# Weighted greatest common divisors and weighted heights

## TL;DR

This paper introduces the concept of weighted greatest common divisors and uses it to define a new height function on weighted projective spaces, extending classical height theory with basic properties and an analogue of Northcott's theorem.

## Contribution

It defines weighted greatest common divisors and develops a corresponding height function on weighted projective spaces, establishing foundational properties and extending height theory.

## Key findings

- Defined weighted greatest common divisors for integer tuples.
- Introduced a height function on weighted projective spaces.
- Proved an analogue of Northcott's theorem for the new height.

## Abstract

We introduce the weighted greatest common divisor of a tuple of integers and explore some of it basic properties. Furthermore, for a set of heights $\mathfrak w=(q_0, \ldots , q_n)$, we use the concept of the weighted greatest common divisor to define a height $\mathfrak{h} (\mathfrak p)$ on weighted projective spaces $\mathbb{WP}_{\mathfrak w}^n (k)$. We prove some of the basic properties of this weighted height, including an analogue of the Northcott's theorem for heights on projective spaces.