 Absolute logarithmic Weil height

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 $f(X)=a_d{\prod }_{i=1}^d(X-{\alpha }_i)$, thenThus 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.