# Greatest common divisors of analytic functions and Nevanlinna theory on algebraic tori

## TL;DR

This paper develops new bounds for the common zeros of meromorphic functions on algebraic tori, integrating Nevanlinna theory with Diophantine approximation techniques to advance the understanding of greatest common divisors in complex analysis.

## Contribution

It introduces a general form of the conjectural asymptotic gcd inequality and extends results to moving targets, combining Nevanlinna theory with Diophantine methods.

## Key findings

- Proved a general asymptotic gcd inequality for meromorphic functions.
- Extended results to moving targets in algebraic tori.
- Enhanced bounds using stronger inequalities from Nevanlinna theory.

## Abstract

We study upper bounds for the counting function of common zeros of two meromorphic functions in various contexts. The proofs and results are inspired by recent work involving greatest common divisors in Diophantine approximation, to which we introduce additional techniques to take advantage of the stronger inequalities available in Nevanlinna theory. In particular, we prove a general version of a conjectural "asymptotic gcd" inequality of Pasten and the second author, and consider moving targets versions of our results.