On some application of the spectral properties of the matrices
type: Item
instance of: scholarly article
zbMATH Open document ID: 1276.11169
publication date: 2012-09-05T00:00:00Z
full work available at URL: NNTDM-17-2-12-17.pdf
Mathematics Subject Classification ID: 11R04
Mathematics Subject Classification ID: 15A42
zbMATH DE Number: 6077464
zbMATH Keywords: conjecture of Schinzel and Zassenhaus
zbMATH Keywords: spectral properties of matrices
author: Aleksander Grytczuk
author: Izabela Kurzydło
MaRDI profile type: Publication
title: On some application of the spectral properties of the matrices
published in: Notes on Number Theory and Discrete Mathematics
review text: In this paper the authors prove that if \(\alpha\) is an algebraic integer of degree \(n\) which is not a root of unity and has norm different from \(\pm 1\) then it has a conjugate \(\alpha'\) whose modulus is greater than or equal to \(1+(\log 2)/n\). This result (Theorem 1) is related to a still open conjecture of Schinzel and Zassenhaus and is very well-known. The proof is using an inequality for the eigenvalues of a matrix that is composed with coefficients of the minimal polynomial of \(\alpha\). The statement of the other result (Theorem 2) asserting that the same holds for \(\alpha\) of norm \(\pm 1\) under the inequality \(n \varepsilon > (1-\varepsilon) \log 2\) is not clear, since \(\varepsilon\) is not defined.
reviewed by: A. Dubickas
description: scientific article; zbMATH DE number 6077464