# zbMATH — the first resource for mathematics

On the Remak height, the Mahler measure and conjugate sets of algebraic numbers lying on two circles. (English) Zbl 0997.11087
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 ${\mathcal R}(\alpha) = |a_0|\prod_{j=1}^{d-1} |\alpha_j|^{(d-j)/(d-1)}.$ They prove two inequalities relating $${\mathcal R}(\alpha)$$ to the better known Mahler measure $$M(\alpha)$$, namely $c M(\alpha)^{d/(2(d-1))} \leq {\mathcal R}(\alpha) \leq M(\alpha),$ 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.

##### MSC:
 11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
##### Keywords:
Mahler measure; Pisot number; Salem number; Remak height
Full Text: