×

zbMATH — the first resource for mathematics

The classification and the computation of the zeros of quaternionic, two-sided polynomials. (English) Zbl 1190.65075
The quaternionic polynomials of the two-side type
\[ p(z):=\sum_{j=0}^n a_j z^j b_j, \quad z,a_j,b_j \in \mathbb{H}, \quad a_0b_0 \neq 0, a_n b_n \neq 0, \] where \(\mathbb{H}\) is the skew field of quaternions are treated. It is shown that there are three more classes of zeros defined by the rank of a certain real \((4\times 4)\) matrix. This information can be used to find all zeros in the same class if only one zero in that class is known. The essential tool is the description of the polynomial \(p\) by a matrix equation \(P(z):=\mathbf{A}(z)+B(z)\), where \(\mathbf{A}(z)\) is a real \((4 \times 4)\) matrix determined by the coefficients of the given polynomial \(p\) and \(P, z, B\) are real column vectors with four rows. The Newton method is applied to \(P(z)=0\). There are various examples in the paper.

MSC:
65H04 Numerical computation of roots of polynomial equations
11R52 Quaternion and other division algebras: arithmetic, zeta functions
12E15 Skew fields, division rings
12Y05 Computational aspects of field theory and polynomials (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aramanovitch L.I.: Quaternion non-linear filter for estimation of rotating body attitude. Math. Methods Appl. Sci. 18, 1239–1255 (1995) · Zbl 0841.93060 · doi:10.1002/mma.1670181504
[2] Eilenberg S., Niven I.: The ”Fundamental Theorem of Algebra” for quaternions. Bull. Am. Math. Soc. 50, 246–248 (1944) · Zbl 0063.01228 · doi:10.1090/S0002-9904-1944-08125-1
[3] Gentili G., Struppa D.C.: On the multiplicity of zeros of polynomials with quaternionic coefficients. Milan J. Math. 76, 1–10 (2007)
[4] 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 · doi:10.1007/s00209-007-0254-9
[5] Gordon B., Motzkin T.S.: 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
[6] Gürlebeck K., Sprössig W.: Quaternionic and Clifford Calculus for Physicists and Engineers, pp. 371. Wiley, Chichester (1997) · Zbl 0897.30023
[7] Horn R.A., Johnson C.R.: Matrix Analysis, pp. 561. Cambridge University Press, Cambridge (1992)
[8] Janovská, D., Opfer, G.: A note on the computation of all zeros of simple quaternionic polynomials. SIAM J. Numer. Anal. (2009, to appear) · Zbl 1247.65060
[9] Janovská D., Opfer G.: Linear equations in quaternionic variables. Mitt. Math. Ges. Hamburg 27, 223–234 (2008) · Zbl 1179.11042
[10] Janovská D., Opfer G.: Givens’ transformation applied to quaternion valued vectors. BIT 43(Suppl.), 991–1002 (2003) · Zbl 1052.65030 · doi:10.1023/B:BITN.0000014561.58141.2c
[11] Lam T.Y.: A first course in noncommutative rings, 2nd edn, pp. 385. Springer, New York (2001) · Zbl 0980.16001
[12] De Leo S., Ducati G., Leonardi V.: Zeros of unilateral quaternionic polynomials. Electron. J. Linear Algebra 15, 297–313 (2006) · Zbl 1151.15303
[13] Niven I.: Equations in quaternions. Am. Math. Monthly 48, 654–661 (1941) · Zbl 0060.08002 · doi:10.2307/2303304
[14] Opfer G.: Polynomials and Vandermonde matrices over the field of quaternions. Electron. Trans. Numer. Anal. 36, 9–16 (2009) · Zbl 1196.11154
[15] Pogorui A., Shapiro M.: On the structure of the set of zeros of quaternionic polynomials. Complex Var. Elliptic Funct. 49, 379–389 (2004) · Zbl 1160.30353 · doi:10.1080/0278107042000220276
[16] Pumplün S., Walcher S.: On the zeros of polynomials over quaternions. Comm. Algebra 30, 4007–4018 (2002) · Zbl 1024.12002 · doi:10.1081/AGB-120005832
[17] Serôdio R., Pereira E., Vitória J.: Computing the zeros of quaternionic polynomials. Comput. Math. Appl. 42, 1229–1237 (2001) · Zbl 1050.30037 · doi:10.1016/S0898-1221(01)00235-8
[18] van der Waerden, B.L.: Algebra I, 5. Aufl., 292 p. Springer, Berlin (1960) · Zbl 0087.25903
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.