Mahler measure

Mahler measure

In mathematics, the Mahler measure M(p) of a polynomial p 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 is assumed complex-valued and

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

is the Lτ norm of p (although this is not a true norm for values of τ < 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 α is defined as the Mahler measure of the minimal polynomial of α over Q.

The measure is named after Kurt Mahler.

Contents

Properties

  • The Mahler measure is multiplicative, i.e. M(pq) = M(p)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.

See also

References

External links


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