# Mahler measure

By [ur](https://paragraph.com/@ur) · 2022-06-08

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In [mathematics](https://en.wikipedia.org/wiki/Mathematics), the **Mahler measure** {\\displaystyle M(p)} **of a** [**polynomial**](https://en.wikipedia.org/wiki/Polynomial) {\\displaystyle p(z)} with [complex](https://en.wikipedia.org/wiki/Complex_number) [coefficients](https://en.wikipedia.org/wiki/Coefficient) 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](https://en.wikipedia.org/wiki/Height_of_a_polynomial). Using [Jensen's formula](https://en.wikipedia.org/wiki/Jensen%27s_formula), it can be proved that this measure is also equal to the [geometric mean](https://en.wikipedia.org/wiki/Geometric_mean) of {\\displaystyle |p(z)|} for {\\displaystyle z} on the [unit circle](https://en.wikipedia.org/wiki/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**](https://en.wikipedia.org/wiki/Algebraic_number) {\\displaystyle \\alpha } is defined as the Mahler measure of the [minimal polynomial](https://en.wikipedia.org/wiki/Minimal_polynomial_\(field_theory\)) of {\\displaystyle \\alpha } over {\\displaystyle \\mathbb {Q} }. In particular, if {\\displaystyle \\alpha } is a [Pisot number](https://en.wikipedia.org/wiki/Pisot_number) or a [Salem number](https://en.wikipedia.org/wiki/Salem_number), then its Mahler measure is simply {\\displaystyle \\alpha }.

The Mahler measure is named after the German-born Australian [mathematician](https://en.wikipedia.org/wiki/Mathematician) [Kurt Mahler](https://en.wikipedia.org/wiki/Kurt_Mahler).

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*Originally published on [ur](https://paragraph.com/@ur/mahler-measure)*