Mahler measure

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

$${\displaystyle M(p)=|a|\prod _{|\alpha _{i}|\geq 1}|\alpha _{i}|=|a|\prod _{i=1}^{n}\max\{1,|\alpha _{i}|\},}$$ where ${\displaystyle p(z)}$ factorizes over the complex numbers ${\displaystyle \mathbb {C} }$ as $${\displaystyle p(z)=a(z-\alpha _{1})(z-\alpha _{2})\cdots (z-\alpha _{n}).}$$

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 ${\displaystyle |p(z)|}$ for ${\displaystyle z}$ on the unit circle (i.e., ${\displaystyle |z|=1}$): $${\displaystyle M(p)=\exp \left(\int _{0}^{1}\ln(|p(e^{2\pi i\theta })|)\,d\theta \right).}$$

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

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