\in \C[X_1, \ldots, X_{\ell}]$, we denote by $Z(F_1,\ldots,F_s)$ their zero set in $\C^{m}$.

Throughout the paper we assume that the greatest common divisor of two (or more) polynomials is monic, so it is well-defined.

We also define here the main concept of this paper.

The polynomials $F_1,\ldots,F_{s}\in\C[X_1,\ldots,X_{\ell}]$ are [*multiplicatively independent*]{} if there exists no nonzero vector $(\nu_1,\ldots,\nu_{s})\in\Z^{s}$ such that $$F_1^{\nu_1}\cdots F_s^{\nu_{s}}=1.$$ Similarly, we say that the polynomials $F_1,\ldots,F_{s}\in\C[X_1,\ldots,X_{\ell}]$ are [*multiplicatively independent in the group $\C(X_1,\ldots,X_{\ell})^*/\C^*$*]{} if there exists no nonzero vector $(\nu_1,\ldots,\nu_{s})\in\Z^{s}$ and $a\in\C^*$ such that $$F_1^{\nu_1}\cdots F_s^{\nu_{s}}=a.$$

We present now in more details the main results of this paper.

Our results: univariate case
----------------------------

Section \[sec:prel\] is dedicated to outlining the tools and results needed along the paper. In particular, in Section \[sec:ART\] we recall the result of [@AR Theorem 1] and, using a uniform bound for the number of points on a curve with coordinates roots of unity due to Beukers and Smyth [@BS], we derive in Lemma \[lem:univAR\] a version of  that gives an upper bound on $\deg \gcd(f^n-1,g^m-1)$ that depends only the degrees of $f$ and $g$ (rather than on the polynomials themselves).

Such a uniform bound is crucial for some of our main results presented 