Skip to main content
SpringerLink
Log in

Testing reducibility of linear differential operators: A group theoretic perspective

  • Published:
Applicable Algebra in Engineering, Communication and Computing Aims and scope

Abstract

Let k[D] be the ring of differential operators with coefficients in a differential fieldk. We say that an elementL ofk[D] isreducible ifL=L 1·L 2 forL 1,L 2gEk[D],L 1,L 2∉k. We show that for a certain class of differential operators (completely reducible operators) there exists a Berlekamp-style algorithm for factorization. Furthermore, we show that operators outside this class can never be irreducible and give an algorithm to test if an operator belongs to the above class. This yields a new reducibility test for linear differential operators. We also give applications of our algorithm to the question of determining Galois groups of linear differential equations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Amitsur, A. S.: Differential polynomials and division algebras. Ann. Math.59, 245–278 (1954)

    Google Scholar 

  2. Appell, P.: Mémoire sur les Équations Différentielles Linéaires. Ann. Scient. Éc. Norm. Sup., Ser. 2,10, 391–424 (1881)

    Google Scholar 

  3. Barkatou, M. A.: An algorithm for computing a companion diagonal form for a system of linear differential equations. AAECC4, 185–195 (1993)

    Google Scholar 

  4. Beke, E.: Die Irreducibilität der homogenen linearen Differentialgleichungen. Math. Ann.45, 278–294 (1994)

    Google Scholar 

  5. Bertrand, D., Beukers, F.: Équations Différentielles Linéaires et Majorations de Multiplicités. Ann. Scient. Ec. Norm. Sup.18, 181–192 (1985)

    Google Scholar 

  6. Björk, J.-E.: Ring of Differential Operators. North Holland, New York 1979

    Google Scholar 

  7. Borel, A.: Linear algebraic groups. In Proc. Symp. Pure Math. vol 9, pp 3–19. Am. Math. Soc., Providence 1966

    Google Scholar 

  8. Bronstein, M.: On solutions of linear ordinary differential equations in their coefficient field. J. Symp. Comp.13, 413–439 (1992)

    Google Scholar 

  9. Cohn, P.: Free Rings and their Relations. Academic Press, London 1985

    Google Scholar 

  10. Deligne, P.: Equations Différentielles à Points Singuliers Réguliers. Lecture Notes in Mathematicsvol 163, Springer: Berlin, Heidelberg, New York 1970

    Google Scholar 

  11. Frobenius, G.: Über den Begriff der Irreducibilität in der Theorie der linearen Differentialgleichungen. J. Math.76, 236–271 (1873)

    Google Scholar 

  12. Giesbrecht, M.: Factoring in skew-polynomial rings. Department of Computer Science, University of Toronto, preprint, 1992

  13. Grigoriev, D. Yu.: Complexity of factoring and calculating the GCD of linear ordinary differential operators. J. Symbolic Computation10, 7–37 (1990)

    Google Scholar 

  14. Grigoriev, D. Yu.: Complexity of irreducibility testing for a system of linear ordinary differential equations. Proc. Int. Symp. on Symb. Alg. Comp., ACM Press 225–230 (1990)

  15. Hilb, E.: Linear Differentialgleichungen im Komplexen Gebiet. In: Encyklopedie der mathematischen Wissenschaften. IIBb, Teubner, Leipzig 1915

    Google Scholar 

  16. Humphreys, J.: Linear Algebraic Groups. Graduate Texts in Mathematics vol21, Springer: Berlin, Heidelberg, New York 1975

    Google Scholar 

  17. Hungerford, T.: Algebra. Graduate Texts in Mathematics vol73, Springer: Berlin, Heidelberg, New York 1974

    Google Scholar 

  18. Jacobson, N.: Pseudo-linear transformations. Ann. Math.38, 484–507 (1937)

    Google Scholar 

  19. Jacobson, N.: Structure of Rings. American Mathematical Society Colloquium Publications, XXXVII, Second Edition, Providence, 1964

  20. Jordan, C.: Sur une application de la théorie des substitutions à l'étude des équations différentielles linéaires. Bull. Soc. Math. France II,100 (1875)

  21. Kaplansky, I.: Introduction to Differential Algebra. Paris: Hermann 1957

    Google Scholar 

  22. Landau, E.: Ein Satz über die Zerlegung homogener linearer Differentialausdrücke in irreducible Factoren. J. Math.24, 115–120 (1902)

    Google Scholar 

  23. Lang, S.: Algebra. Second Edition, Addition-Wesley 1984

  24. Loewy, A.: Über reduzible lineare homogene Differentialgleichungen. Math. Ann.56, 549–584 (1903)

    Google Scholar 

  25. Loewy, A.: Über vollständig reduzible lineare homogene Differentialgleichungen. Math. Ann.62, 89–117 (1906)

    Google Scholar 

  26. Loewy, A.: Über lineare homogene Differentialgleichungen derselben Art. Math. Ann.70, 551–560 (1911)

    Google Scholar 

  27. Loewy, A.: Zur Theorie der linearen homogenen Differentialausdrücke. Math. Ann.72, 203–210 (1912)

    Google Scholar 

  28. Loewy, A.: Über die Zerlegungen eines linearen homogenen Differentialausdruckes in grösste volständig reduzible Faktoren. Sitz. der Heideberger Akad. der Wiss.,8, 1–20 (1917)

    Google Scholar 

  29. Loewy, A.: Über Matrizen- und Differentialkomplexe. Math Ann.78, 1:1–51; II: 343–358; III: 359–368 (1917)

    Google Scholar 

  30. Loewy, A.: Begleitmatrizen und lineare homogene Differentialausdrücke. Math. Zeit.7, 58–125 (1920)

    Google Scholar 

  31. MacDonald, B.: Finite Rings with Identity. Marcel Dekker, New York 1974

    Google Scholar 

  32. Mignotte, M.: Mathematics for Computer Algebra. Springer: Berlin, Heidelberg, New York 1992

    Google Scholar 

  33. Ore, O.: Formale Theorie der linearen Differentialgleichungen. J. Math.167, 221–234, (1932) II, ibid.,168, 233–252 (1932)

    Google Scholar 

  34. Ore, O.: Theory of non-commutative polynomial rings. Ann. Math.34, 480–508 (1993)

    Google Scholar 

  35. Poole, E. G. C.: Introducion to the Theory of Linear Differential Equations. Dover Publications, New York 1960

    Google Scholar 

  36. Poincaré, H.: Mémoire sur les fonctions Zétafuchsiennes. Acta Math.5, 209–278 (1984)

    Google Scholar 

  37. Ritt, J. F.: Differential Algebra, Dover Publications, New York 1966

    Google Scholar 

  38. Schlesinger, L.: Handbuch für Theorie der linearen Differentialgleichungen II, Leipzig: Teubner 1887

    Google Scholar 

  39. Schwarz, F.: A factorization algorithm for linear ordinary differential equations. Proc. of the ACM-SIGSAM 1989 ISSAC, ACM Press 17–25 (1989)

  40. Singer, M. F.: Algebraic solutions of nth order linear differential equations. Proceedings of the 1979 Queens Conference on Number Theory, Queens Papers in Pure and Applied Mathematics54, (1980)

  41. Singer, M. F.: Liouvillian solutions of linear differential equations with liouvillian coefficients, J. Symb. Comp.11, 251–273 (1991)

    Google Scholar 

  42. Singer, M. F.: Moduli of linear differential equations on the Riemann Sphere with fixed Galois groups. Pacific Math.160, No. 2, 343–395 (1993)

    Google Scholar 

  43. Singer, M. F., Ulmer, F.: Galois groups of second and third order linear differential equations. J. Symb. Comp.16, July 9–36 (1993)

    Google Scholar 

  44. van der Waerden, B. L.: Modern Algebra, Vol. I, Second Edition, Frederick Ungar, New York 1953

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, North Carolina State University, Box 8205, 27695-8205, Raleigh, NC, USA

    Michael F. Singer

Authors
  1. Michael F. Singer
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Partially supported by NSF Grant 90-24624

Rights and permissions

Reprints and permissions

About this article

Cite this article

Singer, M.F. Testing reducibility of linear differential operators: A group theoretic perspective. AAECC 7, 77–104 (1996). https://doi.org/10.1007/BF01191378

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01191378

Keywords

Search

Navigation