The number of zeros of unilateral polynomials over coquaternions revisited

Abstract

The literature on quaternionic polynomials and, in particular, on methods for determining and classifying their zero-sets, is fast developing and reveals a growing interest on this subject. In contrast, polynomials defined over the algebra of coquaternions have received very little attention from researchers. One of the few exceptions is the very recent paper by Janovská and Opfer [Electronic Transactions on Numerical Analysis, Volume 46, pp. 55-70, 2017], where, among other results, we can find a first attempt to prove that a unilateral coquaternionic polynomial of degree n has, at most, n(2n − 1) zeros.In this paper we present a full proof of the referred result, using a totally different and, from our point of view, much simpler approach. Also, we give a complete characterization of the zero-sets of such polynomials and present a new result giving conditions which guarantee the existence of a special type of zeros. An algorithm to compute and classify all the zeros of a coquaternionic polynomial is proposed and several numerical examples are carefully constructed.

On the Roots of Coquaternions

“…An admissible class of a general polynomial of the form (4.1) may also contain an infinite number of zeros forming a strict subset of the class. As proved in [10], this set of zeros (considered as points in R 4 ) form a straight line, and are therefore called linear zeros. However, as we will see, linear zeros never occur in the case of polynomials of the form x n − q that we are considering here.…”

“…In this paper we give a complete description of the roots of any coquaternion, extending in this way the work of [17]. Recent results on the structure of the zeros of coquaternionic unilateral polynomials [9,10] allow to express the nth roots of a real number in terms of similarity classes.…”

“…We first recall the concepts of similarity and quasi-similarity for coquaternions; see, e.g., [9,10] and references therein. We say that a coquaternion q is similar to a coquaternion p if there exists an invertible coquaternion h such that q = h −1 ph.…”

“…In [17,Theorem 6], the author was aware of the possibility of the existence of infinitely many roots of the equation x n = α, but did not fully discuss this case. The key ingredient to solve this problem was the use of the recent results obtained in [9,10].…”

“…Although coquaternions, also known in the literature as split quaternions, are not as popular as the well-known Hamilton's quaternions, recent studies show an increasing interest in these hypercomplex numbers, not only from the theoretical point of view, but also from the perspective of applications. Algebraic properties of coquaternions and coquaternionic polynomials are considered in e.g., [9,10,19] while [13] contains a study of coquaternionic analysis based on representation theory. Some geometric applications of coquaternions can be found in [16,18,20] and the relation between coquaternions and complexified mechanics is discussed in [3].…”