Advertisement

Programming and Computer Software

, Volume 44, Issue 2, pp 105–111 | Cite as

Satellite Unknowns in Irreducible Differential Systems

  • A. A. Panferov
Article
  • 34 Downloads

Abstract

In this paper, the concept of satellite unknowns in differential systems with selected unknowns is considered in the context of irreducible systems. It is proved that any unselected unknown in an irreducible differential system is linearly satellite for any nonempty set of selected unknowns. An algorithm for factorization of differential systems is proposed that is not always applicable but executable in polynomial time. Cases where the algorithm cannot be applied are also recognized in polynomial time.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Panferov, A.A., On determination of satellite unknowns in linear differential systems, Materialy mezhdunarodnoi konferentsii Komp’yuternaya algebra (Proc. Int. Conf. Computer Algebra), Moscow, 2016, pp. 78–80.Google Scholar
  2. 2.
    Panferov, A.A., Selected and satellite unknowns in linear differential systems, Adv. Appl. Math., 2017, vol. 85, pp. 1–11.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Panferov, A.A., Partial algorithms for satellite unknowns determination, Program. Comput. Software, 2017, vol. 43, no. 2, pp. 119–125.MathSciNetCrossRefGoogle Scholar
  4. 4.
    van der Put, M. and Singer, M.F., Galois theory of linear differential equations, Grundlehren der mathematischen Wissenschaften, Berlin: Springer, 2003.Google Scholar
  5. 5.
    Minchenko, A., Ovchinnikov, A., and Singer, M.F., Reductive linear differential algebraic groups and the Galois groups of parameterized linear differential equations, Int. Math. Res. Notices, 2015, vol. 215, no. 7, pp. 1733–1793.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hrushovski, E., Computing the Galois group of a linear differential equation, Banach Center Publications, 2002, vol. 58, pp. 97–138.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Panferov, A.A., Symbolic algorithm for recognition of satellite unknowns in linear differential systems with selected unknowns, Nauchnaya konferentsiya Lomonosovskie chteniya (Sci. Conf. Lomonosov Readings) (Moscow, 2017), Moscow: MAKS Press, 2017, pp. 122–122.Google Scholar
  8. 8.
    Abramov, S.A. and Bronshtein, M., Solution of linear differential and difference systems with respect to a part of the unknowns, Comput. Math. Math.Phys., 2006, vol. 46, no. 2, pp. 218–230.MathSciNetCrossRefGoogle Scholar
  9. 9.
    Feng, R., Hrushovski’s algorithm for computing the Galois group of a linear differential equation, Adv. Appl. Math., 2015, vol. 65, pp. 1–37.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Panferov, A.A., Differential equation systems with selected part of the unknowns, Program. Comput. Software, 2015, vol. 41, no. 2, pp. 90–97.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Panferov, A.A., Partitions of the set of selected unknowns in linear differential–algebraic systems, Program. Comput. Software, 2016, vol. 42, no. 2, pp. 84–89.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Compoint, E. and Singer, M.F., Computing Galois groups of completely reducible differential equations, J. Symbol. Comput., 1999, vol. 28, pp. 473–494.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Cluzeau, T., Factorization of differential systems in characteristic p, Proc. ISSAC, 2003, pp. 58–65.Google Scholar
  14. 14.
    Bolibrukh, A.A., Obratnye zadachi monodromii v analiticheskoi teorii differentsial’nykh uravnenii (Inverse Monodromy Problems in the Analytical Theory of Differential Equations), Moscow: Moscow Cent. Contin. Math. Educ., 2009.Google Scholar
  15. 15.
    Beke, E., Die irreducibilität der homogenen linearen differentialgleichungen, Mathematische Annalen, 1894, vol. 45, pp. 185–195.zbMATHGoogle Scholar
  16. 16.
    Schwarz, F., A factorization algorithm for linear ordinary differential equations, Proc. ISSAC, 1989, pp. 17–25.Google Scholar
  17. 17.
    Bronstein, M., On solutions of linear differential equations in their coefficient field, J. Symbol. Comput., 1992, vol. 13, pp. 413–439.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Grigoriev, D.Yu., Complexity of factoring and calculating the gcd of linear ordinary differential operators, J. Symbol. Comput., 1990, vol. 10, pp. 7–37.MathSciNetCrossRefGoogle Scholar
  19. 19.
    Singer, M.F., Testing reducibility of linear differential operators: A group theory perspective, Appl. Algebra Eng. Commun. Comput., 1996, vol. 7, no. 2, pp. 77–104.CrossRefzbMATHGoogle Scholar
  20. 20.
    Tsarev, S.P., An algorithm for complete enumeration of all factorizations of a linear ordinary differential operator, Proc. ISSAC, 1996, pp. 226–231.Google Scholar
  21. 21.
    van Hoeij, M., Factorization of differential operators with rational functions coefficients, J. Symbol. Comput., 1997, vol. 24, pp. 537–561.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Grigoriev, D.Yu., Complexity of irreducibility testing for a system of linear ordinary differential equations, Proc. ISSAC, 1990, pp. 225–230.CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Faculty of Computational Mathematics and CyberneticsMoscow State UniversityMoscowRussia
  2. 2.Dorodnicyn Computing Centre, Federal Research Center Computer Science and ControlRussian Academy of SciencesMoscowRussia

Personalised recommendations