On parameterized differential Galois extensions

Sánchez, Omar León; Nagloo, Joel

doi:10.1016/j.jpaa.2015.12.001

Cited by 6 publications

(2 citation statements)

References 14 publications

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“…Let K be a ∆ = {∂, δ}-field and k = K ∂ . We assume for simplicity that (k, δ) is a differentially closed field (this assumption was relaxed in [13,33,28]).…”

Section: Differential Modules and Their Galois Groupsmentioning

confidence: 99%

Calculating Galois groups of third-order linear differential equations with parameters

Minchenko

Ovchinnikov

2018

Commun. Contemp. Math.

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Motivated by developing algorithms that decide hypertranscendence of solutions of extensions of the Bessel differential equation, algorithms computing the unipotent radical of a parameterized differential Galois group have been recently developed. Extensions of Bessel's equation, such as the Lommel equation, can be viewed as homogeneous parameterized linear differential equations of the third order. In the present paper, we give the first known algorithm that calculates the differential Galois group of a third order parameterized linear differential equation.

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“…Let K be a ∆ = {∂, δ}-field and k = K ∂ . We assume for simplicity that (k, δ) is a differentially closed field (this assumption was relaxed in [13,33,28]).…”

Section: Differential Modules and Their Galois Groupsmentioning

confidence: 99%

Calculating Galois groups of third-order linear differential equations with parameters

Minchenko

Ovchinnikov

2018

Commun. Contemp. Math.

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“…This is not usual for the fields of functions that appear in the realm of algebraic geometry or complex analysis. It has been shown that, under some assumptions, this condition can be weakened by algebraic means as it is shown in [9,20,27].…”

Section: Introductionmentioning

confidence: 99%

Differential Galois Theory and Isomonodromic Deformations

Blázquez-Sanz¹,

Casale²,

Arboleda³

2019

SIGMA

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We present a geometric setting for the differential Galois theory of G-invariant connections with parameters. As an application of some classical results on differential algebraic groups and Lie algebra bundles, we see that the Galois group of a connection with parameters with simple structural group G is determined by its isomonodromic deformations. This allows us to compute the Galois groups with parameters of the general Fuchsian special linear system and of Gauss hypergeometric equation.

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Calculating differential Galois groups of parametrized differential equations, with applications to hypertranscendence

Hardouin¹,

Minchenko²,

Ovchinnikov³

2016

Math. Ann.

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The main motivation of our work is to create an efficient algorithm that decides hypertranscendence of solutions of linear differential equations, via the parameterized differential and Galois theories. To achieve this, we expand the representation theory of linear differential algebraic groups and develop new algorithms that calculate unipotent radicals of parameterized differential Galois groups for differential equations whose coefficients are rational functions. P. Berman and M.F. Singer presented an algorithm calculating the differential Galois group for differential equations without parameters whose differential operator is a composition of two completely reducible differential operators. We use their algorithm as a part of our algorithm. As a result, we find an effective criterion for the algebraic independence of the solutions of parameterized differential equations and all of their derivatives with respect to the parameter.

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On parameterized differential Galois extensions

Cited by 6 publications

References 14 publications

Calculating Galois groups of third-order linear differential equations with parameters

Calculating Galois groups of third-order linear differential equations with parameters

Differential Galois Theory and Isomonodromic Deformations

Calculating differential Galois groups of parametrized differential equations, with applications to hypertranscendence

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