respectively, we get $$\deg \gcd\(h_1\(f^n\),h_2\(g^m\)\) 
\le d_{h_1}d_{h_2} \(11(d_f+d_g)^{2}\)^{\min(d_f,d_g)}$$ which concludes the proof.

Proof of Theorem \[thm:genAR2\] {#sec:univar2}
-------------------------------

We use the same idea as in the proof of [@AR Theorem1] combined with Lemma \[lem:bmz\].

First, we note that for any zero $t\in \C$ of $$\gcd\(\prod_{i=1}^{\ell}f_i^{n_i}-\prod_{i=1}^k\varphi_i^{\nu_i},\prod_{i=1}^rg_i^{m_i}-\prod_{i=1}^s\psi_i^{\mu_i}\)$$ the condition  ensures that $\varphi_i(t),\psi_j(t)\ne0$, $i=1,\ldots,l$, $j=1,\ldots,k$. Therefore each such zero $t$ satisfies $$\label{eq:ffgg}
\prod_{i=1}^{\ell}f_i(t)^{n_i}\cdot\prod_{i=1}^k\varphi_i(t)^{-\nu_i}=\prod_{i=1}^rg_i(t)^{m_i}\cdot\prod_{i=1}^s\psi_i(t)^{-\mu_i}=1.$$

We apply Lemma \[lem:bmz\] with $k$ replaced by $L={\ell}+k+r+s$ and with the curve $$\begin{split}
C= \bigl\{(f_1(t),\ldots,f_{\ell}(t),\varphi_1(t)&,\ldots,\varphi_{k}(t),g_1(t),\ldots,g_r(t),\\
&\psi_1(t),\ldots,\psi_s(t))~:~  t\in\C\bigr\} \subseteq\G_m^{L}.
\end{split}$$ Indeed, we denote $$\vec{v}=(n_1,\ldots,n_{\ell},-\nu_1,\ldots,-\nu_k),\ \vec{w}=(m_1,\ldots,m_r,-\mu_1,\ldots,-\mu_s).$$ As the vectors $$(\vec{v},\vec{0}),\ (\vec{0},\vec{w})\in \Z^L$$ are linearly independent, by Lemma \[lem:bmz\] we obtain that there are finitely many $t\in\C$ such that  holds for some vectors $\vec{v},\vec{w}$ as above.

We denote by $S$ the set of such $t\in\C$. For $\vec{v},\vec{w}$, we denote $$\cD_{\vec{v},\vec{w}}=\gcd\(\prod_{i=1}^{\ell}f_i^{n_i}-\prod_{i=1}^k\varphi_i^{\nu_i},\prod_{i=1}^rg_i^{m_i}-\prod_{i=1}^s\psi_i^{\mu_i}\).$$ We see from the above that set of zeros $Z(\cD_{\vec{v},\vec{w}})$ belongs to some fixed set that depends only on the above curve $C$ and thus only on the polynomials in the initial data. To give the upper bound for $\deg \cD_{\vec{v},\vec{w}}$ we only need to prove that the multiplicity of the roots $t\in S$ of $\cD_{\vec{v},\vec{w}}$ can be bounded uniformly for all integer vectors $\vec{v},\vec{w}$. This is given by Lemma \[lem:ABCpoly\] applied 