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Algebraic numbers close to both 0 and 1. (English) Zbl 0786.11063

Let \(\alpha\) be a number algebraic over the rationals and let \(H(\alpha)\) denote the absolute logarithmic height of \(\alpha\), which can be defined as \(H(\alpha)= \log M(f)^{1/n}\), where \(\alpha\) is a root of the irreducible polynomial \(f(x)\) with rational coefficients and degree \(n\), and where \(M(f)\) denotes the Mahler measure of \(f\). The author gives an elementary proof of a sharp version of a remarkable inequality of Shouwu Zhang. He shows that, for all \(\alpha\neq 0,1\), \((1\pm\sqrt{-3})/2\), the following inequality holds: \[ H(\alpha)+ H(1-\alpha)\geq {1\over 2} \log {{1+\sqrt{5}}\over 2}= 0.2406059\dots, \] with equality if and only if \(\alpha\) or \(1-\alpha\) is a primitive 10th root of unity. He also proves a sharp projective version of this inequality for the curve \(x+y+z=0\) and gives an outline of how to prove similar results for other curves.

MSC:

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