Linear Manifolds in Sets of Solutions of Quaternionic Polynomial Equations of Several Types

Abstract

Abstract. We give complete description of possible shapes of the set of the solutions of any quaternionic equation of the form ax + xb = c. Moreover we study the set of the solutions of a quaternionic equation of the form ax 2 + x 2 b = c by the method of sections by hyperplanes perpendicular to the real axis; for every case where such section is an unbounded linear manifold a necessary and sufficient condition is found. Mathematics Subject Classification (2010). 11R52.

An Efficient Method for Solving Equations in Generalized Quaternion and Octonion Algebras

“…is also a root of this equation. Using Theorem 3 from [12] and Theorem 3.1, we just proved the following theorem: More results on the structure of roots of the quadratic quaternionic equations can be found in see [22], [13], [14], [15].…”

“…More results on the structure of roots of the quadratic quaternionic equations can be found in see [22], [13], [14], [15].…”

On Reduction of Quaternionic Polynomials and Their Identical Equality

“…The point is that the process of looking for zeros of a quaternionic polynomial often demands very long calculations, and every additional term leads to a great additional piece of work. Therefore this paper may be considered as one on the topic about zeros of quaternionic polynomials and may be included in a list of works containing, for example, [11], [1], [12], [14], [3], [4], [10], [5], [15], [16], [6], [7], [8], [9]. Every work from this list includes investigations of some questions about zeros of quaternionic polynomials of some type(s).…”

“…The corresponding works are, for example, [14], [5], [15], [16], [6], [7], [8], [9]. These works contain much information about linear polynomials and some information about several kinds of quadratic ones.…”

“…For a great majority of types of such equations we do not know any method of solution better than rewriting by the components, although sometimes with some additional manipulations (see [14], [5], [15], [16], [6], [7], [8], [9]). This paper proposes one more type of additional manipulations making it possible to decrease the number of the terms in a given equation.…”