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The number of zeros of unilateral polynomials over coquaternions revisited. (English) Zbl 1411.15013
Summary: The literature on quaternionic polynomials and, in particular, on methods for finding and classifying their zero sets, is fast developing and reveals a growing interest in this subject. In contrast, polynomials defined over the algebra of coquaternions have received very little attention from researchers. One of the few exceptions is the very recent paper by D. Janovská and G. Opfer [ETNA, Electron. Trans. Numer. Anal. 46, 55–70 (2017; Zbl 1368.65069)], where, among other results, we can find a first attempt to prove that a unilateral coquaternionic polynomial of degree \(n\) has, at most, \( n(2n-1)\) zeros. In this paper we present a full proof of this result, using a totally different and, from our point of view, much simpler approach. Also, we give a complete characterization of the zero sets of such polynomials and present a new result giving conditions which guarantee the existence of a special type of zeros. An algorithm to compute and classify all the zeros of a coquaternionic polynomial is proposed and several numerical examples are carefully constructed.
Reviewer: Reviewer (Berlin)

MSC:
15A66 Clifford algebras, spinors
12E05 Polynomials in general fields (irreducibility, etc.)
65H04 Numerical computation of roots of polynomial equations
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References:
[1] Niven, I., Equations in quaternions, Amer Math Monthly, 48, 654-661, (1941) · Zbl 0060.08002
[2] Serôdio, R.; Pereira, E.; Vitória, J., Computing the zeros of quaternion polynomials, Comput Math Appl, 42, 8, 1229-1237, (2001) · Zbl 1050.30037
[3] De Leo, S.; Ducati, G.; Leonardi, V., Zeros of unilateral quaternionic polynomials, Electron J Linear Algebra, 15, 297-313, (2006) · Zbl 1151.15303
[4] Falcão, MI, Newton method in the context of quaternion analysis, Appl Math Comput, 236, 458-470, (2014) · Zbl 1334.65085
[5] Miranda, F.; Falcão, MI; Murgante, B., Lecture notes in computer science, 8579, Modified quaternion Newton’s method, 146-161, (2014), Cham: Springer, Cham
[6] Pogorui, AA; 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] Serôdio, R.; Siu, L-S, Zeros of quaternion polynomials, Appl Math Lett, 14, 2, 237-239, (2001) · Zbl 0979.30030
[8] Janovská, D.; Opfer, G., A note on the computation of all zeros of simple quaternionic polynomials, SIAM J Numer Anal, 48, 1, 244-256, (2010) · Zbl 1247.65060
[9] Erdo\vgdu, M.; Özdemir, M., Two-sided linear split quaternionic equations with n unknowns, Linear Multilinear Algebra, 63, 1, 97-106, (2015) · Zbl 1311.15005
[10] Janovská, D.; Opfer, G., Linear equations and the Kronecker product in coquaternions, Mitt Math Ges Hamburg, 33, 181-196, (2013) · Zbl 1298.15006
[11] Janovská, D.; Opfer, G., Zeros and singular points for one-sided coquaternionic polynomials with an extension to other ℝ4 algebras, Electron Trans Numer Anal, 41, 133-158, (2014) · Zbl 1307.65060
[12] Özdemir, M., The roots of a split quaternion, Appl Math Lett, 22, 2, 258-263, (2009) · Zbl 1163.15303
[13] Pogoruy, AA; Rodríguez-Dagnino, R., Some algebraic and analytical properties of coquaternion algebra, Adv Appl Clifford Algebra, 20, 1, 79-84, (2010) · Zbl 1186.37053
[14] Janovská, D.; Opfer, G., The number of zeros of unilateral polynomials over coquaternions and related algebras, Electron Trans Numer Anal, 46, 55-70, (2017) · Zbl 1368.65069
[15] Antonuccio, F., Split-quaternions and the Dirac equation, Adv Appl Clifford Algebr, 25, 1, 13-29, (2015) · Zbl 1325.35184
[16] Erdo\vgdu, M.; Özdemir, M., On eigenvalues of split quaternion matrices, Adv Appl Clifford Algebr, 23, 3, 615-623, (2013) · Zbl 1296.15005
[17] Brenner, JL, Matrices of quaternions, Pacific J Math, 1, 3, 329-335, (1951) · Zbl 0043.01402
[18] Falcão, MI; Miranda, F.; Severino, R.; Gervasi, O., Lecture notes in computer science, 10405, Polynomials over quaternions and coquaternions: a unified approach, 379-393, (2017), Cham: Springer, Cham
[19] Kula, L.; Yayli, Y., Split quaternions and rotations in semi euclidean space E4_2, J Korean Math Soc, 44, 1313-1327, (2007) · Zbl 1140.15016
[20] Lam, TY, A first course in noncommutative rings, (1991), New York: Springer, New York
[21] Gordon, B.; Motzkin, TS, On the zeros of polynomials over division rings, Trans Amer Math Soc, 116, 218-226, (1965) · Zbl 0141.03002
[22] Smoktunowicz, A.; Wróbel, I., On improving the accuracy of Horner’s and Goertzel’s algorithms, Numer Algorithms, 38, 4, 243-258, (2005) · Zbl 1075.65034
[23] Falcão, MI; Miranda, F.; Severino, R., Evaluation schemes in the ring of quaternionic polynomials, Bit Numer Math, 58, 1, 51-72, (2018) · Zbl 06858769
[24] Eilenberg, S.; Niven, I., The ‘fundamental theorem of algebra’ for quaternions, Bull Amer Math Soc, 50, 4, 246-248, (1944) · Zbl 0063.01228
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