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Mathematica Tools for Quaternionic Polynomials

  • M. Irene Falcão
  • Fernando Miranda
  • Ricardo Severino
  • M. Joana Soares
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10405)

Abstract

In this paper we revisit the ring of (left) one-sided quaternionic polynomials with special focus on its zero structure. This area of research has attracted the attention of several authors and therefore it is natural to develop computational tools for working in this setting. The main contribution of this paper is a Mathematica collection of functions QPolynomial for solving polynomial problems that we frequently find in applications.

Keywords

Quaternions Polynomial ring Factorization Symbolic computation 

Notes

Acknowledgments

Research at CMAT was financed by Portuguese funds through Fundação para a Ciência e a Tecnologia, within the Project UID/MAT/00013/2013. Research at NIPE was carried out within the funding with COMPETE reference number POCI-01-0145-FEDER-006683 (UID/ECO/ 03182/2013), with the FCT/MEC’s (Fundação para a Ciência e a Tecnologia, I.P.) financial support through national funding and by the ERDF through the Operational Programme on “Competitiveness and Internationalization - COMPETE 2020" under the PT2020 Partnership Agreement.

References

  1. 1.
    Beck, B.: Sur les équations polynomiales dans les quaternions. Enseign. Math. 25, 193–201 (1979)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bolotnikov, V.: Zeros and factorizations of quaternion polynomials: the algorithmic approach. arXiv:1505.03573 (2015)
  3. 3.
    Brenner, J.L.: Matrices of quaternions. Pacific J. Math. 1(3), 329–335 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chapman, A., Machen, C.: Standard polynomial equations over division algebras. Adv. Appl. Clifford Algebras 27, 1065–1072 (2016). doi: 10.1007/s00006-016-0740-4 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Damiano, A., Gentili, G., Struppa, D.: Computations in the ring of quaternionic polynomials. J. Symbolic Comput. 45(1), 38–45 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    De Leo, S., Ducati, G., Leonardi, V.: Zeros of unilateral quaternionic polynomials. Electron. J. Linear Algebra 15, 297–313 (2006)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Eilenberg, S., Niven, I.: The “fundamental theorem of algebra” for quaternions. Bull. Am. Math. Soc. 50, 246–248 (1944)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Falcão, M.I., Miranda, F.: Quaternions: a Mathematica package for quaternionic analysis. In: Murgante, B., Gervasi, O., Iglesias, A., Taniar, D., Apduhan, B.O. (eds.) ICCSA 2011. LNCS, vol. 6784, pp. 200–214. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-21931-3_17 CrossRefGoogle Scholar
  9. 9.
    Falcão, M.I., Miranda, F., Severino, R., Soares, M.J.: Quaternionic polynomials with multiple zeros: a numerical point of view. In: AIP Conference Proceedings, vol. 1798, no. 1, p. 020099 (2017)Google Scholar
  10. 10.
    Falcão, M.I., Miranda, F., Severino, R., Soares, M.J.: Weierstrass method for quaternionic polynomial root-finding. arXiv:1702.04935 (2017)
  11. 11.
    Farouki, R.T., Gentili, G., Giannelli, C., Sestini, A., Stoppato, C.: A comprehensive characterization of the set of polynomial curves with rational rotation-minimizing frames. Adv. Comput. Math. 43(1), 1–24 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gentili, G., Stoppato, C.: Zeros of regular functions and polynomials of a quaternionic variable. Mich. Math. J. 56(3), 655–667 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gentili, G., Struppa, D.C.: On the multiplicity of zeroes of polynomials with quaternionic coefficients. Milan J. Math. 76, 15–25 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gordon, B., Motzkin, T.: On the zeros of polynomials over division rings I. Trans. Am. Math. Soc. 116, 218–226 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Jacobson, N.: The Theory of Rings. Mathematical Surveys and Monographs. American Mathematical Society, New York (1943)CrossRefzbMATHGoogle Scholar
  16. 16.
    Janovská, D., Opfer, G.: The classification and the computation of the zeros of quaternionic, two-sided polynomials. Numer. Math. 115(1), 81–100 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kalantari, B.: Algorithms for quaternion polynomial root-finding. J. Complex. 29(3–4), 302–322 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lam, T.Y.: A First Course in Noncommutative Rings. Graduate Texts in Mathematics. Springer, New York (1991)CrossRefzbMATHGoogle Scholar
  20. 20.
    Lianggui, F., Kaiming, Z.: Classifying zeros of two-sided quaternionic polynomials and computing zeros of two-sided polynomials with complex coefficients. Pacific J. Math 262(2), 317–337 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Miranda, F., Falcão, M.I.: Modified quaternion newton methods. In: Murgante, B., Misra, S., Rocha, A.M.A.C., Torre, C., Rocha, J.G., Falcão, M.I., Taniar, D., Apduhan, B.O., Gervasi, O. (eds.) ICCSA 2014. LNCS, vol. 8579, pp. 146–161. Springer, Cham (2014). doi: 10.1007/978-3-319-09144-0_11 Google Scholar
  22. 22.
    Niven, I.: Equations in quaternions. Am. Math. Monthly 48, 654–661 (1941)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ore, O.: Theory of non-commutative polynomials. Ann. Math. 34(3), 480–508 (1933)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Pereira, R.: Quaternionic Polynomials and Behavioral Systems. Ph. D. Thesis, Universidade de Aveiro (2006)Google Scholar
  25. 25.
    Pereira, R., Rocha, P.: On the determinant of quaternionic polynomial matrices and its application to system stability. Math. Methods Appl. Sci. 31(1), 99–122 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Pereira, R., Rocha, P., Vettori, P.: Algebraic tools for the study of quaternionic behavioral systems. Linear Algebra Appl. 400(1–3), 121–140 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Pogorui, A., Shapiro, M.: On the structure of the set of zeros of quaternionic polynomials. Complex Variables Theor. Appl. 49(6), 379–389 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Pumplün, S., Walcher, S.: On the zeros of polynomials over quaternions. Commun. Algebra 30(8), 4007–4018 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Serôdio, R., Pereira, E., Vitória, J.: Computing the zeros of quaternion polynomials. Comput. Math. Appl. 42(8–9), 1229–1237 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Serôdio, R., Siu, L.S.: Zeros of quaternion polynomials. Appl. Math. Lett. 14(2), 237–239 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Topuridze, N.: On the roots of polynomials over division algebras. Georgian Math. J. 10(4), 745–762 (2003)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Xu, W., Feng, L., Yao, B.: Zeros of two-sided quadratic quaternion polynomials. Adv. Appl. Clifford Algebras 24(3), 883–902 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • M. Irene Falcão
    • 1
  • Fernando Miranda
    • 1
  • Ricardo Severino
    • 2
  • M. Joana Soares
    • 3
  1. 1.CMAT and Departamento de Matemática e AplicaçõesUniversidade do MinhoBragaPortugal
  2. 2.Departamento de Matemática e AplicaçõesUniversidade do MinhoBragaPortugal
  3. 3.NIPE and Departamento de Matemática e AplicaçõesUniversidade do MinhoBragaPortugal

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