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An effective lower bound for the height of algebraic numbers. (English) Zbl 0838.11065

Let \(\alpha\) be a non-zero algebraic number of degree \(d\). We make effective the argument of Cantor and Straus to show that if \(\alpha\) is not a root of unity and \(d\geq 2\) then \(dh(\alpha)= \log M(\alpha)> (1/4) (\log (d)/ \log \log (d))^3\), where \(h(\cdot)\) denotes the absolute logarithmic height and \(M(\cdot)\) denotes the Mahler measure. Some related corollaries are also given.

MSC:

11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
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