Abstract
This paper aims to present, in a unified manner, results which are valid on both the algebras of quaternions and coquaternions and, simultaneously, call the attention to the main differences between these two algebras. The rings of one-sided polynomials over each of these algebras are studied and some important differences in what concerns the structure of the set of their zeros are remarked. Examples illustrating this different behavior of the zero-sets of quaternionic and coquaternionic polynomials are also presented.
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Notes
- 1.
Also known, in the literature, as split-quaternions, para-quaternions, anti-quaternions or hyperbolic quaternions.
- 2.
This coincides with the Euclidean norm of the 4-vector \((q_0,q_1,q_2, q_3)\).
- 3.
Right one-sided polynomials are defined in an analogous manner, by considering the coefficients on the right of the variable; all the results for left one-sided polynomials have corresponding results for right one-sided polynomials and hence we restrict our study to polynomials of the first type.
- 4.
Recall that, in the case of quaternions, quasi-similarity and similarity classes coincide, so, in that case, we can also say that \(\mathrm{\Psi }_q\) is an invariant of [q].
- 5.
Recall that this can only happen in the case of coquaternions.
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Acknowledgments
Research at CMAT was financed by Portuguese Funds through FCT - Fundação para a Ciência e a Tecnologia, within the Project UID/MAT/00013/2013. Research at NIPE was carried out within the funding with COMPETE reference number POCI-01-0145-FEDER-006683 (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.
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Falcão, M.I., Miranda, F., Severino, R., Soares, M.J. (2017). Polynomials over Quaternions and Coquaternions: A Unified Approach. In: Gervasi, O., et al. Computational Science and Its Applications – ICCSA 2017. ICCSA 2017. Lecture Notes in Computer Science(), vol 10405. Springer, Cham. https://doi.org/10.1007/978-3-319-62395-5_26
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DOI: https://doi.org/10.1007/978-3-319-62395-5_26
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