On roots of quaternion polynomials.
type: Item
instance of: scholarly article
title: On roots of quaternion polynomials.
zbMATH Open document ID: 1243.16022
author: N. Topuridze
published in: Journal of Mathematical Sciences (New York)
publication date: 2010-01-19T00:00:00Z
review text: Let \(\mathbb H\) be the quaternion skew-field with basis \(\{1,i,j,k\}\) over the real numbers \(\mathbb R\), where \(ijk=i^2=j^2=k^2=-1\). General polynomials over \(\mathbb H\) and some its properties are studied by \textit{S. Eilenberg} and \textit{I. Niven} [in Bull. Am. Math. Soc. 50, 246-248 (1944; Zbl 0063.01228)]. It is known that a general polynomial may have no roots, but every canonical polynomial \(f(x)=x^n+ a_1x^{n-1}+\cdots+a_{n-1}x+a_n\) of degree \(n\geq 1\) with coefficients \(a_m\in\mathbb H\) always has a non-empty set of roots \(X(f)\). In this paper the author studies the geometric and topological structure of the set \(X(f)\). He proves that the set \(X(f)\) consists of \(t\) points and \(s\) two-dimensional spheres with centers on the real line, where \(t+2s\leq n\). It is also shown that one can define multiplicities of components of each type in such way that their sum be \(n\). Analogous problem is also studied by \textit{G. Khimshiashvili} [Bull. Georgian Acad. Sci. 165, No. 3, 465-468 (2002; Zbl 1032.16014)] and \textit{N. Topuridze} [ibid. 164, No. 2, 228-231 (2001; Zbl 1046.16501)].
reviewed by: S. V. Mihovski
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 30G35
Mathematics Subject Classification ID: 30C15
Mathematics Subject Classification ID: 12E15
Mathematics Subject Classification ID: 12E12
zbMATH DE Number: 5660096
zbMATH Keywords: quaternions
zbMATH Keywords: roots of quaternion polynomials
MaRDI profile type: Publication
full work available at URL: s10958-009-9530-z
OpenAlex ID: W2008123222
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DOI: 10.1007/S10958-009-9530-Z
description: scientific article; zbMATH DE number 5660096