On a theorem of Garza regarding algebraic numbers with real conjugates

TL;DR

This note provides a new, simple proof of Garza's bound on the height (or Mahler measure) of an algebraic number with real conjugates. The approach adapts Ho-Schinzel-style techniques and relies on a carefully chosen auxiliary function $f$ and a lemma giving exact maxima on the archimedean places. The resulting bound $H(\\alpha)\\ge\\left(\\frac{2^{1-1/R}+\\sqrt{4^{1-1/R}+4}}{2}\\right)^{R/2}$ depends only on the real-conjugate fraction $R=r/d$, and extends from algebraic integers to all algebraic numbers via standard place-height arguments with $H(\\alpha)=H(\\alpha^{-1})$. The work clarifies optimality for $R=1$ and presents an elementary, self-contained alternative to Garza's original argument.

Abstract

We give a new and simple proof of a theorem of Garza estimating the height (or Mahler measure) of an algebraic number with real conjugates.