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The Lehmer constants of an annulus. (English) Zbl 1030.11057
The Lehmer constant \(L(V)\) of an open set \(V\) in the complex plane is defined [M. Langevin, C. R. Acad. Sci., Paris, Sér I. 301, 463-466 (1985; Zbl 0585.12013)] by \(L(V)=\inf M(\alpha)^{1/\deg(\alpha)}\), the infimum taken over all non-zero non-cyclotomic algebraic integers \(\alpha\) lying with all conjugates outside \(V\), and \(M(\alpha)\) denotes the Mahler measure of \(\alpha\). The authors compute \(L(V)\) in the case when \(V\) is an annulus centered at the origin and prove that for any open set \(V\) the inequality \(L(V)>1\) implies that \(V\) has a non-empty intersection with the unit circle (the converse implication has been earlier established by Langevin (loc.cit.)) Moreover they consider a variant of Lehmer’s constant, namely \(L_\infty(V)=\lim_{d\to\infty}\inf M(\alpha)^{1/\deg(\alpha)}\), the infimum being taken over all \(\alpha\) of degree \(\geq d\) lying with their conjugates outside \(V\), and obtain similar results.

MSC:
11R04 Algebraic numbers; rings of algebraic integers
11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
11R09 Polynomials (irreducibility, etc.)
12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)
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References:
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