Self-inversive polynomials of odd degree
type: Item
instance of: scholarly article
title: Self-inversive polynomials of odd degree
zbMATH Open document ID: 1149.30010
author: László Losonczi
author: Andrzej Schinzel
published in: The Ramanujan Journal
publication date: 2008-05-20T00:00:00Z
review text: Let \(P(z) =\sum_{k=0}^m A_k z^k\). The polynomial \(P(z)\) is called a self-inversial polynomial of degree \(m\) if \(A_m\neq 0\), \(A_{m-k} = \varepsilon \overline{A_k}\) \((k=0, 1, \ldots m) \) with fixed complex \(\varepsilon , | \varepsilon | =1.\) The authors prove the following. Let \(P(z)\) be a self-inversial polynomial of odd degree \(m \geq 3.\) Then the inequality \[ | A_m| \geq \cos \left(\frac{\pi}{2(m+1)} \right) \inf _{c, d \in \mathbb{C}, | d| =1}\sum_{k=0}^m | c A_k -d^{m-k}A_m| \] implies that all zeros of \(P\) are on the unit circle and ones are simple.
Mathematics Subject Classification ID: 30C10
Mathematics Subject Classification ID: 26C15
zbMATH DE Number: 5277384
zbMATH Keywords: zeros
zbMATH Keywords: identities for trigonometric sums
zbMATH Keywords: zeros of polynomials
MaRDI profile type: Publication
full work available at URL: s11139-007-9029-5
OpenAlex ID: W1968305778
cites work: Zeros of Self-Inversive Polynomials
cites work: Q4795324
cites work: Q4821446
cites work: Self-inversive polynomials with all zeros on the unit circle
DOI: 10.1007/S11139-007-9029-5
description: scientific article; zbMATH DE number 5277384