Bounded Height in Pencils of Finitely Generated Subgroups

TL;DR

This paper establishes new height bounds for intersections of finitely generated subgroups in a torus with algebraic varieties within a pencil, significantly extending previous constant case results using novel techniques.

Contribution

It introduces a comprehensive approach to bounding heights in pencils of subgroups, advancing the understanding of their intersection properties.

Findings

Height bounds for subgroup intersections in a pencil

Extension from constant to variable case

Development of new analytical techniques

Abstract

We prove height bounds concerning intersections of finitely generated subgroups in a torus with algebraic subvarieties, all varying in a pencil. This vastly extends the previously treated constant case and involves entirely different, and more delicate, techniques.