Mahler measure

In mathematics, the Mahler measure formula_1 of a polynomial formula_2 with complex coefficients is defined as
where formula_2 factorizes over the complex numbers formula_5 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 formula_7 for formula_8 on the unit circle (i.e., formula_9):
By extension, the Mahler measure of an algebraic number formula_11 is defined as the Mahler measure of the minimal polynomial of formula_11 over formula_13. In particular, if formula_11 is a Pisot number or a Salem number, then its Mahler measure is simply formula_11.
The Mahler measure is named after the German-born Australian mathematician Kurt Mahler.
Higher-dimensional Mahler measure.
The Mahler measure formula_1 of a multi-variable polynomial formula_30 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 formula_32-functions. For example, in 1981, Smyth proved the formulas
where formula_34 is the Dirichlet L-function, and
where formula_36 is the Riemann zeta function. Here formula_37 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 (also see Lehmer's conjecture). If formula_20 vanishes on the torus formula_39, then the convergence of the integral defining formula_1 is not obvious, but it is known that formula_1 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 formula_42 denote the integers and define formula_43 . If formula_44 is a polynomial in formula_45 variables and formula_46 define the polynomial formula_47 of one variable by
and define formula_49 by
where formula_51 .
Theorem (Lawton) : Let formula_44 be a polynomial in "N" variables with complex coefficients. Then the following limit is valid (even if the condition that formula_53 is relaxed):
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 formula_56 is the "m"-th cyclotomic polynomial, the formula_57 are integers, and the formula_58 are chosen minimally so that formula_59 is a polynomial in the formula_60. Let formula_61 be the set of polynomials that are products of monomials formula_62 and extended cyclotomic polynomials.
Theorem (Boyd) : Let formula_63 be a polynomial with integer coefficients. Then formula_64 if and only if formula_65 is an element of formula_61.
This led Boyd to consider the set of values
and the union formula_68. He made the far-reaching conjecture that the set of formula_69 is a closed subset of formula_70. 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 formula_71 , Boyd further conjectures that