The Siegel norm of algebraic numbers (Q2891118)

Let \(\alpha\) be an algebraic number with conjugates \(\alpha_1=\alpha,\dots,\alpha_d\) over \(\mathbb Q\). The authors consider the map \(A:\overline{\mathbb Q}\mapsto[0,\infty)\) given by NEWLINE\[NEWLINEA(\alpha)=(|\alpha_1|^2+\dots+|\alpha_d|^2)/dNEWLINE\]NEWLINE and define the ''Siegel norm'' of an algebraic number \(\alpha\) by \(\|\alpha\|_{\text{Si}}=\sqrt{A(\alpha)}\). The ''spectral norm'' is defined as \(\|\alpha\|=\max_{1\leq j\leq d}|\alpha_j|\). (Usually, the spectral norm is called the ''house'' of an algebraic number.) Evidently, \(\|\alpha\|_{\text{Si}}\leq\|\alpha\|\). The paper is devoted to the study of some relations between the Siegel norm and the spectral norm of an algebraic number and their extensions to the spectral completion of \(\overline{\mathbb Q}\). In particular, they give an upper bound for \(\|\alpha\|_{\text{Si}}\) in terms of \(\|\alpha\|\) and test its sharpness on powers of Pisot numbers.