Abstract
The structure and computation are given for the zeros of a two-sided quadratic quaternion polynomial. In particular, the computation formulae and the sharp bound of essential number are established for the zeros of a two-sided quaternion polynomial of the form \({u^2 + puq + r (p, q, r \in \mathbb{H})}\). As applications, some known results are improved and a conjecture presented by Janovská and Opfer is answered.
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References
Aramanovitch L. I.: Quaternion non-linear filter for estimation of rotating body attitude. Math. Methods Appl. Sci. 18, 1239–1255 (1995)
S. De Leo, G. Ducati and V. Leonardi, Zeros of unilateral quaternionic polynomials. Electron. J. Linear Algebra 15 (2006), 297–313
Eilenberg S, Niven I: The fundamental theorem of algebra for quaternions. Bull. Amer. Math. Soc. 50, 246–248 (1944)
F. Farid, Q. Wang and F. Zhang, The eigenvalues of quaternion matrices. Linear and Multilinear Algebra 59 (2011), 451–473.
Feng G. L.: Decompositions of some types of quaternionic matrices. Linear and Multilinear Algbra 58, 431–444 (2010)
L. G. Feng and W. Cheng, Estimation of the double determinant of quaternion matrices. Acta Mathematica Sicentia , Series B, 30 (2010), 1189–1198.
L. G. Feng and W. Cheng, Solution set of matrix equation over quaternion field. Alg. Collq. 19 (2012) 175–180.
Feng L. G, K. M Zhao: Classifying zeros of two-sided quaternionic polynomials and computing zeros of two-sided polynomials with complex coefficients. Pacific Journal of Mathematics 262, 317–337 (2013)
G. Gentili and D. C. Stoppato, Zeros of regular functions and polynomials of a quaternionic variable. Michigan Math. J. 56 (2008) 655–667.
G Gentili, Struppa D.C: On the multiplicity of zeros of polynomials with quaternionic coefficients. Milan J. of Math 76, 15–25 (2008)
D. Janovská and G. Opfer, The classification and the computation of the zeros of quaternionic, two-sided polynomials. Numer. Math. 115 (2010), 81–100.
D. Janovská and G. Opfer, A note on the computation of all zeros of simple quaternionic polynomials. SIAM J. Numer. Anal. 48 (2010), 244–256.
Z. Jia, X. Cheng and M. Zhao, A new method for roots of monic quaternionic quadratic polynomial. Comput. Math. Appl. 58 (2007), 1852–1858.
T. Y. Lam, A First Course in Noncommutative Rings, volume 131 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 2001.
Niven I.: Equations in quaternions. Amer. Math. Monthly 48, 654–661 (1941)
A Pogorui, M Shapiro: On the structrue of the set of zeros of quaternionic polynomials. Complex Var. Theory Appl. 49, 379–389 (2004)
S. Pumplün and S. Walcher, On the zeros of polynomials over quaternions. Comm. Algebra 30 (2002), 4007–4018.
R. Serôdio, E. Pereira and J. Vit´oria, Computing the zeros of quaternion polynomials. Comput. Math. Appl. 42 (2001), 1229–1237.
R Serôdio, Siu L.S: Zeros of quaternion polynomials. Appl. Math. Lett. 14, 237–239 (2001)
Zhang F.: Quaternions and matrices of quaternions. Linear Algebra Appl. 251, 21–57 (1997)
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The authors are partially supported by NSF of china (Grant 11301539), NUDT and HNSF of China (Grant 11JJ7002).
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Xu, W., Feng, L. & Yao, B. Zeros of Two-Sided Quadratic Quaternion Polynomials. Adv. Appl. Clifford Algebras 24, 883–902 (2014). https://doi.org/10.1007/s00006-014-0473-1
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DOI: https://doi.org/10.1007/s00006-014-0473-1