Rational points on Grassmannians and unlikely intersections in tori

In this paper, we present an alternative proof of a finiteness theorem due to Bombieri, Masser and Zannier concerning intersections of a curve in $\mathbb{G}_m^n$ with algebraic subgroups of dimension $n−2$. Actually, the present conclusion will give more uniform bounds which respect to the former statement. The proof uses a method introduced for the first time by Pila and Zannier to give an alternative proof of Manin-Mumford conjecture and a theorem to count points that satisfy a certain number of linear conditions with rational coefficients. This method has been largely used in many different problems in the context of “unlikely intersections”.