Greatest Common Divisors on the Complement of Numerically Parallel Divisors

TL;DR

This paper extends inequalities involving greatest common divisors of functions at integral points to the setting of numerically parallel divisors, utilizing geometric and height theory methods to generalize previous results.

Contribution

It generalizes existing GCD inequalities to numerically parallel divisors and analyzes the exceptional set using advanced geometric and height techniques.

Findings

Proved new inequalities for GCDs with numerically parallel divisors.

Reduced complex cases to the known case of $\

Analyzed the exceptional set for counting and proximity functions.

Abstract

We prove inequalities involving greatest common divisors of functions at integral points with respect to numerically parallel divisors, generalizing a result of Wang and Yasufuku (after work of Bugeaud-Corvaja-Zannier, Corvaja-Zannier, and the second author). After applying a result of Vojta on integral points on subvarieties of semiabelian varieties, we use geometry and the theory of heights to reduce to the (known) case of $\mathbb{G}_m^n$. In addition to proving results in a broader context than previously considered, we also study the exceptional set in this setting, for both the counting function and the proximity function.