Weighted greatest common divisors and weighted heights

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 $\mathfrak{w}=(q_0,\dots ,q_n)$, we use the concept of the weighted greatest common divisor to define a height $\mathfrak{h}(\mathfrak{p})$ on weighted projective spaces ${\mathbb{W}\mathbb{P}}_{\mathfrak{w}}^n(k)$. We prove some of the basic properties of this weighted height, including an analogue of the Northcott’s theorem for heights on projective spaces.