Mahler measure


In mathematics, the Mahler measure of a polynomial with complex coefficients is defined as
where factorizes over the complex numbers 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 for on the unit circle :
By extension, the Mahler measure of an algebraic number is defined as the Mahler measure of the minimal polynomial of over. In particular, if is a Pisot number or a Salem number, then its Mahler measure is simply.
The Mahler measure is named after the German-born Australian mathematician Kurt Mahler.

Properties

The Mahler measure of a multi-variable polynomial is defined similarly by the formula
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 -functions. For example, in 1981, Smyth proved the formulas
where is the Dirichlet L-function, and
where is the Riemann zeta function. Here is called the logarithmic Mahler measure.

Some results by Lawton and Boyd

From the definition, the Mahler measure is viewed as the integrated values of polynomials over the torus. If vanishes on the torus, then the convergence of the integral defining is not obvious, but it is known that does converge and is equal to a limit of one-variable Mahler measures, which had been conjectured by Boyd.
This is formulated as follows: Let denote the integers and define . If is a polynomial in variables and define the polynomial of one variable by
and define by
where .
Theorem : Let be a polynomial in N variables with complex coefficients. Then the following limit is valid :

Boyd's proposal

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.
Define an extended cyclotomic polynomial to be a polynomial of the form
where is the m-th cyclotomic polynomial, the are integers, and the are chosen minimally so that is a polynomial in the. Let be the set of polynomials that are products of monomials and extended cyclotomic polynomials.
Theorem : Let be a polynomial with integer coefficients. Then if and only if is an element of.
This led Boyd to consider the set of values
and the union. He made the far-reaching conjecture that the set of is a closed subset 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 , Boyd further conjectures that