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On the absolute Mahler measure of polynomials having all zeros in a sector


Authors: Georges Rhin and Christopher Smyth
Journal: Math. Comp. 64 (1995), 295-304
MSC: Primary 11R04; Secondary 11C08, 12D10
DOI: https://doi.org/10.1090/S0025-5718-1995-1257579-6
MathSciNet review: 1257579
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Abstract: Let $ \alpha $ be an algebraic integer of degree d, not 0 or a root of unity, all of whose conjugates $ {\alpha _i}$, are confined to a sector $ \vert\arg z\vert \leq \theta $. We compute the greatest lower bound $ c(\theta )$ of the absolute Mahler measure $ (\prod\nolimits_{i = 1}^d {\max (1,\vert{\alpha _i}\vert){)^{1/d}}} $ of $ \alpha $, for $ \theta $ belonging to nine subintervals of $ [0,2\pi /3]$. In particular, we show that $ c(\pi /2) = 1.12933793$, from which it follows that any integer $ \alpha \ne 1$ and $ \alpha \ne {e^{ \pm i\pi /3}}$ all of whose conjugates have positive real part has absolute Mahler measure at least $ c(\pi /2)$. This value is achieved for $ \alpha $ satisfying $ \alpha + 1/\alpha = \beta _0^2$, where $ {\beta _0} = 1.3247 \ldots $ is the smallest Pisot number (the real root of $ \beta _0^3 = {\beta _0} + 1$).


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DOI: https://doi.org/10.1090/S0025-5718-1995-1257579-6
Article copyright: © Copyright 1995 American Mathematical Society

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