Weighted greatest common divisors and weighted heights

Abstract

We introduce the weighted greatest common divisor of a tuple of integers and explore some of its basic properties. Furthermore, for a set of weights w=(q0,...,qn), we use the concept of the weighted greatest common divisor to define a height h(p) on weighted projective spaces WPwn(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.