= Absolute logarithmic Weil height
{c}
{title2=$h(\alpha)$}
= Logarithmic height
{synonym}
The <Absolute logarithmic Weil height> is the logarithm $h(\alpha)=\log H(\alpha)$ of the absolute multiplicative height of an <algebraic number>. If its primitive <integer> <minimal polynomial> is $f(X)=a_d\prod_{i=1}^d(X-\alpha_i)$, then
$$
h(\alpha)=\frac1d\left(\log|a_d|+\sum_{i=1}^d\log\max(1,|\alpha_i|)\right).
$$
Thus it is the logarithm of the <Mahler measure> divided by the degree.
For a reduced rational $a/b$, $h(a/b)=\log\max(|a|,|b|)$. It satisfies $h(\alpha^m)=|m|h(\alpha)$ for nonzero $\alpha$ and integral $m$. Unlike a single absolute value, it records denominator complexity and the behavior of all conjugates, making it suitable for arithmetic bounds that should not depend on a chosen embedding.
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