The ${\rm R}_2$ measure for totally positive algebraic integers

V. Flammang¹

¹ UMR CNRS 7502, IECL Université de Lorraine, site de Metz Département de Mathématiques UFR MIM Ile du Saulcy, CS 50128 57045 Metz Cedex 01, France

Colloquium Mathematicum, Tome 144 (2016) no. 1, pp. 45-53.

Résumé

Let $\alpha $ be a totally positive algebraic integer of degree $d$, i.e., all of its conjugates $\alpha _1= \alpha , \ldots ,\alpha _d$ are positive real numbers. We study the set ${\cal R}_2$ of the quantities $(\prod _{i=1}^d (1 + \alpha _i^2)^{1/2})^{1/d}$. We first show that $\sqrt 2$ is the smallest point of ${\cal R}_2$. Then, we prove that there exists a number $l$ such that ${\cal R}_2$ is dense in $(l, \infty )$. Finally, using the method of auxiliary functions, we find the six smallest points of ${\cal R}_2$ in $(\sqrt 2, l)$. The polynomials involved in the auxiliary function are found by a recursive algorithm.

Détail

DOI : 10.4064/cm6221-1-2016

Keywords: alpha totally positive algebraic integer degree its conjugates alpha alpha ldots alpha positive real numbers study set cal quantities prod alpha first sqrt smallest point cal prove there exists number cal dense infty finally using method auxiliary functions six smallest points cal sqrt polynomials involved auxiliary function found recursive algorithm