Closed sets of Mahler measures

Abstract

Given a $k$-variable Laurent polynomial $F$, any $l\times k$ integer matrix $A$ naturally defines an $l$-variable Laurent polynomial $F_A.$ I prove that for fixed $F$ the set $\mathcal M(F)$ 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 Primitive Hermite Normal Form. Furthermore, if $F$ has integer coefficients and $\mathcal M(F)$ contains $0,$ then $0$ is an isolated point of this set. I also show that, for a given bound $B>0$, the set ${\mathcal M}_B$ 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 ${\mathcal M}_B$. These results constitute evidence consistent with a conjecture of Boyd from 1980 to the effect that the union $\mathcal L$ of all sets ${\mathcal M}_B$ for $B>0$ is closed, with $0$ an isolated point of $\mathcal L$.