## 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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