\ldots, \ell+1,$$ where the product runs over all primes $p\le s(R)$, that contain all the points in $Z(R)$ with coordinates roots of unity. In particular, all points $\(F_1({{\boldsymbol{\alpha}}}),\ldots,F_{\ell+1}({{\boldsymbol{\alpha}}})\)$ such that $F_i({{\boldsymbol{\alpha}}})^{n_i}=1$, $i=1,\ldots,\ell+1$, lie in these algebraic subgroups. It remains to estimate $s(R)$.

As $R$ is a polynomial in $\ell+1$ variables and $\deg R\le D^{\ell}$, we have that $s(R)\le  \gamma_{\ell}(D)$, where $ \gamma_{\ell}(D)$ is defined by .

Thus, the points ${{\boldsymbol{\alpha}}}$ such that $F_i({{\boldsymbol{\alpha}}})^{n_i}=1$, $i=1,\ldots,\ell+1$, lie in at most $N(H)$ algebraic varieties, each defined by at most $\ell+1$ polynomials of the form $F_1^{b_{j,1}}F_2^{b_{j,2}}\cdots F_{\ell+1}^{b_{j,\ell+1}}-1$ (note that these polynomials are non constant since $F_1,\ldots,F_{\ell+1}$ are multiplicatively independent) of degree at most $$\sum_{i=1}^{\ell+1}b_{j,i} \deg F_i
\le (\ell+1)D^{\ell+1}\prod_{p\le  \gamma_{\ell}(D)}p,\qquad j = 1, \ldots, \ell+1,$$ where the product runs over all primes $p\le  \gamma_{\ell}(D)$.

Final Comments and Questions
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Extensions over $\C$
--------------------

Lemma \[lem:bmz\] gives only the finiteness of the intersection of curves in $\G_m^{\ell}$ with algebraic subgroups. As already mentioned after Theorem \[thm:genAR2\], it is of high interest to have available uniform bounds for the size of this intersection. This implies uniform bounds on the degree of $h$ in Theorem \[thm:genAR2\].

More generally, one can ask for the number of solutions to $f(x,y) 