# Generalized GCD for toric Fano varieties

## TL;DR

This paper investigates the greatest common divisor problem within the context of toric Fano varieties, employing advanced geometric and arithmetic tools to establish bounds and deepen understanding of GCD behavior in algebraic geometry.

## Contribution

It introduces a novel approach combining Okounkov bodies and Vojta's theorem to analyze GCD problems for toric Fano varieties, extending classical results to a broader geometric setting.

## Key findings

- Established bounds for GCD of pairs of algebraic numbers
- Applied Okounkov bodies and Vojta's theorem in a new context
- Extended classical GCD results to toric Fano varieties

## Abstract

We study the greatest common divisor problem for torus invariant blowing-up morphisms of nonsingular toric Fano varieties. Our main result applies the theory of Okounkov bodies together with an arithmetic form of Cartan's Second Main theorem, which has been established by Ru and Vojta. It also builds on Silverman's geometric concept of greatest common divisor. As a special case of our results, we deduce a bound for the generalized greatest common divisor of pairs of nonzero algebraic numbers.