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A new method for roots of monic quaternionic quadratic polynomial. (English) Zbl 1189.65090
Summary: The purpose of this paper is to show how the problem of finding roots (or zeros) of the monic quaternionic quadratic polynomials (MQQP) can be solved by its equivalent real quadratic form. The real quadratic form matrices, firstly defined in this paper, are used to form a simple equivalent real quadratic form of MQQP. Some necessary and sufficient conditions for the existence of roots of MQQP are also presented. The main idea of the practical method proposed in this work can be summarized in two steps: translating MQQP into its equivalent real quadratic form, and giving directly the quaternionic roots of MQQP by solving its equivalent real quadratic form.

##### MSC:
 65H04 Numerical computation of roots of polynomial equations 16K20 Finite-dimensional division rings
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##### References:
 [1] Adler, S., Quaternionic quantum mechanics and quantum fields, (1995), Oxford University Press New York · Zbl 0885.00019 [2] R. Carvalho, S. Leo, Quaternions in algebra and analysis, Technical report (Unicamp), Campinas(2002) [3] Niven, J., Equations in quaternions, Amer. math. monthly, 48, 654-661, (1941) · Zbl 0060.08002 [4] Niven, J., The roots of a quaternion, Amer. math. monthly, 49, 386-388, (1942) · Zbl 0061.01407 [5] Eilenberg, S.; Niven, I., The “fundamental theorem of algebra” for quaternions, Bull. amer. math. soc., 50, 246-248, (1944) · Zbl 0063.01228 [6] Pogorui, A.; Shapiro, M., On the structure of the set of zeros of quaternionic polynomials, Complex var. theory appl., 49, 6, 379-389, (2004) · Zbl 1160.30353 [7] Gentili, G.; Struppa, D.C., A new theory of regular functions of a quaternionic variable, Adv. math., 216, 279-301, (2007) · Zbl 1124.30015 [8] Pumplün, S.; Walcher, S., On the zeros of polynomials over quaternions, Comm. algebra, 30, 4007-4018, (2002) · Zbl 1024.12002 [9] Gentili, G.; Struppa, D.C.; Vlacci, F., The fundamental theorem of algebra for Hamilton and Cayley numbers, Math. Z., 259, 895-902, (2008) · Zbl 1144.30004 [10] Janovská, D.; Opfer, G., Computing quaternionic roots by newton’s method, Electron. trans. numer. anal., 26, 82-102, (2007) · Zbl 1160.65016 [11] Leo, S.; Ducati, G.; Leonardi, V., Zeros of unilateral quaternionic polynomials, Electron. J. linear algebra, 15, 297-313, (2006) · Zbl 1151.15303 [12] Serôdio, R.; Pereira, E.; Vitoria, J., Computing the zeros of quaternion polynomials, Comput. math. appl., 42, 1229-1237, (2001) · Zbl 1050.30037 [13] Gordon, B.; Motzkin, T.S., On the zeros of polynomials over division rings, Trans. amer. math. soc., 116, 218-226, (1965) · Zbl 0141.03002 [14] Röhrl, H., On the zeros of polynomials over arbitrary finite dimensional algebras, Manuscripta math., 25, 259-390, (1978) · Zbl 0425.17001
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