# All Science Fair Projects

## Science Fair Project Encyclopedia for Schools!

 Search    Browse    Forum  Coach    Links    Editor    Help    Tell-a-Friend    Encyclopedia    Dictionary

# Science Fair Project Encyclopedia

For information on any area of science that interests you,
enter a keyword (eg. scientific method, molecule, cloud, carbohydrate etc.).
Or else, you can start by choosing any of the categories below.

# 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 | | α

is the lα 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 name is for Kurt Mahler .

### 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.
03-10-2013 05:06:04