$p\le s(H)$, such that if $\xi\in\cH$ is a torsion point, then $\xi\in\cup_{B\in\cB}W_B$, where $$W_B=\bigcap_{j=1}^{k}Z\(X_1^{b_{j,1}}X_2^{b_{j,2}}\cdots X_k^{b_{j,k}}-1\).$$

The proof is essentially given in [@GranRud Corollary 3.1]. Indeed, each matrix $B$ corresponds to a partition of the set $\{1,2,\ldots,s(H)\}$, and thus, the number of matrices $B$ in the set $\cB$ is given by the number of such partitions, which, by [@BT Theorem 2.1], is at most $N(H)$.

The number of rows $n_B$ of a matrix $B \in \cB$ is not specified in [@GranRud Corollary 3.1]. However, we can choose the largest linear independent set of these vectors $\vec{b}_j$, which is of cardinality at most $k$, and all other varieties of the form $Z\(X_1^{b_{i,1}}X_2^{b_{i,2}}\cdots 
        X_k^{b_{i,k}}-1\)$ are defined by combinations of these vectors. Thus, we can consider $n_B\le k$. Repeating some rows if necessary we can take $n_B=k$ which concludes the proof.

Hilbertian fields and multiplicative independence {#sec:Hilb}
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For the first multivariate generalisation of [@AR Theorem 1] we need a result which says that given $F_1,\ldots,F_s\in\C[X_1,\ldots,X_{\ell}]$ that are multiplicatively independent in $\C(X_1,\ldots,X_{\ell})^*/\C^*$, there exists a specialisation $(\alpha_2,\ldots,\alpha_{\ell})\in\C^{\ell-1}$ such that $F_i(X_1,\alpha_2,\ldots,\alpha_{\ell})$, $i=1,\ldots,s$, are multiplicatively independent in $\C(X_1)^*/\C^*$ (see Lemma \[lem:multindep\] below). Such a result follows directly from [@BMZ99 Theorem 1] which says that the points 