Quasi-Spherical and Multi-Quasi-Spherical Polynomial Quaternionic Equations: Introduction of the Notions and Some Examples

We introduce notions of quasi-spherical and multi-quasi-spherical polynomial quaternionic equations defined in terms of the shape of the set of the solutions of the equation. We establish that every quaternionic equation of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sum\limits_{\ell = 1}^m {a^{(\ell)}} x^2 b^{(\ell)} + \sum\limits_{p = 1}^q {c^{(p)}} xd^{(p)} + h = 0}$$\end{document}is quasi-spherical. We get some sufficient conditions under which a quadratic quaternionic polynomial can be represented as a product of linear quaternionic polynomials (and thus the corresponding equation is multi-quasi-spherical).