Isomonodromic differential equations and differential categories

Gorchinskiy, Sergey; Ovchinnikov, Alexey

doi:10.1016/j.matpur.2013.11.001

Cited by 21 publications

(35 citation statements)

References 46 publications

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“…0.6] gives an algebraic recipe to produce a commuting basis Π ′ for L from the (possibly non-commuting) basis Π ′′ . This recipe was generalized and applied towards an algorithm to decide isomonodromy in [13]…”

Section: Resultsmentioning

confidence: 99%

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

Arreche

2014

Advances in Applied Mathematics

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Abstract. We propose a new method to compute the unipotent radical Ru(H) of the differential Galois group H associated to a parameterized second-order homogeneous linear differential equation of the formis a rational function in x with coefficients in a Π-field F of characteristic zero, and Π is a commuting set of parametric derivations. The procedure developed by Dreyfus reduces the computation of Ru(H) to solving a creative telescoping problem, whose effective solution requires the assumption that the maximal reductive quotient H/Ru(H) is a Π-constant linear differential algebraic group. When this condition is not satisfied, we compute a new set of parametric derivations Π ′ such that the associated differential Galois group H ′ has the property that H ′ /Ru(H ′ ) is Π ′ -constant, and such that Ru(H) is defined by the same differential equations as Ru(H ′ ). Thus the computation of Ru(H) is reduced to the effective computation of Ru(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.

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Section: Resultsmentioning

confidence: 99%

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

Arreche

2014

Advances in Applied Mathematics

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“…In case IV, there is a finite subset Π of the F -span of Π consisting of F -linearly independent, pairwise commuting derivations such that H is conjugate to the simple group SL2(F Π ); see [4,6,8]. As the only abelian quotient of H in this case is Λ = {1}, we obtain G from Corollary 3.3.…”

Section: Remark 42mentioning

confidence: 95%

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

Arreche

2014

Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation

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We develop algorithms to compute the differential Galois group G associated to a parameterized second-order homogeneous linear differential equation of the formx with coefficients in a partial differential field F of characteristic zero. This work relies on earlier procedures developed by Dreyfus and by the present author to compute G when r1 = 0. By reinterpreting a classical change-ofvariables procedure in Galois-theoretic terms, we complete these algorithms to compute G with no restrictions on r1.

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“…See Theorem A.5.2.3 in [Sib90]. Integrability has been studied from a Galoisian point of view in [CS06], [HS08] and [GO12] for equations depending on differential parameters (see also [MS12] and [Dre12]). Here we consider the dependence on a difference parameter.…”

Section: Discrete Integrabilitymentioning

confidence: 99%

Difference Algebraic Relations Among Solutions of Linear Differential Equations

Vizio¹,

Hardouin²,

Wibmer³

2015

J. Inst. Math. Jussieu

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We extend and apply the Galois theory of linear differential equations equipped with the action of an endomorphism. The Galois groups in this Galois theory are difference algebraic groups and we use structure theorems for these groups to characterize the possible difference algebraic relations among solutions of linear differential equations. This yields tools to show that certain special functions are difference transcendent. One of our main results is a characterization of discrete integrability of linear differential equations with almost simple usual Galois group, based on a structure theorem for the Zariski dense difference algebraic subgroups of almost simple algebraic groups, which is a schematic version, in characteristic zero, of a result due to Z. Chatzidakis, E. Hrushovski and Y. Peterzil.

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Isomonodromic differential equations and differential categories

Cited by 21 publications

References 46 publications

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

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

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

Difference Algebraic Relations Among Solutions of Linear Differential Equations

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