are multiplicatively independent.

Recalling the definition of the polynomials $G_i$, $i=1,\ldots,s$, and applying again Lemma \[lem:multconst\], we get that the polynomials $F_i(X_1,\alpha_2,\ldots,\alpha_{\ell})$, $i=1,\ldots,s$, are multiplicatively independent in $\K(X_1)^*/\K^*$, which concludes the proof.

Multiplicities of zeroes
------------------------

To prove Theorem \[thm:genAR2\] we need a uniform bound for the multiplicities of zeros of polynomials of the form $f_1^{n_1}\cdots f_{\ell}^{n_{\ell}}-g_1^{m_1}\cdots g_r^{m_r}$. We present such a result below, as well as deduce as a consequence a similar uniform bound for rational functions, which we hope to be of independent interest.

For a rational function $h\in\C(T)$, we denote by $\Mult(h)$ the largest multiplicity and by $Z(h)$ the set of zeros of $h$ in $\C$, respectively. We also recall that for a polynomial $f\in\C[T]$, we use the notation $d_f$ for the degree of $f$.

\[lem:ABCpoly\] Let $f_1,\ldots,f_{\ell},g_1,\ldots,g_r\in\C[T]$ be polynomials satisfying $Z(f_1\cdots f_{\ell})\cap Z(g_1\cdots g_{\ell})=\emptyset$. Then, for all $n_1,\ldots,n_{\ell},m_1,\ldots,m_r\ge 0$, we have $$\Mult\(f_1^{n_1}\cdots f_{\ell}^{n_{\ell}}-g_1^{m_1}\cdots g_r^{m_r}\)\le \sum_{i=1}^{\ell}d_{f_i}+\sum_{j=1}^rd_{g_j}.$$

We denote $\vec{n}=(n_1,\ldots,n_{\ell})\in\N^{\ell}$ and $\vec{m}=(m_1,\ldots,m_r)\in\N^r$.

Writing the factorisation into linear factors, we have $$f_1^{n_1}\cdots f_{\ell}^{n_{\ell}}-g_1^{m_1}\cdots g_{r}^{m_r}=a_{\vec{n},\vec{m}}\prod_{t\in Z\(f_1^{n_1}\cdots f_{\ell}^{n_{\ell}}-g_1^{m_1}\cdots g_{r}^{m_r}\)}(T-t)^{e_t},$$ where $a_{\vec{n},\vec{m}}\in\C$ is the leading coefficient of $f_1^{n_1}\cdots f_{\ell}^{n_{\ell}}-g_1^{m_1}\cdots g_{r}^{m_r}$.

For simplicity we denote by $$S_{\vec{n},\vec{m}}=Z\(f_1^{n_1}\cdots f_{\ell}^{n_{\ell}}-g_1^{m_1}\cdots g_{r}^{m_r}\).$$

Let $\Mult=\max_{t\in S_{\vec{n},\vec{m}}} e_t$ be the largest multiplicity of the zeros of $f_1^{n_1}\cdots f_{\ell}^{n_{\ell}}-g_1^{m_1}\cdots 