Small totally $p$-adic algebraic numbers

TL;DR

This paper provides an elementary proof that totally p-adic algebraic numbers, excluding zero and roots of unity, have a positive lower bound on their absolute logarithmic Weil-height, improving known bounds.

Contribution

It offers the best known lower bounds for the Weil-height of totally p-adic numbers, based on a simplified proof inspired by C. Petsche.

Findings

Lower bounds differ from the true value by less than log(3)/p

Provides an elementary proof of height bounds for totally p-adic numbers

Improves existing bounds in this setting

Abstract

The purpose of this note is to give a short and elementary proof of the fact, that the absolute logarithmic Weil-height is bounded from below by a positive constant for all totally p-adic numbers which are neither zero nor a root of unity. The proof is based on an idea of C. Petsche and gives the best known lower bounds in this setting. These bounds differ from the truth by a term of less than $\log(3)/p$.