Mahler measure

In mathematics, the Mahler measure $M(p)$ of a polynomial $p(z)$ with complex coefficients is defined as

M(p)=|a|\prod _{|\alpha _{i}|\geq 1}|\alpha _{i}|=|a|\prod _{i=1}^{n}\max\{1,|\alpha _{i}|\},

where $p(z)$ factorizes over the complex numbers $\mathbb {C}$ as

p(z)=a(z-\alpha _{1})(z-\alpha _{2})\cdots (z-\alpha _{n}).

The Mahler measure can be viewed as a kind of height function. Using Jensen's formula, it can be proved that this measure is also equal to the geometric mean of $|p(z)|$ for $z$ on the unit circle (i.e., $|z|=1$ ):

M(p)=\exp \left(\int _{0}^{1}\ln(|p(e^{2\pi i\theta })|)\,d\theta \right).

By extension, the Mahler measure of an algebraic number $\alpha$ is defined as the Mahler measure of the minimal polynomial of $\alpha$ over $\mathbb {Q}$ . In particular, if $\alpha$ is a Pisot number or a Salem number, then its Mahler measure is simply $\alpha$ .

The Mahler measure is named after the German-born Australian mathematician Kurt Mahler.

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