Mahler measure

In mathematics, the Mahler measure M ( p ) {displaystyle M(p)} of a polynomial p ( z ) {displaystyle p(z)} with complex coefficients is defined as where p ( z ) {displaystyle p(z)} factorizes over the complex numbers C {displaystyle mathbb {C} } as 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 ) | {displaystyle |p(z)|} for z {displaystyle z} on the unit circle (i.e., | z | = 1 {displaystyle |z|=1} ): By extension, the Mahler measure of an algebraic number α {displaystyle alpha } is defined as the Mahler measure of the minimal polynomial of α {displaystyle alpha } over Q {displaystyle mathbb {Q} } . In particular, if α {displaystyle alpha } is a Pisot number or a Salem number, then its Mahler measure is simply α {displaystyle alpha } . The Mahler measure is named after the German-born Australian mathematician Kurt Mahler. The Mahler measure M ( p ) {displaystyle M(p)} of a multi-variable polynomial p ( x 1 , … , x n ) ∈ C [ x 1 , … , x n ] {displaystyle p(x_{1},ldots ,x_{n})in mathbb {C} } is defined similarly by the formula