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).

Here

 \|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)

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 \alpha is defined as the Mahler measure of the minimal polynomial of \alpha over \mathbb{Q}.

The measure is named after Kurt Mahler.

Properties[edit]

  • 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[1]

M(p) = \exp\left( \frac{1}{(2\pi)^n} \int_0^{2\pi} \int_0^{2\pi} \cdots \int_0^{2\pi} \ln \Bigl( \bigl |p(e^{i\theta_1}, e^{i\theta_2}, \ldots, e^{i\theta_n}) \bigr| \Bigr) \, d\theta_1\, d\theta_2\cdots d\theta_n \right).

If p vanishes on the torus (S^1)^n, then the convergence of the integral defining M(p) is not obvious, but it is known that M(p) does converge and is equal to a limit of one-variable Mahler measures.[2] For \boldsymbol r=(r_1,\ldots,r_n)\in\mathbb{N}^n, let q(\boldsymbol r) be the smallest number such that there is a nontrivial relation r_1s_1+r_2s_2+\cdots+r_ns_n with s_i\in\mathbb{Z} and all |s_i|\le q(\boldsymbol r). Then


  M\bigl(p(x_1,\ldots,x_n)\bigr) = \lim_{\boldsymbol r\in\mathbb{N}^n\atop q(\boldsymbol r)\to\infty} M\bigl(p(x^{r_1},x^{r_2},\ldots,x^{r_n})\bigr).

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

See also[edit]

References[edit]

  1. ^ Schinzel (2000) p.224
  2. ^ Lawton (1983)

External links[edit]