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Computing Burchnall–Chaundy Polynomials with Determinants

  • Johan RichterEmail author
  • Sergei Silvestrov
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 179)

Abstract

In this expository paper we discuss a way of computing the Burchnall–Chaundy polynomial of two commuting differential operators using a determinant. We describe how the algorithm can be generalized to general Ore extensions, and which properties of the algorithm that are preserved.

Keywords

Ore extensions Burchnall-Chaundy theory Determinants 

References

  1. 1.
    Amitsur, S.A.: Commutative linear differential operators. Pac. J. Math. 8, 1–10 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Burchnall, J.L., Chaundy, T.W.: Commutative ordinary differential operators. Proc. Lond. Math. Soc. (Ser. 2) 21, 420–440 (1922)Google Scholar
  3. 3.
    Burchnall, J.L., Chaundy, T.W.: Commutative ordinary differential operators. Proc. Roy. Soc. Lond. (Ser. A) 118, 557–583 (1928)CrossRefzbMATHGoogle Scholar
  4. 4.
    Burchnall, J.L., Chaundy, T.W.: Commutative ordinary differential operators. II. The Identity \(P^n=Q^m\). Proc. Roy. Soc. Lond. (Ser. A). 134, 471–485 (1932)Google Scholar
  5. 5.
    Carlson, R.C., Goodearl, K.R.: Commutants of ordinary differential operators. J. Differ. Equs. 35, 339–365 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    De Jeu, M., Svensson, C., Silvestrov, S.: Algebraic curves for commuting elements in the \(q\)-deformed Heisenberg algebra. J. Algebr. 321(4), 1239–1255 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Flanders, H.: Commutative linear differential operators. Technical Report 1, University of California, Berkely (1955)Google Scholar
  8. 8.
    Goodearl, K.R.: Centralizers in differential, pseudodifferential, and fractional differential operator rings. Rocky Mt J. Math. 13, 573–618 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Krichever, I.M.: Integration of non-linear equations by the methods of algebraic geometry. Funktz. Anal. Priloz. 11(1), 15–31 (1977)zbMATHGoogle Scholar
  10. 10.
    Krichever, I.M.: Methods of algebraic geometry in the theory of nonlinear equations. Uspekhi Mat. Nauk. 32(6), 183–208 (1977)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Larsson, D.: Burchnall–Chaundy theory, Ore extensions and \(\sigma \)-differential operators. Preprint U.U.D.M. Report vol. 45. Department of Mathematics, Uppsala University 13 pp (2008)Google Scholar
  12. 12.
    Mumford, D.: An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg–de Vries equation and related non-linear equations. In: Proceedings of International Symposium on Algebraic Geometry, Kyoto, Japan, pp. 115–153 (1978)Google Scholar
  13. 13.
    Ore, O.: Theory of non-commutative polynomials. Ann. Math. 34(2, 3), 480–508 (1933)Google Scholar
  14. 14.
    Richter, J.: Burchnall–Chaundy theory for Ore extensions. In: Makhlouf, A., Paal, E., Silvestrov, S.D., Stolin, A. (eds.) Algebra, Geometry and Mathematical Physics, vol. 85, pp. 61–70. Springer Proceedings in Mathematics & Statistics, Mulhouse, France (2014)Google Scholar
  15. 15.
    Richter, J., Silvestrov, S.: Burchnall-Chaundy annihilating polynomials for commuting elements in Ore extension rings. J. Phys.: Conf. Ser. 346 (2012). doi: 10.1088/1742-6596/346/1/012021

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Division of Applied Mathematics, School of EducationCulture and Communication, Mälardalen UniversityVästeråsSweden

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