Relative Northcott numbers for the weighted Weil heights

TL;DR

This paper introduces a relative Northcott number for weighted Weil heights, extending previous concepts to provide bounds and field extensions related to height functions in number theory.

Contribution

It defines a new relative Northcott number for weighted Weil heights and constructs field extensions with prescribed Northcott numbers, advancing the understanding of height bounds.

Findings

Defined a relative Northcott number for weighted Weil heights

Constructed field extensions with specific Northcott numbers

Extended previous work on Northcott numbers to a relative setting

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.