A relationship between the Mahler measure and the discriminant of algebraic numbers (Q1358492)

Let \(\alpha\) be an algebraic number of degree \(d(\alpha)\), disciminant \(\Delta(\alpha)\) and Mahler measure \(M(\alpha)\). Let \(\Omega\) denote the set of nonzero algebraic integers \(\alpha\) such that for each prime \(p\), \(d(\alpha^p) = d(\alpha)\). Note that nontrivial roots of unity are not in \(\Omega\). Let \(\delta(\alpha) = d(\alpha)/\Delta(\alpha)\). The author shows that the quantity \(\delta(\alpha)\) can replace \(d(\alpha)\) in Dobrowolski's famous estimate. That is, he shows that, given \(\varepsilon > 0\), there is a \(\delta_0(\varepsilon) > 0\) such that if \(\alpha \in \Omega\) and \(\delta = \max(\delta(\alpha),\delta_0)\), then \[ \ln M(\alpha) \geq (2 - \varepsilon) (\ln \ln \delta/\ln \delta)^3. \] The proof uses the method of \textit{D. C. Cantor} and \textit{E. G. Straus} [Acta. Arith. 42 97-100 (1982; Zbl 0504.12002)]. A second result deals with small values of \(\delta(\alpha)\). Namely, if \(p > 2\) is a prime and \(d(\alpha^p) = d(\alpha)\) and if \(\delta(\alpha) \leq 1\), then \(\ln M(\alpha) \geq \ln(p/2)/(2(p+1)).\)