Comments on Salem polynomials
type: Item
instance of: scholarly article
title: Comments on Salem polynomials
zbMATH Open document ID: 1480.11135
published in: Archiv der Mathematik
publication date: 2021-07-13T00:00:00Z
review text: The author gives a sufficient condition on a monic irreducible reciprocal polynomial \(P \in \mathbb{Z}[x]\) of even degree \(d \geq 4\) to have \(d-2\) roots on the unit circle, refining a result of \textit{R. S. Vieira} [Ramanujan J. 42, No. 2, 363--369 (2017; Zbl 1422.30013)]. More precisely, write \(P = x^d + c_1 x^{d-1} + \cdots + c_1 x + 1\), and denote by \(L\) be the length of \(P\), that is, the sum of the absolute values of the coefficients of \(P\). The author shows that if \(|c_1| > L/4\), then \(P\) has \(d-2\) roots on the unit circle, and two real roots \(\alpha\) and \(\alpha^{-1}\), with \(|\alpha|>1\). In particular, \(|\alpha|\) is a Salem number. Furthermore, the roots of \(P\) on the unit circle interlace with the \((d-2)\)-th roots of unity. The proof is a simple application of the intermediate value theorem and Hurwitz's theorem in complex analysis. The author also gives a variant of a result of \textit{D. Stankov} [J. Théor. Nombres Bordx. 31, No. 1, 215--226 (2019; Zbl 1453.11135)], by characterising Salem polynomials among reciprocal polynomials, with no irreducibility assumption.
reviewed by: François Brunault
Mathematics Subject Classification ID: 11R06
Mathematics Subject Classification ID: 12D10
Mathematics Subject Classification ID: 11C08
Mathematics Subject Classification ID: 30C15
zbMATH DE Number: 7371067
zbMATH Keywords: Salem polynomials
zbMATH Keywords: Salem numbers
zbMATH Keywords: reciprocal polynomials
zbMATH Keywords: totally real Pisot numbers
zbMATH Keywords: Pisot polynomials
author: Toufik Zaïmi
describes a project that uses: PARI/GP
MaRDI profile type: Publication
full work available at URL: s00013-021-01598-w
OpenAlex ID: W3159474180
cites work: Constructions of Pisot and Salem numbers with flat palindromes
cites work: A characterization of two related classes of Salem numbers
cites work: Q4038523
cites work: Small Salem numbers
cites work: Q4735248
cites work: Distribution modulo one and Diophantine approximation
cites work: Algorithms for localizing roots of a polynomial and the Pisot Vijayaraghavan numbers
cites work: Q3980848
cites work: Une caractérisation de certaines classes d'entiers algébriques généralisant les nombres de Salem
cites work: Q3023601
cites work: Q5237773
cites work: Conjugate algebraic integers on a circle
cites work: Q5332701
cites work: A necessary and sufficient condition for an algebraic integer to be a Salem number
cites work: On the number of roots of self-inversive polynomials on the complex unit circle
cites work: Q4317301
cites work: On totally real Pisot numbers
cites work: A remark on an algorithm for testing Pisot polynomials
DOI: 10.1007/S00013-021-01598-W
description: scientific article; zbMATH DE number 7371067