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Closed sets of Mahler measures


Author: Chris Smyth
Journal: Proc. Amer. Math. Soc. 146 (2018), 2359-2372
MSC (2010): Primary 11R06
DOI: https://doi.org/10.1090/proc/13951
Published electronically: February 16, 2018
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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
Email: C.Smyth@ed.ac.uk

DOI: https://doi.org/10.1090/proc/13951
Keywords: Mahler measure, closure
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
Article copyright: © Copyright 2018 American Mathematical Society

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