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Mathematica Tools for Quaternionic Polynomials

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Book cover Computational Science and Its Applications – ICCSA 2017 (ICCSA 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10405))

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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.

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Notes

  1. 1.

    In such case, it can be proved that all quaternions in [q] are in fact zeros of P and therefore the choice of the term spherical to designate this type of zeros is natural.

  2. 2.

    In [2], such a chain was called a spherical chain.

  3. 3.

    More precisely, q is a mixed zero of P if \(m_P([q])>0\) and \(m_P(q)> m_P(q')\), for all \(q'\in [q]\).

  4. 4.

    In [2] the same problem is also addressed and one can find a particular (different) choice of \(\tilde{x}_l\) (\(l=2,\dots ,n\)) and \( y_l\) (\(l=2,\dots ,m\)).

References

  1. Beck, B.: Sur les équations polynomiales dans les quaternions. Enseign. Math. 25, 193–201 (1979)

    MathSciNet  MATH  Google Scholar 

  2. Bolotnikov, V.: Zeros and factorizations of quaternion polynomials: the algorithmic approach. arXiv:1505.03573 (2015)

  3. Brenner, J.L.: Matrices of quaternions. Pacific J. Math. 1(3), 329–335 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  5. Damiano, A., Gentili, G., Struppa, D.: Computations in the ring of quaternionic polynomials. J. Symbolic Comput. 45(1), 38–45 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. De Leo, S., Ducati, G., Leonardi, V.: Zeros of unilateral quaternionic polynomials. Electron. J. Linear Algebra 15, 297–313 (2006)

    MathSciNet  MATH  Google Scholar 

  7. Eilenberg, S., Niven, I.: The “fundamental theorem of algebra” for quaternions. Bull. Am. Math. Soc. 50, 246–248 (1944)

    Article  MathSciNet  MATH  Google Scholar 

  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

    Chapter  Google Scholar 

  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. Falcão, M.I., Miranda, F., Severino, R., Soares, M.J.: Weierstrass method for quaternionic polynomial root-finding. arXiv:1702.04935 (2017)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gentili, G., Stoppato, C.: Zeros of regular functions and polynomials of a quaternionic variable. Mich. Math. J. 56(3), 655–667 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gentili, G., Struppa, D.C.: On the multiplicity of zeroes of polynomials with quaternionic coefficients. Milan J. Math. 76, 15–25 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gordon, B., Motzkin, T.: On the zeros of polynomials over division rings I. Trans. Am. Math. Soc. 116, 218–226 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jacobson, N.: The Theory of Rings. Mathematical Surveys and Monographs. American Mathematical Society, New York (1943)

    Book  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kalantari, B.: Algorithms for quaternion polynomial root-finding. J. Complex. 29(3–4), 302–322 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lam, T.Y.: A First Course in Noncommutative Rings. Graduate Texts in Mathematics. Springer, New York (1991)

    Book  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Niven, I.: Equations in quaternions. Am. Math. Monthly 48, 654–661 (1941)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ore, O.: Theory of non-commutative polynomials. Ann. Math. 34(3), 480–508 (1933)

    Article  MathSciNet  MATH  Google Scholar 

  24. Pereira, R.: Quaternionic Polynomials and Behavioral Systems. Ph. D. Thesis, Universidade de Aveiro (2006)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  28. Pumplün, S., Walcher, S.: On the zeros of polynomials over quaternions. Commun. Algebra 30(8), 4007–4018 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  29. Serôdio, R., Pereira, E., Vitória, J.: Computing the zeros of quaternion polynomials. Comput. Math. Appl. 42(8–9), 1229–1237 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  30. Serôdio, R., Siu, L.S.: Zeros of quaternion polynomials. Appl. Math. Lett. 14(2), 237–239 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  31. Topuridze, N.: On the roots of polynomials over division algebras. Georgian Math. J. 10(4), 745–762 (2003)

    MathSciNet  MATH  Google Scholar 

  32. Xu, W., Feng, L., Yao, B.: Zeros of two-sided quadratic quaternion polynomials. Adv. Appl. Clifford Algebras 24(3), 883–902 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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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.

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Authors and Affiliations

  1. CMAT and Departamento de Matemática e Aplicações, Universidade do Minho, Braga, Portugal

    M. Irene Falcão & Fernando Miranda

  2. Departamento de Matemática e Aplicações, Universidade do Minho, Braga, Portugal

    Ricardo Severino

  3. NIPE and Departamento de Matemática e Aplicações, Universidade do Minho, Braga, Portugal

    M. Joana Soares

Authors
  1. M. Irene Falcão
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  2. Fernando Miranda
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  3. Ricardo Severino
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  4. M. Joana Soares
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Corresponding author

Correspondence to M. Irene Falcão .

Editor information

Editors and Affiliations

  1. University of Perugia, Perugia, Italy

    Osvaldo Gervasi

  2. University of Basilicata, Potenza, Italy

    Beniamino Murgante

  3. Covenant University, Ota, Nigeria

    Sanjay Misra

  4. University of Trieste, Trieste, Italy

    Giuseppe Borruso

  5. Polytechnic University of Bari, Bari, Italy

    Carmelo M. Torre

  6. University of Minho, Braga, Portugal

    Ana Maria A.C. Rocha

  7. Monash University, Clayton, Victoria, Australia

    David Taniar

  8. Kyushu Sangyo University, Fukuoka, Japan

    Bernady O. Apduhan

  9. Saint Petersburg State University, Saint Petersburg, Russia

    Elena Stankova

  10. University of Trieste, Trieste, Italy

    Alfredo Cuzzocrea

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Falcão, M.I., Miranda, F., Severino, R., Soares, M.J. (2017). Mathematica Tools for Quaternionic Polynomials. In: Gervasi, O., et al. Computational Science and Its Applications – ICCSA 2017. ICCSA 2017. Lecture Notes in Computer Science(), vol 10405. Springer, Cham. https://doi.org/10.1007/978-3-319-62395-5_27

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  • DOI: https://doi.org/10.1007/978-3-319-62395-5_27

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