On the length of totally positive algebraic integers
type: Item
instance of: scholarly article
title: On the length of totally positive algebraic integers
zbMATH Open document ID: 0831.11057
published in: Journal of Number Theory
publication date: 1996-02-18T00:00:00Z
review text: For a polynomial \[ P(X)= X^d+ a_{d-1} X^{d-1}+ \cdots+ a_0= \prod_{k=1}^d (X- \alpha) \] put \(R_1 (P)= \prod_{k=1}^d (1+|\alpha_k |)\) and \(R(P)= R_1 (P)^{1/d}\). The author studies the set \({\mathcal R}\) of the values of \(R(P)\), where \(P\) runs over irreducible polynomials \(\neq X, X-1\) with integral coefficients having all roots positive. He determines the five smallest numbers in \({\mathcal R}\) (which correspond to certain polynomials considered by \textit{C. J. Smyth} [Math. Comput. 37, 205-208 (1981; Zbl 0475.12001)]) and shows that this set is dense in a certain halfline \([l,\infty ]\) with \(l= 2.3768 \dots\;\).
Mathematics Subject Classification ID: 11R04
Mathematics Subject Classification ID: 11R09
Mathematics Subject Classification ID: 11R80
zbMATH DE Number: 809107
zbMATH Keywords: totally positive integer
zbMATH Keywords: minimal polynomial
author: Valérie Flammang
reviewed by: Władysław Narkiewicz
MaRDI profile type: Publication
full work available at URL: jnth.1995.1101
OpenAlex ID: W2034443245
DOI: 10.1006/JNTH.1995.1101
description: scientific article; zbMATH DE number 809107