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

Voutier, Paul M.

Acta Arith. 74, No. 1, 81-95 (1996).

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.

Reviewer: Paul M. Voutier (London)

MSC:

11R06

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

Keywords:

algebraic number; absolute logarithmic height; Mahler measure