An upper bound for the generalized greatest common divisor of rational points

TL;DR

This paper establishes an upper bound for the generalized greatest common divisor of rational points on a smooth projective variety over a number field, using a uniform Riemann--Roch type inequality.

Contribution

It introduces a new upper bound for the gcd of rational points relative to subvarieties, based on a uniform Riemann--Roch inequality.

Findings

Provides an explicit upper bound for the gcd of rational points.

Develops a uniform Riemann--Roch type inequality applicable to the problem.

Enhances understanding of the distribution of rational points on algebraic varieties.

Abstract

Let $X$ be a smooth projective variety defined over a number field $K$. We give an upper bound for the generalized greatest common divisor of a point $x\in X$ with respect to an irreducible subvariety $Y\subseteq X$ also defined over $K$. To prove the result, we stablish a rather uniform Riemann--Roch type inequality.