Mahler measure

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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).


 ||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)


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.


  • 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[edit]

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

M(p) = \exp\left( \frac{1}{(2\pi)^n} \int_{0}^{2\pi} \int_{0}^{2\pi} \cdots \int_{0}^{2\pi}
      d\theta_1\, d\theta_2\cdots d\theta_n \right).

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

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