: Mathematics: Mahler measure

mathematical analysis > function > elementary function > polynomial > Mahler measure

algebra > polynomial > Mahler measure

number > number theory > analytic number theory > Mahler measure

Mahler measure

In mathematics, the Mahler measure ${\\displaystyle M(p)}$ of a polynomial ${\\displaystyle p(z)}$ with complex coefficients is defined as ${\\displaystyle M(p)=|a|\\prod _{|\\alpha _{i}|\\geq 1}|\\alpha _{i}|=|a|\\prod _{i=1}^{n}\\max\\{1,|\\alpha _{i}|\\},}$ where ${\\displaystyle p(z)}$ factorizes over the complex numbers ${\\displaystyle \\mathbb {C} }$ as ${\\displaystyle p(z)=a(z-\\alpha _{1})(z-\\alpha _{2})\\cdots (z-\\alpha _{n}).}$ 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 ${\\displaystyle |p(z)|}$ for ${\\displaystyle z}$ on the unit circle (i.e., ${\\displaystyle |z|=1}$ ): ${\\displaystyle M(p)=\\exp \\left(\\int _{0}^{1}\\ln(|p(e^{2\\pi i\ heta })|)\\,d\ heta \ ight).}$ By extension, the Mahler measure of an algebraic number ${\\displaystyle \\alpha }$ is defined as the Mahler measure of the minimal polynomial of ${\\displaystyle \\alpha }$ over ${\\displaystyle \\mathbb {Q} }$ . In particular, if ${\\displaystyle \\alpha }$ is a Pisot number or a Salem number, then its Mahler measure is simply ${\\displaystyle \\alpha }$ . The Mahler measure is named after the German-born Australian mathematician Kurt Mahler.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Mahler_measure)