Schanuel's theorem for heights defined via extension fields

TL;DR

This paper establishes asymptotic formulas for counting algebraic numbers with bounded heights, extending Schanuel's theorem to new height functions defined via extension fields, and analyzes related constants.

Contribution

It introduces a general framework for heights defined through extension fields and provides asymptotic counting formulas, addressing questions posed by Loher and Masser.

Findings

Asymptotic formula for the number of algebraic numbers with bounded height involving extension fields.

A new class of height functions is analyzed and incorporated into counting problems.

Sharp bounds on the leading constant in the asymptotic formula are established.

Abstract

Let $k$ be a number field, let $\theta$ be a nonzero algebraic number, and let $H(\cdot)$ be the Weil height on the algebraic numbers. In response to a question by T. Loher and D. W. Masser, we prove an asymptotic formula for the number of $\alpha \in k$ with $H(\alpha \theta)\leq X$. We also prove an asymptotic counting result for a new class of height functions defined via extension fields of $k$. This provides a conceptual framework for Loher and Masser's problem and generalizations thereof. Moreover, we analyze the leading constant in our asymptotic formula for Loher and Masser's problem. In particular, we prove a sharp upper bound in terms of the classical Schanuel constant.