Closed sets of Mahler measures

TL;DR

This paper proves that the set of Mahler measures derived from fixed Laurent polynomials and bounded coefficient polynomials is closed, with zero as an isolated point, supporting Boyd's conjecture on the structure of Mahler measures.

Contribution

It establishes the closedness of Mahler measure sets for fixed polynomials and bounded length polynomials, and characterizes the isolated nature of zero within these sets.

Findings

The set of Mahler measures for fixed Laurent polynomial is closed.

Zero is an isolated point in Mahler measure sets when they contain zero.

The union of Mahler measures of bounded-length polynomials is also closed.

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 $\mathcal L$.