Rational points on Grassmannians and unlikely intersections in tori

TL;DR

This paper offers an alternative proof of a finiteness theorem related to intersections of curves with algebraic subgroups in the multiplicative group, utilizing Pila and Zannier's method for unlikely intersections.

Contribution

It introduces a new proof technique for a classical finiteness result, expanding the toolkit for studying unlikely intersections in algebraic groups.

Findings

Provides an alternative proof of Bombieri, Masser, and Zannier's finiteness theorem.

Demonstrates the application of Pila and Zannier's method to unlikely intersections.

Enhances understanding of rational points on Grassmannians and their intersections.

Abstract

In this paper, we present an alternative proof of a finiteness theorem due to Bombieri, Masser and Zannier concerning intersections of a curve in the multiplicative group of dimension n with algebraic subgroups of dimension n-2. 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".