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AN ANALOGUE OF THE SCHUR–SIEGEL–SMYTH TRACE PROBLEM

Part of: Algebraic number theory: global fields Polynomials and matrices Computational number theory

Published online by Cambridge University Press:  30 August 2022

V. FLAMMANG
V. FLAMMANG*
Affiliation:
Département de Mathématiques, UFR MIM, UMR CNRS 7502. IECL, Université de Lorraine, site de Metz, 3 rue Augustin Fresnel BP 45112, Metz cedex 3 57073, France
*
e-mail: valerie.flammang@univ-lorraine.fr
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Abstract

By analogy with the trace of an algebraic integer $\alpha $ with conjugates $\alpha _1=\alpha , \ldots , \alpha _d$, we define the G-measure $ {\mathrm {G}} (\alpha )= \sum _{i=1}^d ( |\alpha _i| + 1/ | \alpha _i | )$ and the absolute ${\mathrm G}$-measure ${\mathrm {g}}(\alpha )={\mathrm {G}}(\alpha )/d$. We establish an analogue of the Schur–Siegel–Smyth trace problem for totally positive algebraic integers. Then we consider the case where $\alpha $ has all its conjugates in a sector $| \arg z | \leq \theta $, $0 < \theta < 90^{\circ }$. We compute the greatest lower bound $c(\theta )$ of the absolute G-measure of $\alpha $, for $\alpha $ belonging to $11$ consecutive subintervals of $]0, 90 [$. This phenomenon appears here for the first time, conforming to a conjecture of Rhin and Smyth on the nature of the function $c(\theta )$. All computations are done by the method of explicit auxiliary functions.

Keywords

totally positive algebraic integertracestaircase function

MSC classification

Primary: 11R06: PV-numbers and generalizations; other special algebraic numbers; Mahler measure
Secondary: 11C08: Polynomials 11R80: Totally real fields 11Y40: Algebraic number theory computations

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 107 , Issue 2 , April 2023 , pp. 227 - 238
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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