On a geometric property of Lemniscates
type: Item
instance of: scholarly article
title: On a geometric property of Lemniscates
zbMATH Open document ID: 0383.30001
DOI: 10.1007/BF01818573
author: K. Appert
published in: Aequationes Mathematicae
publication date: 1978-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/136753
review text: Motivated by a property of polynomials of a complex variable, the authors prove the theorem below and discuses related open questions. Theorem. Let \(p_n(w,w_k) = \prod_{k=1}^n|w-w_k|\quad (w,w_k\in\mathbb R^3)\) and \(E(p_n)=\{w: p_n(w,w_k)\leq1\}\). If \(p_n(w,w_k)\) and \(p_n^{^*}(w,w_k^{^*})\) are such that \(E(p_n)\subseteq E(p_n^{^*})\) and if all the zeros \(w_k\) of \(p_n\) lie on the same plane, then \(p_n(w,w_k)\equiv p_n^{^*}(w,w_k^{^*})\). Moreover, the hypothesis \(E(p_n)\subseteq E(p_n{^*})\) is not sufficient to deduce \(p_n=p_n^{^*}\). [For further properties of products \(p_n(w,w_k)\), see \textit{J.B.Diaz} and \textit{D.B.Schaffer}, Appl. Anal. 6, 109-117 (1977; Zbl 0346.30003).]
Mathematics Subject Classification ID: 30C10
zbMATH DE Number: 3595483
MaRDI profile type: Publication
OpenAlex ID: W2074334528
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description: scientific article; zbMATH DE number 3595483