Mahler measure
In mathematics, the Mahler measure formula_1 of a polynomial formula_2 with complex coefficients is defined as formula_3 where formula_2 factorizes over the complex numbers formula_5 as formula_6 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): formula_10 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 formula_31 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 formula_33 where formula_34 is the Dirichlet L -function, and formula_35 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 formula_48 and define formula_49 by formula_50 where formula_51. 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 formula_52 where formula_53 is the "m"-th cyclotomic polynomial, the formula_54 are integers, and the formula_55 are chosen minimally so that formula_56 is a polynomial in the formula_57. Let formula_58 be the set of polynomials that are products of monomials formula_59 and extended cyclotomic polynomials. This led Boyd to consider the set of values formula_60 and the union formula_61. He made the far-reaching conjecture that the set of formula_62 is a closed subset of formula_63. 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_64 , Boyd further conjectures that formula_65 Mahler measure and entropy. An action formula_66 of formula_67 by automorphisms of a compact metrizable abelian group may be associated via duality to any countable module formula_45 over the ring formula_69. The topological entropy (which is equal to the measure-theoretic entropy) of this action, formula_70, is given by a Mahler measure (or is infinite). In the case of a cyclic module formula_71 for a non-zero polynomial formula_72 the formula proved by Lind, Schmidt, and Ward gives formula_73, the logarithmic Mahler measure of formula_74. 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 formula_75 of a single compact group automorphism, this means that the set of possible values of the entropy of such actions is either all of formula_76 or a countable set depending on the solution to Lehmer's problem. Lind also showed that the infinite-dimensional torus formula_77 either has ergodic automorphisms of finite positive entropy or only has automorphisms of infinite entropy depending on the solution to  Lehmer's problem.