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.