have $$\deg \cD_{n,m}\le \deg E_{n,m}\le \deg \gcd\(h_1\(f^n\),h_2\(g^m\)\).$$ We make here the remark that using the automorphism  was essential to have these degree inequalities, as if one just uses Hilbert’s irreducibility theorem applied directly with the polynomials $F$ and $G$, we cannot guarantee that when we make specialisations we get that $\deg \cD_{n,m}\le \deg \gcd\(h_1\(f^n\),h_2\(g^m\)\)$.

We apply now Theorem \[thm:genAR1\] and using the fact that $\deg f,\deg g< (D+1)^{\ell}$ we conclude that $$\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}},$$ which finishes the proof.

Proof of Theorem \[thm:CommonZeros\] {#sec:multivar2}
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We define $$\cH=\{(F_1({{\boldsymbol{\alpha}}}),\ldots,F_{\ell+1}({{\boldsymbol{\alpha}}}))\mid {{\boldsymbol{\alpha}}}\in\C^{\ell}\}.$$ By Lemma \[lem:perron\] there exists a polynomial $R\in\C[Z_1,\ldots,Z_{\ell+1}]$ of degree at most $D^{\ell}$ such that $R(F_1,\ldots,F_{\ell+1})=0$. In other words, any point of $\cH$ is a point on the hypersurface defined by the zero set of $R$ in $\C^{{\ell+1}}$. In particular, any point ${{\boldsymbol{\alpha}}}\in\C^{\ell}$ such that $F_i({{\boldsymbol{\alpha}}})^{n_i}=1$, $i=1,\ldots,\ell+1$ gives a point on the hypersurface defined by the zero set of $R$ with coordinates roots of unity.

From Lemma \[lem:GranRud\] we get that there are at most $$N \le N(R)  \le  (0.792s(R)/\log \(s(R)+1\))^{s(R)}$$ algebraic subgroups, each defined by the zero set of at most $\ell+1$ polynomials of the form $$Z_1^{b_{j,1}}Z_2^{b_{j,2}}\cdots Z_{\ell+1}^{b_{j,\ell+1}}-1\in 
\C[Z_1, \ldots Z_{\ell+1}]$$ with $$\sum_{i=1}^{\ell+1}b_{j,i}\le (\ell+1)D^{\ell}\prod_{p\le s(R)}p,
\qquad j = 1, 