Mahler measure

In mathematics, the Mahler measure $M (p)$ of a polynomial p is

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

Here p is assumed complex-valued and

$||p||_{\alpha}\,$

is the l_α norm of p.

It can be shown that if

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

then

$M(p) = |a| \prod_{|\alpha_i| \ge 1} |\alpha_i|.$

The measure is named after Kurt Mahler.

[edit] Properties

The Mahler measure is multiplicative, i.e. M(pq) = M(p)M(q).
If p is an irreducible polynomial with $p(0) \ne 0$ and $M (p) = 1$ , then p is a cyclotomic polynomial.

Jensen, J. L. "Sur un nouvel et important théorème de la théorie des fonctions." Acta Mathematica 22, 359–364, 1899.
Mossinghoff, M. J. "Polynomials with Small Mahler Measure." Mathematics of Computation 67, 1697–1705 and S11–S14, 1998.