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Quadratic formulas for quaternions. (English) Zbl 1011.15010

Let \(\mathbb{R}\) denote the field of real numbers and \(\mathbb{H}\) denote the algebra of real quaternions. The authors of the paper provide the explicit formulas for computing all roots of a quadratic polynomial of the form \(x^2+bx+c\), for all \(b,c\in \mathbb{H}\). The general situation is divided into several particular cases depending on the relations between parameters \(b\) and \(c\). In each case the direct expressions of all roots through the parameters \(b\) and \(c\) are derived. As a corollary, the necessary and sufficient conditions on parameters \(b,c\) that guarantee the existence of exactly one, two, or infinitely many different roots are given.
The results of this paper unify several particular results on this subject [see, S. Zhang and D. Mu, J. Math. Res. Expo. 14, No. 2, 260-264 (1994; Zbl 0815.16007), R. M. Porter, J. Nat. Geom. 11, No. 2, 101-106 (1997; Zbl 0869.11024), I. Niven, Am. Math. Mon. 48, 654-661 (1941; Zbl 0060.08002)].

MSC:

15B33 Matrices over special rings (quaternions, finite fields, etc.)
11D09 Quadratic and bilinear Diophantine equations
12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)
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References:

[1] Zhang, F., Quaternions and matrices of quaternion, Linear Algebra and Its Application, 251, 21-57 (1997) · Zbl 0873.15008
[2] Zhang, S. Z.; Mu, D. L., Quadratic equations over noncommutative division rings, J. Math. Res. Exposition, 14, 260-264 (1994) · Zbl 0815.16007
[3] Porter, R. Michael, Quaternionic linear and quadratic equations, Journal of Natural Geometry, 11, 101-106 (1997) · Zbl 0869.11024
[4] Niven, I., Equations in quaternions, American Math. Monthly, 48, 654-661 (1941) · Zbl 0060.08002
[5] Huang, L.; So, W., On left eigenvalues of quaternionic matrices, Linear Algebra and Its Application, 323, 105-116 (2001) · Zbl 0976.15014
[6] Heidrich, R.; Jank, G., On the iteration of quaternionic Moebius transformations, Complex Variables Theory Application, 29, 313-318 (1996) · Zbl 0860.30041
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