The t-metric Mahler measures of surds and rational numbers

A. Dubickas and C. Smyth introduced the metric Mahler measure where M ( α ) denotes the usual (logarithmic) Mahler measure of . This definition extends in a natural way to the t -metric Mahler measure by replacing the sum with the usual L t norm of the vector ( M ( α 1 ),…, M ( α N )) for any t ≧1....

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Bibliographic Details
Published in:Acta mathematica Hungarica 2012-03, Vol.134 (4), p.481-498
Main Authors: Jankauskas, Jonas, Samuels, Charles L.
Format: Article
Language:English
Subjects:
Mathematics
Mathematics and Statistics
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Summary:A. Dubickas and C. Smyth introduced the metric Mahler measure where M ( α ) denotes the usual (logarithmic) Mahler measure of . This definition extends in a natural way to the t -metric Mahler measure by replacing the sum with the usual L t norm of the vector ( M ( α 1 ),…, M ( α N )) for any t ≧1. For α ∈ℚ, we prove that the infimum in M t ( α ) may be attained using only rational points, establishing an earlier conjecture of the second author. We show that the natural analogue of this result fails for general by giving an infinite family of quadratic counterexamples. As part of this construction, we provide an explicit formula to compute M t ( D 1/ k ) for a squarefree D ∈ℕ.
ISSN:0236-5294
1588-2632
DOI:10.1007/s10474-011-0126-y