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[1]

$M(p) = \exp\left( \frac{1}{(2\pi)^n} \int_0^{2\pi} \int_0^{2\pi} \cdots \int_0^{2\pi} \ln \Bigl( \bigl |p(e^{i\theta_1}, e^{i\theta_2}, \ldots, e^{i\theta_n}) \bigr| \Bigr) \, d\theta_1\, d\theta_2\cdots d\theta_n \right).$

If $p$ vanishes on the torus $(S^1)^n$, then the convergence of the integral defining $M(p)$ is not obvious, but it is known that $M(p)$ does converge and is equal to a limit of one-variable Mahler measures.[2] For $\boldsymbol r=(r_1,\ldots,r_n)\in\mathbb{N}^n$, let $q(\boldsymbol r)$ be the smallest number such that there is a nontrivial relation $r_1s_1+r_2s_2+\cdots+r_ns_n$ with $s_i\in\mathbb{Z}$ and all $|s_i|\le q(\boldsymbol r)$. Then

$M\bigl(p(x_1,\ldots,x_n)\bigr) = \lim_{\boldsymbol r\in\mathbb{N}^n\atop q(\boldsymbol r)\to\infty} M\bigl(p(x^{r_1},x^{r_2},\ldots,x^{r_n})\bigr).$

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