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Necessary conditions for liouvillian solutions of (third order) linear differential equations

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Abstract

In this paper we show how group theoretic information can be used to derive a set of necessary conditions on the coefficients ofL(y) forL(y=0 to have a liouvillian solution. The method is used to derive (and improve in one case) the necessary conditions of the Kovacic algorithm and to derive an explicit set of necessary conditions for third order differential equations.

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References

  1. Balser, W., Jurkat, W., Lutz, D. A.: A General Theory of Invariants for Meromorphic Differential Equations; Part I, Formal Invariants. Funkcialaj Ekvacioj22, 197–221 (1979)

    Google Scholar 

  2. Bronstein, M.: On Solutions of Linear Differential Equations in their Coefficient Field. J. Symb. Comp.13 (1992)

  3. Bronstein, M.: Linear Ordinary Differential Equations: breaking through the order 2 barrier. Proceedings of ISSAC '92. Wang, P. S. (ed). ACM Pess, New York, 1992, pp. 42–48

    Google Scholar 

  4. Cannon, J. J.: An introduction to the group theory language Cayley. In: Computational Group Theory, Atkinson, M. D. (ed). New York: Academic Press 1984

    Google Scholar 

  5. Coddington, E. A., Levinson, N.: Theory of Ordinary Differential Equations. McGraw-Hill: New York 1955

    Google Scholar 

  6. Duval, A., Loday-Richaud, M.: Kovacic's algorithm and its application to some families of special functions. AAECC3, 211–246 (1992)

    Google Scholar 

  7. Frölich, A., Shepherdson, J. C.: Effective Procedures in Field Theory. Phil. Trans. R. Soc. Ser. A248, 407–432 (1955–56)

    Google Scholar 

  8. Fuchs, L.: Uber die linearen Differentialgleichungen zweiter Ordnung, welche algebraische Integrale besitzen, und eine neue Anwendung der Invarianten-theorie. J. Math.81 (1875)

  9. Hurwitz, A.: Uber einige besondere homogene lineare Differentialgleichungen. Math. Ann.26 (1886)

  10. Kaplansky, I.: Introduction to differential algebra. Paris: Hermann 1957

    Google Scholar 

  11. Kolchin, E. R.: Algebraic groups and algebraic dependence. Am. J. Math.90 (1968)

  12. Kovacic, J.: An algorithm for solving second order linear homogeneous differential equations. J. Symb. Comp.2 (1986)

  13. Lang, S.: Algebra, Second Edition. New York: Addison Wesley 1984

    Google Scholar 

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

    Google Scholar 

  15. Ritt, J. F.: Differential Algebra. New York: Dover Publications 1950

    Google Scholar 

  16. Schlesinger, L.: Handbuch der Theorie der linearen Differentialgleichungen. Leipzig: Teubner 1885

    Google Scholar 

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

  18. Singer, M. F.: Solving Homogeneous Linear Differential Equations in Terms of Second Order Linear Differential Equations. Am. J. Math.107 (1985)

  19. Singer, M. F.: Moduli of Linear Differential Equations on the Riemann Sphere with Fixed Galois Group. Pac. J. Math.160 (1993)

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

  21. Singer, M. F., Ulmer, F.: Liouvillian and algebraic solutions of second and third order linear differential equations. J. Symb. Comp.16 (1993)

  22. Singer, M. F., Ulmer, F.: On a third order differential equation whose differential Galois group is the simple group of 168 elements. Proceedings of the 1993 Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, LNCS vol. 673. Berlin, Heidelberg, New York: Springer

  23. Tretkoff, C., Tretkoff, M.: Solution of the inverse problem of differential Galois theory in the classical case. Am. J. Math.101 (1979)

  24. Ulmer, F.: On liouvillian solutions of differential equations. AAECC2, 171–193 (1992)

    Google Scholar 

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Author notes

  1. Felix Ulmer

    Present address: IRMAR, Université de Rennes I, F-35042, Rennes Cédex

Authors and Affiliations

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

    Michael F. Singer & Felix Ulmer

Authors
  1. Michael F. Singer
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  2. Felix Ulmer
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Additional information

A weaker version of these results were announced inLiouvillian Solutions of Third Order Linear Differential Equations; New Bounds and Necessary Conditions, Proceedings of the 1992 International Symposium on Symbolic and Algebraic Computation, ACM Press

Partially supported by NSF Grant 90-24624

Partially supported by Deutsche Forschungsgemeinschaft, while on leave from Universität Karlsruhe. This paper was written during the two year visit of the second author at North Carolina State University. The second author would like to thank North Carolina State University for its hospitality and support during the preparation of this paper

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Singer, M.F., Ulmer, F. Necessary conditions for liouvillian solutions of (third order) linear differential equations. AAECC 6, 1–22 (1995). https://doi.org/10.1007/BF01270928

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  • DOI: https://doi.org/10.1007/BF01270928

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