4 research outputs found

    Computing the differential Galois group of a parameterized second-order linear differential equation

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    We develop algorithms to compute the differential Galois group GG associated to a parameterized second-order homogeneous linear differential equation of the form βˆ‚2βˆ‚x2Y+r1βˆ‚βˆ‚xY+r0Y=0, \tfrac{\partial^2}{\partial x^2} Y + r_1 \tfrac{\partial}{\partial x} Y + r_0 Y = 0, where the coefficients r1,r0∈F(x)r_1, r_0 \in F(x) are rational functions in xx with coefficients in a partial differential field FF of characteristic zero. Our work relies on the procedure developed by Dreyfus to compute GG under the assumption that r1=0r_1 = 0. We show how to complete this procedure to cover the cases where r1β‰ 0r_1 \neq 0, by reinterpreting a classical change of variables procedure in Galois-theoretic terms.Comment: 14 page

    Computation of the unipotent radical of the differential Galois group for a parameterized second-order linear differential equation

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    We propose a new method to compute the unipotent radical Ru(H)R_u(H) of the differential Galois group HH associated to a parameterized second-order homogeneous linear differential equation of the form βˆ‚2βˆ‚x2Yβˆ’qY=0,\tfrac{\partial^2}{\partial x^2}Y-qY=0, where q∈F(x)q \in F(x) is a rational function in xx with coefficients in a Ξ \Pi-field FF of characteristic zero, and Ξ \Pi is a commuting set of parametric derivations. The procedure developed by Dreyfus reduces the computation of Ru(H)R_u(H) to solving a creative telescoping problem, whose effective solution requires the assumption that the maximal reductive quotient H/Ru(H)H / R_u(H) is a Ξ \Pi-constant linear differential algebraic group. When this condition is not satisfied, we compute a new set of parametric derivations Ξ β€²\Pi' such that the associated differential Galois group Hβ€²H' has the property that Hβ€²/Ru(Hβ€²)H'/ R_u(H') is Ξ β€²\Pi'-constant, and such that Ru(H)R_u(H) is defined by the same differential equations as Ru(Hβ€²)R_u(H'). Thus the computation of Ru(H)R_u(H) is reduced to the effective computation of Ru(Hβ€²)R_u(H'). We expect that an elaboration of this method will be successful in extending the applicability of some recent algorithms developed by Minchenko, Ovchinnikov, and Singer to compute unipotent radicals for higher order equations.Comment: 12 page
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