CLOSED SETS OF MAHLER MEASURES

Given a k-variable Laurent polynomial F, any l x k integer matrix A naturally defines an l-variable Laurent polynomial F-A. I prove that for fixed F the set M(F) of all the logarithmic Mahler measures m(F-A) of F-A for all A is a closed subset of the real line. Moreover, the matrices A can be assumed to be of a special form, which I call Saturated Hermite Normal Form. Furthermore, if F has integer coefficients and M(F) contains 0, then 0 is an isolated point of this set. I also show that, for a given bound B > 0, the set M-B of all Mahler measures of integer polynomials in any number of variables and having length (sum of the moduli of its coefficients) at most B is closed. Again, 0 is an isolated point of M-B. These results constitute evidence consistent with a conjecture of Boyd from 1980 to the effect that the union L of all sets M-B for B > 0 is closed, with 0 an isolated point of L.