-X_1^{d^{{\ell}-1}}\)$$ we obtain that the polynomials $F,G$ are multiplicatively dependent in $\C(X_1,\ldots,X_{\ell})^*/\C^*$ and thus we get a contradiction.

Let $\K$ be the finite extension of $\Q$ by the coefficients of the polynomials $F,G$. By the Hilbert’s irreducibility theorem, see [@Sch Theorem 46], $\K$ is a Hilbertian field. We apply now Lemma \[lem:multindep\] with the polynomials $\widetilde{F},\widetilde{G}$, and thus there exists a specialisation $(\alpha_2,\ldots,\alpha_{\ell})\in\K^{{\ell}-1}$ such that $\widetilde{F}(X_1,\alpha_2,\ldots,\alpha_{\ell})$ and $\widetilde{G}(X_1,\alpha_2,\ldots,\alpha_{\ell})$ are multiplicatively independent in $\C(X_1)^*/\C^*$. For simplicity, we denote $f=\widetilde{F}(X_1,\alpha_2,\ldots,\alpha_{\ell})$ and $g=\widetilde{G}(X_1,\alpha_2,\ldots,\alpha_{\ell})$.

We denote $\cD_{n,m}=\gcd\(h_1\(F^n\),h_2\(G^m\)\)$. Moreover, we note that $$\cD_{n,m}\(X_1,X_2+X_1^d,\ldots,X_{\ell}+X_1^{d^{{\ell}-1}}\)=\gcd\(h_1\(\widetilde{F}^n\),h_2\(\widetilde{G}^m\)\).$$ We denote $E_{n,m}=\gcd\(h_1\(\widetilde{F}^n\),h_2\(\widetilde{G}^m\)\)$, and for the specialisation $(\alpha_2,\ldots,\alpha_{\ell})$ one has $$E_{n,m}(X_1,\alpha_2,\ldots,\alpha_{\ell})\mid \gcd\(h_1\(f^n\),h_2\(g^m\)\).$$ In particular, we 