g_{r}^{m_r}$.

The bound for $\Mult$ follows immediately from the polynomial $ABC$ theorem (proved first by Stothers [@St], and then independently by Mason [@Mas] and Silverman [@Si84]). Indeed, we apply the polynomial $ABC$ theorem with $A=a_{\vec{n},\vec{m}}\prod_{t\in S_{\vec{n},\vec{m}}}(T-t)^{e_t}$, $B=f_1^{n_1}\cdots f_{\ell}^{n_{\ell}}$ and $C=g_1^{m_1}\cdots g_{r}^{m_r}$, which are pairwise corpime. We get $$\label{eq:l}
\sum_{t\in S_{\vec{n},\vec{m}}}e_t\le \sum_{i=1}^{\ell}d_{f_i}+\sum_{j=1}^rd_{g_j}+\#S_{\vec{n},\vec{m}}-1.$$ Taking into account that $$\sum_{t\in S_{\vec{n},\vec{m}}}e_t\ge \Mult +\#S_{\vec{n},\vec{m}}-1,$$ from  we obtain $$\Mult\le \sum_{i=1}^{\ell}d_{f_i}+\sum_{j=1}^rd_{g_j},$$ which concludes the proof.

We present now a similar result for rational functions.

\[cor:ABCrat\] Let $h_1,\ldots,h_{\ell}\in\C(T)$, $h_i=f_i/g_i$, $f_i,g_i\in\C[T]$, $i=1,\ldots,\ell$, with $Z(f_1\cdots f_{\ell})\cap Z(g_1\cdots g_{\ell})=\emptyset$. Then, for all $n_1,\ldots,n_{\ell}\ge 0$, we have $$\Mult\(h_1^{n_1}\cdots h_{\ell}^{n_{\ell}}-1\)\le \sum_{i=1}^{\ell} \(\deg f_i+\deg g_i\).$$

We note that $Z\(h_1^{n_1}\cdots h_{\ell}^{n_{\ell}}-1\)=Z\(f_1^{n_1}\cdots f_{\ell}^{n_{\ell}}-g_1^{n_1}\cdots g_{\ell}^{n_{\ell}}\)$. The result now follows directly from Lemma \[lem:ABCpoly\] applied with $r=\ell$ and $m_i=n_i$, $i=1,\ldots,\ell$.

Algebraic dependence
--------------------

We need the following result [@Plo Theorem 1.1] which gives a degree bound for the annihilating polynomial of algebraically dependent polynomials, which is always the case when the number of polynomials exceeds the number of variables. The result holds over any field, but we present it only over $\C$.

\[lem:perron\] Let $F_1,\ldots,F_{\ell+1}\in\C[X_1,\ldots,X_{\ell}]$ be of degree at most $D$. Then there exists a nonzero polynomial $R\in\C[Z_1,\ldots,Z_{\ell+1}]$ of degree at most $D^{\ell}$ such that $R(F_1,\ldots,F_{\ell+1})=0$.

Proofs of 