# Mahler measure

In mathematics, the Mahler measure $M(p)$ of a polynomial $p(x) \in \mathbb{C}[x]$ with complex coefficients is

$M(p)=\lim_{\tau \rightarrow 0} \|p\|_{\tau} = \exp\left( \frac{1}{2\pi} \int_{0}^{2\pi} \ln(|p(e^{i\theta})|)\, d\theta \right).$

Here

$||p||_{\tau} =\left( { \frac{1}{2\pi} \int_{0}^{2\pi} |p(e^{i\theta})|^\tau \, d\theta } \right)^{1/\tau} \,$

is the $L_\tau$ norm of $p$ (although this is not a true norm for values of $\tau < 1$).

It can be shown that if

$p(z) = a(z-\alpha_1)(z-\alpha_2)\cdots(z-\alpha_n)$

then

$M(p) = |a| \prod_{i=1}^{n}\max\{1,|\alpha_i|\}=|a|\prod_{|\alpha_i| \ge 1} |\alpha_i|.$

The Mahler measure of an algebraic number $\alpha$ is defined as the Mahler measure of the minimal polynomial of $\alpha$ over $\mathbb{Q}$.

The measure is named after Kurt Mahler.

## Properties

• The Mahler measure is multiplicative, i.e. $M(p\,q) = M(p) \cdot M(q).$
• (Kronecker's Theorem) If $p$ is an irreducible monic integer polynomial with $M(p) = 1$, then either $p(z) = z,$ or $p$ is a cyclotomic polynomial.
• Lehmer's conjecture asserts that there is a constant $\mu>1$ such that if $p$ is an irreducible integer polynomial, then either $M(p)=1$ or $M(p)>\mu$.
• The Mahler measure of a monic integer polynomial is a Perron number.

## Higher Dimensional Mahler Measure

The Mahler measure $M(p)$ of a multi-variable polynomial $p(x_1,\ldots,x_n) \in \mathbb{C}[x_1,\ldots,x_n]$ is defined similarly by the formula

$M(p) = \exp\left( \frac{1}{(2\pi)^n} \int_{0}^{2\pi} \int_{0}^{2\pi} \cdots \int_{0}^{2\pi} \ln\Bigl(\bigl|p(e^{i\theta_1},e^{i\theta_2},\ldots,e^{i\theta_n})\bigr|\Bigr)\, d\theta_1\, d\theta_2\cdots d\theta_n \right).$

Multi-variable Mahler measures have been shown, in some cases, to be related to special values of zeta-functions and $L$-functions.