On the zeros of the derivatives of certain polynomials
type: Item
instance of: scholarly article
title: On the zeros of the derivatives of certain polynomials
published in: Moscow Journal of Combinatorics and Number Theory
publication date: 2022-10-18T00:00:00Z
review text: In this paper the author obtains some bounds on the zeros of the derivative \(P'\), where \(P\) is the minimal polynomial of a degree \(d\) Salem number or a Pisot number. In particular, he shows that if \(\alpha\) is a Salem number of degree \(d\) then the derivative \(P'\) of the minimal of polynomial \(P\) of \(\alpha\) has a unique root \(\theta\) outside the unit circle. For this root \(\theta\), the following inequalities hold: \[ \max\Big\{\frac{\alpha+\alpha^{-1}}{2}, \frac{(d-1)\alpha-1}{d}\Big\} <\theta<\frac{(d-1)\alpha+1}{d}. \] For Pisot number \(\alpha\) the situation is similar. The derivative \(P'\) has a unique root \(\theta\) outside the unit circle. It satisfies \(|\theta-(d-1)\alpha/d|<1/d\) except when the minimal polynomial \(P\) of \(\alpha\) is one of \(9\) explicitly listed polynomials (of degrees at most \(5\)).
Mathematics Subject Classification ID: 11R06
Mathematics Subject Classification ID: 12D10
Mathematics Subject Classification ID: 11C08
Mathematics Subject Classification ID: 30C15
zbMATH DE Number: 7602920
zbMATH Keywords: Salem polynomials
zbMATH Keywords: Pisot polynomials
zbMATH Keywords: Gauss-Lucas theorem
zbMATH Keywords: Cohn's theorem
author: Toufik Zaïmi
reviewed by: A. Dubickas
MaRDI profile type: Publication
full work available at URL: moscow.2022.11.205
OpenAlex ID: W4306290529
cites work: Complex Pisot numbers in algebraic number fields
cites work: Zeros of Self-Inversive Polynomials
cites work: Small Salem numbers
cites work: On the beta expansion for Salem numbers of degree 6
cites work: Fonctions méromorphes dans le cercle-unité et leurs séries de Taylor
cites work: �ber die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreise
cites work: On Mahler measures of a self-inversive polynomial and its derivative
cites work: Étude de certaines fonctions méromorphes bornées sur le cercle unité. Application à un ensemble fermé d'entiers algébriques
cites work: Complex Pisot numbers of small modulus.
cites work: On the zeros of the derivative of a polynomial
cites work: On the zeros of a polynomial and its derivative
cites work: Q4740523
cites work: Algebraic integers whose conjugates lie in the unit circle
cites work: A STABILITY VERSION OF THE GAUSS–LUCAS THEOREM AND APPLICATIONS
cites work: An effective lower bound for the height of algebraic numbers
cites work: A remark on an algorithm for testing Pisot polynomials
cites work: Comments on Salem polynomials
DOI: 10.2140/MOSCOW.2022.11.205
description: scientific article; zbMATH DE number 7602920