On roots of quaternion polynomials.
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)].
scientific article; zbMATH DE number 5660096