A lower bound for the height of a rational function at $S$-unit points

Let $\mathrm{\Gamma }$ be a finitely generated subgroup of the multiplicative group ${\text{G}}_m^2(\bar{Q})$. Let $p(X,Y),q(X,Y)\in \text{bat}Q$ be two coprime polynomials not both vanishing at $(0,0)$; let $ϵ>0$. We prove that, for all $(u,v)\in \mathrm{\Gamma }$ outside a proper Zariski closed subset of $G_m^2$, the height of $p(u,v)\mathrm{/}q(u,v)$ verifies $h(p(u,v)/q(u,v))>h(1:p(u,v):q(u,v))-\epsilon \max(h(uu),h(v))$. As a consequence, we deduce upper bounds for (a generalized notion of) the g.c.d. of $u-1,v-1$ for $u,v$ running over $\mathrm{\Gamma }$.