Lower bounds on the projective heights of algebraic points

TL;DR

This paper extends a known lower bound on the sum of Weil heights of algebraic numbers to cases where the sum equals any totally real algebraic number, broadening the scope of previous results.

Contribution

It generalizes the Beukers-Zagier lower bound to include arbitrary totally real algebraic numbers, expanding its applicability.

Findings

Extended the lower bound to all totally real algebraic numbers

Connected the result to Schinzel's theorem on totally real algebraic integers

Provided a broader framework for height bounds in algebraic number theory

Abstract

If $\alpha_1,\ldots,\alpha_r$ are algebraic numbers such that $$N=\sum_{i=1}^r\alpha_i \ne \sum_{i=1}^r\alpha_i^{-1}$$ for some integer $N$, then a theorem of Beukers and Zagier gives the best possible lower bound on $$\sum_{i=1}^r\log h(\alpha_i)$$ where $h$ denotes the Weil Height. We will extend this result to allow $N$ to be any totally real algebraic number. Our generalization includes a consequence of a theorem of Schinzel which bounds the height of a totally real algebraic integer.