for any vector $$(n_1,\ldots,n_{\ell},\nu_1,\ldots,\nu_k,m_1,\ldots,m_r,\mu_1,\ldots,\mu_s)\in\cN$$ we have $$\gcd\(\prod_{i=1}^{\ell}f_i^{n_i}-\prod_{i=1}^k\varphi_i^{\nu_i},\prod_{i=1}^rg_i^{m_i}-\prod_{i=1}^s\psi_i^{\mu_i}\)=1.$$

Although we prefer to keep the language of polynomials, one can easily see that Theorem \[thm:genAR2\] can be reformulated in terms of $S$-units in $\C[T]$ and implies that for any set of $S$-units, there exists a polynomial $h\in \C[T]$ such that for any multiplicatively independent $S$-units $U,V$ we have $$\gcd\(U-1,V-1\)\mid h.$$ In particular, this extension of [@AR] is fully analogous to the aforementioned extension of [@CZ05] over [@BCZ].

We also compare Theorem \[thm:genAR2\], which for multiplicatively independent $S$-units $U,V$, gives a uniform bound for $\deg\gcd(U-1,V-1)$, while the result of Corvaja and Zannier [@CZ1-08 Corollary 2.3] gives $$\deg \gcd(U-1,V-1)\ll \max(\deg U,\deg V)^{2/3}.$$ However, [@CZ1-08 Corollary 2.3] applies to more general situations.

It is interesting to unify Theorems \[thm:genAR1\] and \[thm:genAR2\] and obtain a similar result for $$\gcd\(h_1\(f_1^{n_1}\cdots f_{\ell}^{n_{\ell}}\),h_2\(g_1^{m_1}\cdots g_r^{m_r}\)\),$$ where $h_1,h_2\in\C[T]$. Similar ideas may work for this case however they require a uniform bound for the number of points on intersections of curves in $\G_{m}^{\ell+r}$ with algebraic subgroups of dimension $k \le \ell+r-2$ in Lemma \[lem:bmz\]. We note that for $\ell=r=1$ this was possible due to the uniform bounds of [@BS]. However, no such bounds are available in the more general case that we 