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. 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 Mahler measure and entropy An action of by automorphisms of a compact metrizable abelian group may be associated via duality to any countable module over the ring . The topological entropy (which is equal to the measure-theoretic entropy) of this action, , is given by a Mahler measure (or is infinite). In the case of a cyclic module for a non-zero polynomial the formula proved by Lind, Schmidt, and Ward gives , the logarithmic Mahler measure of . 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. See also Bombieri norm Height of a polynomial Notes References Everest, Graham and Ward, Thomas (1999). "Heights of polynomials and entropy in algebraic dynamics". Springer-Verlag London, Ltd., London. xii+211 pp. ISBN: 1-85233-125-9 . External links Mahler Measure on MathWorld Jensen's Formula on MathWorld Analytic number theory Polynomials