Abstract
Given a linear differential equation with known finite differential Galois group, we discuss methods to construct the minimal polynomial of a solution. We first outline a well known general method involving a basis transformation of the basis of formal solutions at a singular point. In the second part we construct directly the minimal polynomial of an eigenvector of the monodromy matrix at a singular point. The method is very efficient for irreducible second and third order linear differential equations where a one dimensional eigenspace of some monodromy matrix always exists.
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References
Conway, J.H., Hulpke, A., McKay, J.: On transitive permutation groups. LMS J. Comput. Math. 1, 1–8 (1998)
Cormier, O., Singer, M.F., Trager, B.M., Ulmer, F.: Linear differential operators for polynomial equations. J. Symb. Comp. 34, 355–398 (2002)
Fakler, W.: Algebraische Algorithmen zur Lösung von linearen Differentialgleichungen. Teubner Verlag, Stuttgart, Leipzig, 1999. Dissertation 1998, Universität Karlsruhe, Reihe MuPAD Reports.
Kovacic, J.: An algorithm for solving second order linear homogeneous differential equations. J. Symb. Comp. 2, 3–43 (1986)
Salvy, B., Zimmermann, P.: Gfun: a maple package for the manipulation of generating and holonomic functions in one variable. ACM Trans. Math. Software, 20, 163–177 (1994)
Singer, M.F.: Algebraic solutions of n-th order linear differential equations. Queens Papers in Pure and Applied Mathematics, 54, 379–420 (1980). Proceedings of the 1979 Queens Conference on Number Theory.
Singer, M.F., Ulmer, F.: Liouvillian and algebraic solutions of second and third order linear differential equations. J. Symb. Comp. 16, 37–73 (1993)
Singer, M.F., Ulmer, F.: Necessary conditions for liouvillian solutions of (third order) linear differential equations. J. of Appl. Alg. Eng. Comm. Comp. 6, 1–22 (1995)
Ulmer, F.: Liouvillian solutions of third order differential equation. J. Symb. Comp. 36, 855–889 (2003)
Ulmer, F., Weil, J.A.: Note on kovacic's algorithm. J. Symb. Comp. 22, 179–200 (1996)
van der Put, M., Singer, M.F.: Galois Theory of Linear Differential equations. Springer, Berlin, Heidelberg, Newyork. (2003)
van der Put, M., Ulmer, F.: Differential equations and finite groups. J. Alg. 226, 920–966 (2000)
van Hoeij, M.: The minimum polynomial of an algebraic solution of abel's problem. preprint Florida State University, 02 (2000)
van Hoeij, M., Ragot, J.F., Ulmer, F., Weil, J.A.: Liouvillian solutions of linear differential equations of order three and higher. J. Symb. Comp. 28, 589–609 (1999)
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Ulmer, F. Note on algebraic solutions of differential equations with known finite Galois group. AAECC 16, 205–218 (2005). https://doi.org/10.1007/s00200-005-0177-9
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DOI: https://doi.org/10.1007/s00200-005-0177-9