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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.
Science Fair Project Encyclopedia Contents Page
09-23-2007 01:00:40
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