Mahler measure information

In mathematics, the Mahler measure $M(p)$ of a polynomial $p(z)$ with complex coefficients is defined as

M(p)=|a|\prod _{|\alpha _{i}|\geq 1}|\alpha _{i}|=|a|\prod _{i=1}^{n}\max\{1,|\alpha _{i}|\},

where

p(z)

factorizes over the complex numbers

\mathbb {C}

as

p(z)=a(z-\alpha _{1})(z-\alpha _{2})\cdots (z-\alpha _{n}).

The Mahler measure can be viewed as a kind of height function. Using Jensen's formula, it can be proved that this measure is also equal to the geometric mean of $|p(z)|$ for $z$ on the unit circle (i.e., $|z|=1$ ):

M(p)=\exp \left(\int _{0}^{1}\ln(|p(e^{2\pi i\theta })|)\,d\theta \right).

By extension, the Mahler measure of an algebraic number $\alpha$ is defined as the Mahler measure of the minimal polynomial of $\alpha$ over $\mathbb {Q}$ . In particular, if $\alpha$ is a Pisot number or a Salem number, then its Mahler measure is simply $\alpha$ .

The Mahler measure is named after the German-born Australian mathematician Kurt Mahler.

Mahler measure information

and 19 Related for: Mahler measure information

Mahler measure

Height function

Kurt Mahler

Gustav Mahler

List of unsolved problems in mathematics

Perron number

Touchard polynomials

Christopher Deninger

List of number theory topics

Salem number

Transcendental number theory

Geometrical properties of polynomial roots

Lehmer

List of women in mathematics

Scientific phenomena named after people

David William Boyd

Eduard Mahler

Das Lied von der Erde

Bombieri norm