Relative Northcott numbers for the weighted Weil heights

Abstract

It is fundamental in number theory to calculate lower bounds for height functions. Grizzard studied lower bounds for the Weil height in a relative setting. Vidaux and Videla introduced the Northcott number for a set $A\subset\bar{\mathbb{Q}}$. It bounds the Weil height on $A$ from below, outside the zero-height points and the finitely many small-height points. Pazuki, Technau, and Widmer introduced the weighted Weil heights. These heights generalize both the absolute and relative Weil heights. In this paper, we introduce a relative version of the Northcott number related to the weighted Weil height. We also give a field extension whose Northcott number equals a given positive number. The work is a relative version of the previous work of the author and Sano on the Northcott numbers for the weighted Weil heights.