Closed sets of Mahler measures
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Abstract:
Given a $k$-variable Laurent polynomial $F$, any $\ell \times k$ integer matrix $A$ naturally defines an $\ell$-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 Saturated 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 $\mathcal L$.
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Additional Information
- Chris Smyth
- Affiliation: School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh EH9 3FD, Scotland, United Kingdom
- MR Author ID: 164180
- Email: C.Smyth@ed.ac.uk
- Received by editor(s): June 14, 2017
- Received by editor(s) in revised form: August 22, 2017
- Published electronically: February 16, 2018
- Communicated by: Matthew A. Papanikolas
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 2359-2372
- MSC (2010): Primary 11R06
- DOI: https://doi.org/10.1090/proc/13951
- MathSciNet review: 3778140