×

Polynomial equations over octonion algebras. (English) Zbl 1450.17012

The paper is about a way how to solve polynomial equations \(c_n z^n + c_{n-1} z^{n-1} + \ldots +c_1 z+c_0=0\) over octonion division algebras. The author continues his recent research on this topic and he gives the complete algorithm how to find all roots of a polynomial equation of degree \(n\) over an octonion division algebra [Algorithm 3.5]. The algorithm is based on a reduction to a specific (companion) polynomial of degree \(2n\) (see, Theorem 3.4), and it is illustrated in a beautiful example [Example 3.6].
The rest of the paper is devoted to a study of monic standard polynomials (with \(c_n=1\)). The companion matrix of a monic standard polynomial is defined and a relation between left and right eigenvalues and roots of the principal polynomial \(\phi\) and his companion polynomial \(\Phi\) has been established.
Namely, it is proved that left eigenvalues can be obtained from roots of a \(\gamma\)-twist of \(\phi\) [Theorem 4.1] and all left eigenvalues are staying between roots of \(\phi\) and roots of \(\Phi\) [Theorem 4.2]; on the other side, all left eigenvalues are coinciding with roots of \(\Phi\) [Corollary 5.5.].

MSC:

17D05 Alternative rings
15A18 Eigenvalues, singular values, and eigenvectors
15B33 Matrices over special rings (quaternions, finite fields, etc.)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abrate, M., Quadratic formulas for generalized quaternions, J. Algebra Appl.8(3) (2009) 289-306. · Zbl 1210.15015
[2] Chapman, A., Quaternion quadratic equations in characteristic 2, J. Algebra Appl.14(3) (2015), Article ID: 1550033, 8 pp. · Zbl 1329.12001
[3] Chapman, A. and Machen, C., Standard polynomial equations over division algebras, Adv. Appl. Clifford Algebr.27(2) (2017) 1065-1072. · Zbl 1366.16014
[4] Janovská, D. and Opfer, G., A note on the computation of all zeros of simple quaternionic polynomials, SIAM J. Numer. Anal.48(1) (2010) 244-256. · Zbl 1247.65060
[5] Knus, M.-A., Merkurjev, A., Rost, M. and Tignol, J.-P., The Book of Involutions, , Vol. 44 (American Mathematical Society, Providence, RI, 1998), With a preface in French by J. Tits. · Zbl 0955.16001
[6] Mironov, V. L. and Mironov, S. V., Sedeonic equations of ideal fluid, J. Math. Phys.58(8) (2017), Article ID: 083101, 12 pp. · Zbl 1370.76009
[7] Springer, T. A. and Veldkamp, F. D., Octonions, Jordan Algebras and Exceptional Groups, (Springer-Verlag, Berlin, 2000). · Zbl 1087.17001
[8] Wang, Q.-W., Zhang, X. and Zhang, Y., Algorithms for finding the roots of some quadratic octonion equations, Comm. Algebra42(8) (2014) 3267-3282. · Zbl 1305.15042
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.