with complex coefficients is defined 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
By extension, the Mahler measure of an algebraic number
is defined as the Mahler measure of the minimal polynomial of
The Mahler measure is named after the German-born Australian mathematician Kurt Mahler.
It inherits the above three properties of the Mahler measure for a one-variable polynomial.
The multi-variable Mahler measure has been shown, in some cases, to be related to special values of zeta-functions and
For example, in 1981, Smyth[3] proved the formulas
is the Riemann zeta function.
is called the logarithmic Mahler measure.
From the definition, the Mahler measure is viewed as the integrated values of polynomials over the torus (also see Lehmer's conjecture).
does converge and is equal to a limit of one-variable Mahler measures,[4] which had been conjectured by Boyd.
denote the integers and define
be a polynomial in N variables with complex coefficients.
Boyd provided more general statements than the above theorem.
He pointed out that the classical Kronecker's theorem, which characterizes monic polynomials with integer coefficients all of whose roots are inside the unit disk, can be regarded as characterizing those polynomials of one variable whose measure is exactly 1, and that this result extends to polynomials in several variables.
be the set of polynomials that are products of monomials
This led Boyd to consider the set of values
He made the far-reaching conjecture[5] that the set of
An immediate consequence of this conjecture would be the truth of Lehmer's conjecture, albeit without an explicit lower bound.
As Smyth's result suggests that
by automorphisms of a compact metrizable abelian group may be associated via duality to any countable module
the formula proved by Lind, Schmidt, and Ward gives
In the general case, the entropy of the action is expressed as a sum of logarithmic Mahler measures over the generators of the principal associated prime ideals of the module.
As pointed out earlier by Lind in the case
of a single compact group automorphism, this means that the set of possible values of the entropy of such actions is either all of
or a countable set depending on the solution to Lehmer's problem.
Lind also showed that the infinite-dimensional torus
either has ergodic automorphisms of finite positive entropy or only has automorphisms of infinite entropy depending on the solution to Lehmer's problem.