An algorithm for the factorization of split quaternion polynomials. (English) Zbl 1470.16051

Summary: We present an algorithm to compute all factorizations into linear factors of univariate polynomials over the split quaternions, provided such a factorization exists. Failure of the algorithm is equivalent to non-factorizability for which we present also geometric interpretations in terms of rulings on the quadric of non-invertible split quaternions. However, suitable real polynomial multiples of split quaternion polynomials can still be factorized and we describe how to find these real polynomials. Split quaternion polynomials describe rational motions in the hyperbolic plane. Factorization with linear factors corresponds to the decomposition of the rational motion into hyperbolic rotations. Since multiplication with a real polynomial does not change the motion, this decomposition is always possible. Some of our ideas can be transferred to the factorization theory of motion polynomials. These are polynomials over the dual quaternions with real norm polynomial and they describe rational motions in Euclidean kinematics. We transfer techniques developed for split quaternions to compute new factorizations of certain dual quaternion polynomials.


16S36 Ordinary and skew polynomial rings and semigroup rings
12D05 Polynomials in real and complex fields: factorization
51M09 Elementary problems in hyperbolic and elliptic geometries
51M10 Hyperbolic and elliptic geometries (general) and generalizations
70B10 Kinematics of a rigid body
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] Alkhaldi, A.H., Alaoui, M.K., Khamsi, M.A.: New Trends in Analysis and Geometry. Cambridge Scholars Publishing (2020). http://www.cambridgescholars.com/new-trends-in-analysis-and-geometry
[3] Cao, W.: Quadratic formulas for split quaternions (2019). arXiv: 1905.08153
[4] Casas-Alvero, E., Analytic Projective Geometry (2014), Zurich: European Mathematical Society, Zurich · Zbl 1292.51002 · doi:10.4171/138
[5] Dorst, L., The construction of 3D conformal motions, Math. Comput. Sci., 10, 97-113 (2016) · Zbl 1341.65006 · doi:10.1007/s11786-016-0250-8
[6] Dorst, L.; Valkenburg, R.; Dorst, L.; Lasenby, J., Square root and logarithm of rotors in 3D conformal geometric algebra using polar decomposition, Guide to Geometric Algebra in Practice, 81-104 (2011), Berlin: Springer, Berlin · Zbl 1290.68121 · doi:10.1007/978-0-85729-811-9_5
[7] Falcão, MI; Miranda, F.; Severino, R.; Soares, MJ, The number of zeros of unilateral polynomials over coquaternions revisited, Linear Multilinear Algebra, 67, 6, 1231-1249 (2019) · Zbl 1411.15013 · doi:10.1080/03081087.2018.1450828
[8] 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
[9] 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
[10] 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
[11] 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
[12] 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
[13] Li, Z., Schicho, J., Schröcker, H.P.: The geometry of quadratic quaternion polynomials in Euclidean and non-Euclidean planes. In: L. Cocchiarella (ed.) ICGG 2018—Proceedings of the 18th International Conference on Geometry and Graphics, pp. 298-309. Springer International Publishing, Cham (2019) · Zbl 1400.51009
[14] Niven, I., Equations in quaternions, Am. Math. Mon., 48, 10, 654-661 (1941) · Zbl 0060.08002 · doi:10.1080/00029890.1941.11991158
[15] Scharler, DF; Siegele, J.; Schröcker, HP, Quadratic split quaternion polynomials: factorization and geometry, Adv. Appl. Clifford Algebras, 30, 23 (2019) · Zbl 1448.12001 · doi:10.1007/s00006-019-1037-1
[16] Siegele, J.; Scharler, DF; Schröcker, HP, Rational motions with generic trajectories of low degree, Comput. Aided Geom. Design, 76, 101793 (2020) · Zbl 1448.70006 · doi:10.1016/j.cagd.2019.101793
[17] Wildberger, N., Universal hyperbolic geometry II: a pictorial overview, KoG, 14, 3-24 (2010) · Zbl 1217.51009
[18] Wildberger, N., Universal hyperbolic geometry III: first steps in projective triangle geometry, KoG, 15, 25-49 (2011) · Zbl 1262.51016
[19] Wildberger, N., Universal hyperbolic geometry I: trigonometry, Geom. Dedicata, 163, 215-274 (2013) · Zbl 1277.51018 · doi:10.1007/s10711-012-9746-9
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.