614 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 ∂2∂x2Y+r1∂∂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 r1≠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 ∂2∂x2Y−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 Π\Pi-field FF of characteristic zero, and Π\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 Π\Pi-constant linear differential algebraic group. When this condition is not satisfied, we compute a new set of parametric derivations Π′\Pi' such that the associated differential Galois group H′H' has the property that H′/Ru(H′)H'/ R_u(H') is Π′\Pi'-constant, and such that Ru(H)R_u(H) is defined by the same differential equations as Ru(H′)R_u(H'). Thus the computation of Ru(H)R_u(H) is reduced to the effective computation of Ru(H′)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

    A density theorem for parameterized differential Galois theory

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    We study parameterized linear differential equations with coefficients depending meromorphically upon the parameters. As a main result, analogously to the unparameterized density theorem of Ramis, we show that the parameterized monodromy, the parameterized exponential torus and the parameterized Stokes operators are topological generators in Kolchin topology, for the parameterized differential Galois group introduced by Cassidy and Singer. We prove an analogous result for the global parameterized differential Galois group, which generalizes a result by Mitschi and Singer. These authors give also a necessary condition on a group for being a global parameterized differential Galois group; as a corollary of the density theorem, we prove that their condition is also sufficient. As an application, we give a characterization of completely integrable equations, and we give a partial answer to a question of Sibuya about the transcendence properties of a given Stokes matrix. Moreover, using a parameterized Hukuhara-Turrittin theorem, we show that the Galois group descends to a smaller field, whose field of constants is not differentially closed.Comment: To appear in Pacific Journal of Mathematic

    Parameterized generic Galois groups for q-difference equations, followed by the appendix "The Galois D-groupoid of a q-difference system" by Anne Granier

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    We introduce the parameterized generic Galois group of a q-difference module, that is a differential group in the sense of Kolchin. It is associated to the smallest differential tannakian category generated by the q-difference module, equipped with the forgetful functor. Our previous results on the Grothendieck conjecture for q-difference equations lead to an adelic description of the parameterized generic Galois group, in the spirit of the Grothendieck-Katz's conjecture on p-curvatures. Using this description, we show that the Malgrange-Granier D-groupoid of a nonlinear q-difference system coincides, in the linear case, with the parameterized generic Galois group introduced here. The paper is followed by an appendix by A. Granier, that provides a quick introduction to the D-groupoid of a non-linear q-difference equation.Comment: The content of this paper was previously included in arXiv:1002.483

    Galois Theory of Parameterized Differential Equations and Linear Differential Algebraic Groups

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    We present a Galois theory of parameterized linear differential equations where the Galois groups are linear differential algebraic groups, that is, groups of matrices whose entries are functions of the parameters and satisfy a set of differential equations with respect to these parameters. We present the basic constructions and results, give examples, discuss how isomonodromic families fit into this theory and show how results from the theory of linear differential algebraic groups may be used to classify systems of second order linear differential equations

    Tannakian categories, linear differential algebraic groups, and parameterized linear differential equations

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    We provide conditions for a category with a fiber functor to be equivalent to the category of representations of a linear differential algebraic group. This generalizes the notion of a neutral Tannakian category used to characterize the category of representations of a linear algebraic group.Comment: 26 pages; corrected misprints; simplified Definition 2; more references adde
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