Lower bounds for heights in cyclotomic extensions
Let \(f(x) = a_{0} \prod_{i=1}^{d} (x-\alpha_{i}) = a_{0}x^{n}+a_{1}x^{n-1}+ \cdots + a_{n}\) be an irreducible polynomial with integer coefficients. The Mahler measure of \(f\) is defined to be \(M(f) =\prod_{i=1}^{d}\max \{ 1, |\alpha_{i}| \}\). If \(\alpha \neq 0\) is a root of \(f\), then the absolute logarithmic height \(h(\alpha)\) of \(\alpha\) is defined by \(h(\alpha) = \frac{\log M(f)}{d}\). Lehmer's problem asks if there exists a positive constant \(c>0\) such that \(h(\alpha)> c/d\) as long as \(\alpha\) is neither zero nor a root of unity. The main result of the paper under review is that if \(\alpha\) is a nonzero element of a finite abelian extension of the rationals and is not equal to a root of unity, then \(h(\alpha) \geq 0.155097\). The authors also consider several special cases in which improved lower bounds are given.
scientific article; zbMATH DE number 5723357