Sur les zéros des fonctions symétriques complètes des corps cubiques. (On the zeros of complete symmetric functions in cubic fields)
type: Item
instance of: scholarly article
title: Sur les zéros des fonctions symétriques complètes des corps cubiques. (On the zeros of complete symmetric functions in cubic fields)
zbMATH Open document ID: 0642.12009
author: Bernadette Deshomme
published in: Pacific Journal of Mathematics
publication date: 1989-01-01T00:00:00Z
review text: Let \({\mathcal K}\) be a real cubic field of negative discriminant, defined by the relation \(\omega\) \(3=\epsilon Norm_{{\mathcal K}/{\mathbb{Q}}}(\omega)\), where \(\omega\) is an integer of \({\mathcal K}\) and the unit \(\epsilon\) is not a cube. The complete symmetric function with \(weight\quad n\) of the roots of the minimal polynomial of \(\omega\) satisfies a cubic linear recurrence relation. We show that the zeros of this sequence are connected to the binomial units of \({\mathcal K}\) and further, that the number of zeros is between two and four, except in two cases, when there are five or six zeros (the sequence of Berstel and Mignotte). As an illustration, a table of zeros is obtained for fields \({\mathcal K}\) of discriminant superior to - 400.
Mathematics Subject Classification ID: 11R16
Mathematics Subject Classification ID: 11R27
Mathematics Subject Classification ID: 11B37
zbMATH DE Number: 4045876
zbMATH Keywords: symmetric function of the roots of minimal polynomial
zbMATH Keywords: real cubic field of negative discriminant
zbMATH Keywords: binomial units
MaRDI profile type: Publication
full work available at URL: pjm.1989.139.17
OpenAlex ID: W2022910597
DOI: 10.2140/PJM.1989.139.17
description: scientific article; zbMATH DE number 4045876