future work.

Preliminaries {#sec:prel}
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The Ailon-Rudnick Theorem {#sec:ART}
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The Ailon-Rudnick theorem is based on a well-known conjecture of Lang, proved by Ihara, Serre and Tate [@L], which says that a plane curve, which does not contain a translate of an algebraic subgroup of $\G_m^2$, contains only finitely many torsion points. In this case, Beukers and Smyth [@BS Section 4.1] give a uniform bound for the number of such points (see Lemma \[lem:BS\] below), and Corvaja and Zannier [@CZ08] give an upper bound (actually for curves in $\G_m^n$) for the maximal order of torsion points on the curve.

We now present the result of Ailon and Rudnick [@AR Theorem 1], coupled with the result of Beukers and Smyth [@BS Section 4.1], which we first mention separately.

\[lem:BS\] An algebraic curve $H(X,Y)=0$ has at most $11(\deg H)^2$ points which are roots of unity, unless $H$ has a factor of the form $X^i-\rho Y^j$ or $X^iY^j-\rho$ for some nonnegative integers $i,j$ not both zero and some root of unity $\rho$.

Using Lemma \[lem:BS\], we obtain the following more precise form of [@AR Theorem 1].

\[lem:univAR\] Let $f,g\in\C[T]$ be non constant polynomials. If $f$ and $g$ are multiplicatively independent, then for all 