An effective lower bound for the height of algebraic numbers

TL;DR

This paper establishes a new lower bound for the height of non-zero algebraic numbers of degree at least two that are not roots of unity, refining understanding of algebraic number complexity.

Contribution

The paper provides a novel lower bound for the height of algebraic numbers, improving previous estimates and contributing to number theory and algebraic geometry.

Findings

Proves a lower bound involving degree and logarithmic functions.

Shows the bound applies to all non-zero algebraic numbers not roots of unity.

Enhances theoretical understanding of algebraic number heights.

Abstract

We prove that if $\alpha$ is a non-zero algebraic number of degree $d \geq 2$ which is not a root of unity, then $dh(\alpha)>(1/4) (\log(\log (d))/\log(d))^3.