The values of Mahler measures

The set M* of numbers which occur as Mahler measures of integer polynomials and the subset M of Mahler measures of algebraic numbers (that is, of irreducible integer polynomials) are investigated. It is proved that every number alpha of degree d in M* is the Mahler measure of a separable integer polynomial of degree at most Sigma(1 <= r <= d/2) (d) with all its roots lying in the Galois closure F of Q(alpha), and every unit in M is the Mahler measure of a unit in F of degree at most ((d)([d/2])) over Q. This is used to show that some numbers considered earlier by Boyd are not Mahler measures. The set of numbers which occur as Mahler measures of both reciprocal and nonreciprocal algebraic numbers is also investigated. In particular, all cubic units in this set are described and it is shown that the smallest Pisot number is not the measure of a reciprocal number.