Rational points on Grassmannians and unlikely intersections in tori (Q2788663)

The authors present a new proof of a finiteness theorem due to \textit{E. Bombieri} et al. [Int. Math. Res. Not. 1999, No. 20, 1119--1140 (1999; Zbl 0938.11031)] concerning intersections of irreducible curves defined over \(\overline{\mathbb{Q}}\) in the algebraic torus \(\mathbb{G}_n^m\) with algebraic subgroups of dimension \(n- 2\). Their proof is an application of a method that was originally introduced by the last two authors [Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei 9, 149--162 (2008; Zbl 1164.11029)] in order to give an alternative proof of the Manin-Mumford conjecture. Furthermore, their proof uses a recent result by the third author [Ann. Math. 173, 1779--1840 (2011; Zbl 1243.14022)] on counting rational points of bounded height on compact subanalytic sets in real Euclidean space via \(o\)-minimal structures.