below and proved in Section \[sec:main\]. In particular, our first extension of [@AR Theorem 1], which is proved in Section \[sec:univar1\], is based on this uniform bound.

\[thm:genAR1\] Let $f,g,h_1,h_2\in\C[T]$. If $f$ and $g$ are multiplicatively independent in $\C(T)^*/\C^*$, 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}  \(11(d_f+d_g)^{2}\)^{\min(d_f,d_g)}.$$

For the second extension of [@AR Theorem 1], which is proved in Section \[sec:univar2\], we apply the finiteness result of [@BMZ08; @Mau], see also [@BMZ99], for the number of points on the intersection of curves in $\G_m^n$ with algebraic subgroups, see Lemma \[lem:bmz\]. No uniform bounds are known so far for such finiteness results.

We recall that for a polynomial $f\in\C[T]$, we denote by $Z(f)$ the set of zeros of $f$ in $\C$.

\[thm:genAR2\] Let $f_1,\ldots,f_{\ell},\varphi_1,\ldots,\varphi_{k},g_1,\ldots,g_r,\psi_1,\ldots,\psi_s\in\C[T]$, $\ell,k,r,s\ge 1$, be multiplicatively independent polynomials such that $$\label{eq:cond}
Z(f_1\cdots f_{\ell})\cap Z(\varphi_1\cdots \varphi_k)=\emptyset,\quad Z(g_1\cdots g_r)\cap Z(\psi_1\cdots \psi_s)=\emptyset.$$ Then we have:

-   For all $n_1,\ldots,n_{\ell},\nu_1,\ldots,\nu_k,m_1,\ldots,m_r,\mu_1,\ldots,\mu_s\ge 0$, there exists a polynomial $h\in \C[T]$ such that $$\gcd\(\prod_{i=1}^{\ell}f_i^{n_i}-\prod_{i=1}^k\varphi_i^{\nu_i},\prod_{i=1}^rg_i^{m_i}-\prod_{i=1}^s\psi_i^{\mu_i}\)\mid h.$$

-   If in addition $$\gcd(f_1\cdots f_{\ell}-1,g_1\cdots g_r-1)=1,$$ then there exists a finite set $S$ and monoids $\cL_t \subseteq \N^{\ell+k+r+s}$, $t  \in S$, such that the remaining set $$\cN = \N^{\ell+k+r+s}\setminus \cup_{t\in S} \cL_t$$ is of positive asymptotic density and 