Liouvillian solutions of third order differential equations

Journal of Symbolic Computation

Volumn 36, Issue 6, 2003, Pages 855-889

  Ulmer, Felix ^a  

^a UNIVERSITÉ DE RENNES 1   (France)

Author keywords

12H05; 68Q40

Indexed keywords

EID: 0344254580     PISSN: 07477171     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0747-7171(03)00065-8     Document Type: Article

References (26)

1. Bronstein, M., 2001. Computer algebra algorithms for linear ordinary differential and difference equations. In: Proceedings of the Third European Congress of Mathematics. Progress in Mathematics, vol. II. p. 202.
2. Bronstein, M., Mulders, T., Weil, J.A., 1997. On symmetric powers of differential operators. In: Küchlin, W. (Ed.), ISSAC'97, Proceedings of the 1997 International Conference on Symbolic Computation and Algebraic Computation. ACM Press.
3. Cormier, O., Singer, M.F., Trager, B.M., Ulmer, F., 2002. Linear differential operators for polynomial equations. J. Symbolic Comput. 34, 355-398.
4. Fakler, W., 1999. Algebraische Algorithmen zur Lösung von linearen Differentialgleichungen, Dissertation. Universität Karlsruhe, MuPAD Reports.
5. Geiselman, W., Ulmer, F., 1997. Constructing a third order linear differential equation. Theoret. Comput. Sci. 187, 3-6.
6. Hendriks, P., van der Put, M., 1995. Galois action on solutions of a differential equation. J. Symbolic Comput. 19, 559-576.
7. Ince, E.L., 1956. Ordinary Differential Equations. Dover.
8. Kolchin, E.R., 1948. Algebraic matrix groups and the Picard-Vessiot theory of homogeneous linear ordinary differential equations. Ann. of Math. 49, 1-48.
9. Kovacic, J., 1986. An algorithm for solving second order linear homogeneous differential equations. J. Symbolic Comput. 2, 3-43.
10. Magid, A., 1994. Lectures on Differential Galois Theory, second ed, American Mathematical Society.
11. Singer, M.F., 1980. Algebraic solutions of nth order linear differential equations. Queens Papers in Pure and Appl. Math. 54, 379-420.
12. Singer, M.F., 1981. Liouvillian Solutions of nth Order Linear Differential Equations. Amer. J. Math. 103, 661-682.
13. Singer, M.F., 1985. Solving homogeneous linear differential equations in terms of second order linear differential equations. Amer. J. Math. 107, 663-696.
14. Singer, M.F., 1996. Testing reducibility of linear differential operators: a group theoretic perspective. Appl. Algebra Engrg. Comm. Comput. 7, 77-104.
15. Singer, M.F., Ulmer, F., 1993a. Galois groups of second and third order linear differential equations. J. Symbolic Comput. 16, 1-36.
16. Singer, M.F., Ulmer, F., 1993b. Liouvillian and algebraic solutions of second and third order linear differential equations. J. Symbolic Comput. 16, 37-73.
17. Singer, M.F., Ulmer, F., 1995. Necessary conditions for Liouvillian solutions of (third order) linear differential equations. J. Appl. Algebra Engrg. Comm. Comp. 6, 1-22.
18. Singer, M.F., Ulmer, F., 1997. Linear differential equations and products of linear forms. J. Pure Appl. Algebra 117-118, 549-564.
19. Ulmer, F., 1994. Linear differential equations of prime order. J. Symbolic Comput. 18, 385-401.
20. Ulmer, F., Weil, J.A., 1996. Note on Kovacic's algorithm. J. Symbolic Comput. 22, 179-200.
21. van der Put, M., Singer, M.F., 2003. Galois Theory of Linear Differential Equations. Springer Verlag, Berlin, Heidelberg
22. van der Put, M., Ulmer, F., 2000. Differential equations and finite groups. J. Algebra 226, 920-966.
23. van Hoeij, M., 2000. The Minimum Polynomial of an Algebraic Solution of Abel's Problem (Preprint FSU00-02).
24. van Hoeij, M., Ragot, J.F., Ulmer, F., Weil, J.A., 1999. Liouvillian solutions of linear differential equations of order three and higher. J. Symbolic Comput. 28, 589-609.
25. van Hoeij, M., Weil, J.A., 1997. An algorithm for computing invariants of differential Galois groups. J. Pure Appl. Algebra 117-118, 353-379.
26. Weil, J.A., 1995. Constantes et polynômes de Darboux en algèbre différentielle: application aux systèmes différentiels linéaires, Ph.D. Dissertation. École Polytechnique.