with the polynomials $\prod_{i=1}^{\ell}f_i^{n_i}-\prod_{i=1}^k\varphi_i^{\nu_i}$ and $\prod_{i=1}^rg_i^{m_i}-\prod_{i=1}^s\psi_i^{\mu_i}$.

Indeed, if we denote by $\Mult_1$ and $\Mult_2$ the largest multiplicity of roots in $S$ of the first and second polynomials, respectively, we get $$\Mult_1\le \sum_{i=1}^{\ell} d_{f_i}+\sum_{i=1}^kd_{\varphi_i},\quad \Mult_2\le  \sum_{i=1}^{r} d_{g_i}+\sum_{i=1}^sd_{\psi_i}.$$

Thus, there exists a polynomial $h\in\C[T]$ defined by $$h=\prod_{t\in S} (T-t)^{d},\quad d=\min\(\sum_{i=1}^{\ell} d_{f_i}+\sum_{i=1}^kd_{\varphi_i},\sum_{i=1}^{r} d_{g_i}+\sum_{i=1}^sd_{\psi_i}\)$$ such that $\cD_{\vec{v},\vec{w}}\mid h$ for every vectors $\vec{v},\vec{w}$ as above. This concludes the proof of Part [**[i]{}**]{}.

For Part [**[ii]{}**]{}, for each $t\in S$, let $$\cL_t=\{\(\vec{v},\vec{w}\)\in \N^L
~:~(T-t)\mid \cD_{\vec{v},\vec{w}}\}.$$ We note that $\cL_s$ is actually a monoid as the sum of any two elements in $\cL_t$ is also an element of $\cL_t$. As the set $S$ is finite, there are finitely many such monoids $\cL_t$, $t\in S$, such that $\deg \cD_{\vec{v},\vec{w}}\ge 1$ for any $\(\vec{v},\vec{w}\)\in\cL_t$.

We are left to show that $\cup_{t\in S}\cL_t$ is not the entire space $\N^{L}$. Indeed this follows directly from [@AR Theorem 1] as for the diagonal case, that is $\vec{v}=n(\underbrace{1,\ldots,1}_{\ell},0,\ldots,0)\in\N^{\ell+k}$ and $\vec{w}=n(\underbrace{1,\ldots,1}_{r},0,\ldots,0)\in\N^{r+s}$, we have $$\gcd\(\(f_1\cdots f_{\ell}\)^n-1,\(g_1\cdots g_r\)^n-1\)=1$$ infinitely often.

Thus, for any $\(\vec{v},\vec{w}\)$ outside $\cup_{t\in S}\cL_t$, we have $\cD_{\vec{v},\vec{w}}=1$, and we conclude the proof.

Proof of Theorem \[thm:multivAR\] {#sec:multivar1}
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The idea of the proof lies in applying Hilbert’s irreducibility theorem, and in particular Lemma 