is at most $\min(d_f,d_g)$. The bound now follows.

Intersection of curves with algebraic groups {#sec:intesect}
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We define $\G_m^k$ as the set of $k$-tuples of non-zero complex numbers equiped with the group law defined by component-wise multiplication. We refer to [@Sch Appendix by Umberto Zannier] for necessary definitions on algebraic subgroups.

One of the main tools in our paper is a result on the finiteness of the number of points on the intersection of a curve in $\G_m^k$ with algebraic subgroups of codimension at least $2$, initially obtained in [@BMZ99] for curves over $\ovQ$, and later on extended over $\C$, see [@BMZ08; @Mau] and references therein. We present it in the following form.

\[lem:bmz\] Let $C\subset \G_m^{k}$, $k\ge 2$, be an irreducible curve over $\C$. Assume that for every nonzero vector $(r_1,\ldots,r_{k})\in\Z^k$ the monomial $X_1^{r_1}\cdots X_{k}^{r_{k}}$ is not identically $1$ on $C$. Then there are finitely many points $(x_1,\ldots,x_{k})\in C(\C)$ for which there exist linearly independent vectors $(a_1,\ldots,a_{k}), (b_1,\ldots,b_{k})$ in $\Z^{k}$ such that $$x_1^{a_1}\cdots x_{k}^{a_{k}}=x_1^{b_1}\cdots x_{k}^{b_{k}}=1.$$

\[rem:translate\] As explained in [@BMZ99], the condition of Lemma \[lem:bmz\] that the monomial $X_1^{r_1}\cdots X_{k}^{r_{k}}$ is not identically $1$ on $C$ is equivalent with the curve not being contained in a proper subtorus of 