multivariate generalisation is based on a result of Granville and Rudnick [@GranRud Corollary 3.1], which describes the structure of torsion points on hypersurfaces, see Lemma \[lem:GranRud\].

\[thm:CommonZeros\] Let $F_1,\ldots,F_{\ell+1}\in\C[X_1,\ldots,X_{\ell}]$ be multiplicatively independent polynomials of degree at most $D$. Then, $$\bigcup_{n_1,\ldots,n_{\ell+1}\in\N}Z\(F_1^{n_1}-1,\ldots,F_{\ell+1}^{n_{\ell+1}}-1\)$$ is contained in at most $$N \le  (0.792\gamma_{\ell}(D)/\log \(\gamma_{\ell}(D)+1\))^{\gamma_{\ell}(D)}$$ algebraic varieties, each defined by at most $\ell+1$ polynomials of degree at most $(\ell+1)D^{\ell}\prod_{p\le  \gamma_{\ell}(D)}p$, where the product runs over all primes $p\le  \gamma_{\ell}(D)$.

We recall that by the prime number theorem, for an integer $k\ge 1$, $$\prod_{p\le k}p = \exp(k + o(k)).$$

We note that the bound on the number of algebraic subgroups that contain the points on $\cH$ with coordinates roots of unity may also follow from [@AS Theorem 1.1 and Lemma 2.6], which says that, for a hypersurface defined by $H\in\C[X_1,\ldots,X_s]$, $s\ge 2$, of degree $D$, the number of maximal torsion cosets contained in $\cH$ is at most $$c_1(s)D^{c_2(s)}$$ with $$\label{eq:c1c2}
c_1(s)=s^{\frac{3}{2}(2+s)5^{s}}\quad \textrm{and}\quad c_2(s)=\frac{1}{16}\(49 \cdot 5^{s-2}-4s-9\).$$

We also note that any argument that is based on the Bezout Theorem ultimately leads to bounds that depend on the exponents $n_1,\ldots,n_{\ell+1}$, while the bounds of Theorem \[thm:CommonZeros\] depend only on the initial data.

We conclude the paper with comments on 