Quadratic split quaternion polynomials: factorization and geometry. (English) Zbl 1448.12001

Summary: We investigate factorizability of a quadratic split quaternion polynomial. In addition to inequality conditions for existence of such factorization, we provide lucid geometric interpretations in the projective space over the split quaternions.


12E15 Skew fields, division rings
16S36 Ordinary and skew polynomial rings and semigroup rings
51N20 Euclidean analytic geometry
Full Text: DOI arXiv


[1] Abrate, M., Quadratic formulas for generalized quaternions, J. Algebra Appl., 8, 3, 289-306 (2009) · Zbl 1210.15015 · doi:10.1142/S0219498809003308
[2] Cao, W.: Quadratic formulas for split quaternions (2019). ArXiv: 1905.08153
[3] Casas-Alvero, E., Analytic Projective Geometry (2014), Zurich: European Mathematical Society, Zurich · Zbl 1292.51002
[4] Gordon, B.; Motzkin, Ts, On the zeros of polynomials over division rings, Trans. Am. Math. Soc., 116, 218-226 (1965) · Zbl 0141.03002 · doi:10.1090/S0002-9947-1965-0195853-2
[5] Hegedüs, G.; Schicho, J.; Schröcker, Hp, Factorization of rational curves in the study quadric and revolute linkages, Mech. Mach. Theory, 69, 1, 142-152 (2013) · doi:10.1016/j.mechmachtheory.2013.05.010
[6] Huang, L.; So, W., Quadratic formulas for quaternions, Appl. Math. Lett., 15, 15, 533-540 (2002) · Zbl 1011.15010 · doi:10.1016/S0893-9659(02)80003-9
[7] Kirby, Kg, Beyond the celestial sphere: oriented projective geometry and computer graphics, Math. Mag., 75, 5, 351-366 (2002) · Zbl 1023.51011 · doi:10.1080/0025570X.2002.11953928
[8] Li, Z.; Scharler, Df; Schröcker, Hp, Factorization results for left polynomials in some associative real algebras: State of the art, applications, and open questions, J. Comput. Appl. Math., 349, 508-522 (2019) · Zbl 1425.12001 · doi:10.1016/j.cam.2018.09.045
[9] Li, Z., Schicho, J., Schröcker, H.P.: Spatial straight-line linkages by factorization of motion polynomials. ASME J. Mech. Robot. 8(2), (2015). 10.1115/1.4031806
[10] Li, Z.; Schicho, J.; Schröcker, Hp, Kempe’s universality theorem for rational space curves, Found. Comput. Math., 18, 2, 509-536 (2018) · Zbl 1430.70006 · doi:10.1007/s10208-017-9348-x
[11] Li, Z.; Schicho, J.; Schröcker, Hp, Factorization of motion polynomials, J. Symb. Comput., 92, 190-202 (2019) · Zbl 1411.16043 · doi:10.1016/j.jsc.2018.02.005
[12] Li, Zijia; Schicho, Josef; Schröcker, Hans-Peter, The Geometry of Quadratic Quaternion Polynomials in Euclidean and Non-euclidean Planes, Advances in Intelligent Systems and Computing, 298-309 (2018), Cham: Springer International Publishing, Cham · Zbl 1400.51009
[13] Niven, I., Equations in quaternions, Am. Math. Mon., 48, 10, 654-661 (1941) · Zbl 0060.08002 · doi:10.1080/00029890.1941.11991158
[14] Stolfi, J.: Oriented projective geometry. In: Proceedings of the 3rd ACM Symposium on Computational Geometry, pp. 76-85 (1987)
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.