On the Remak height, the Mahler measure and conjugate sets of algebraic numbers lying on two circles (Q2716973)

Let \(\alpha\) be an algebraic number of degree \(d \geq 2\) with minimal polynomial \(P(z) = a_0 z^d + \dots + a_d \in {\mathbb Z}[z]\) over the rationals, and with conjugates \(\alpha_1, \dots, \alpha_d\) ordered so that \(|\alpha_1|\geq |\alpha_2|\geq \dots |\alpha_d|\). Based on the form of Remak's upper bound for the discriminant of \(P(z)\), the authors define the Remak height of \(\alpha\) by NEWLINE\[NEWLINE{\mathcal R}(\alpha) = |a_0|\prod_{j=1}^{d-1} |\alpha_j|^{(d-j)/(d-1)}.NEWLINE\]NEWLINE They prove two inequalities relating \({\mathcal R}(\alpha)\) to the better known Mahler measure \(M(\alpha)\), namely NEWLINE\[NEWLINEc M(\alpha)^{d/(2(d-1))} \leq {\mathcal R}(\alpha) \leq M(\alpha),NEWLINE\]NEWLINE where \(c = \min(|a_0|,|a_d|)^{1/2}/\text{ max}(|a_0|,|a_d|)^ {1/(2(d-1))}\). The first of these inequalities becomes an equality if and only if \(|a_0|= |a_d|\) and all of the \(\alpha_j\) lie on one or two circles centred at \(0\). They give a complete characterization of the algebraic numbers that satisfy these conditions. It turns out that either \(d/3\) or \(d/2\) of the conjugates lie on one of the circles. Although the final result is easy to state, the proof is non-trivial.