The number of zeros of unilateral polynomials over coquaternions revisited

Abstract

The literature on quaternionic polynomials and, in particular, on methods for finding and classifying their zero sets, is fast developing and reveals a growing interest in 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 [Electron Trans Numer Anal. 2017;46:55–70], 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 this 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.

Keywords:

• Coquaternions
• coquaternionic polynomials
• companion polynomial
• admissible classes

AMS Subject Classifications:

• 15A66
• 12E05
• 65H04

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 We should remark that the results contained in [14] are obtained for polynomials defined not only over the algebra of coquaternions but also over two other algebras: the algebra of nectarines ${\mathbb{H}}_{\mathrm{nec}}$ and the algebra of conectarines ${\mathbb{H}}_{\mathrm{con}}$. Since these two algebras are isomorphic to the algebra of coquaternions, we decided to consider only the coquaternionic case.

2 It is important to remark that we use different notations from the ones introduced in [11], where the symbol $\stackrel{q}{\sim }$ is used for the quasi-similarity relation and the quasi-similarity class of $u\in {\mathbb{H}}_{\mathrm{coq}}$ is denoted by ${[u]}_q$. Since we frequently use q for a coquaternion, we found convenient to adopt different notations.

3 A coquaternion q is usually classified as time-like, space-like or light-like according to $\det (q)>0$, $\det (q)<0$ or $\det (q)=0$, respectively. Hence, Type 1, Type 2 and Type 3 coquaternions can also be described as coquaternions whose vector part is time-like, space-like or light-like, respectively.

4 Right unilateral polynomials are defined in an analogous manner, by considering the coefficients on the right of the variable; all the results for left unilateral polynomials have corresponding results for right unilateral polynomials and hence we restrict our study to polynomials of the first type. The case of general polynomials is out of the scope of this paper. We refer the interested readers to [9], where the case of linear equations with terms of the form axb was considered.

5 This polynomial is more commonly called the characteristic polynomial of the coquaternion q. We prefer to use our definition to emphasize the biunivocal relation between quasi-similarity classes and characteristic polynomials.

6 The fact that P is monic (or with non-singular leading coefficient) guarantees that the companion polynomial has degree 2n and avoids pathological situations as having a non-zero polynomial whose companion polynomial is the zero polynomial; see [11] for such an example.

7 This result was later extended to more general quaternionic polynomials in [24].

8 This result is obtained in a different way in [13] and also in [11], where a different computational procedure for obtaining A and B is proposed; counting the number of arithmetic operations involved, one can conclude (cf. [23] for complexity and stability analysis) that, for $n>3$, the process given by (3.8) involves less computational effort than the method proposed in [11,13].

Additional information

Funding

Research at CMAT was financed by Portuguese Funds through FCT - Fundação para a Ciência e a Tecnologia, within the [project number UID/MAT/00013/2013]. Research at NIPE was carried out within the funding with COMPETE reference number POCI-01-0145-FEDER-006683 [project number UID/ECO/03182/2013], with the FCT/MEC’s (Fundação para a Ciência e a Tecnologia, I.P.) financial support through national funding and by the ERDF through the Operational Programme on ‘Competitiveness and Internationalization - COMPETE 2020’ under the PT2020 Partnership Agreement.