Computing the Lie Algebra of the Differential Galois Group of a Linear Differential System

Barkatou, Moulay A.; Cluzeau, Thomas; Weil, Jacques-Arthur; Vizio, Lucia Di

doi:10.1145/2930889.2930932

Cited by 14 publications

(20 citation statements)

References 17 publications

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“…En m'appuyant sur l'analyse de l'oeuvre scientifique de Paul Painlevé par Jacques Hadamard 5 [57] (cf. aussi [94]) je vais décrire l'état de la recherche sur les équations différentielles vers 1880 et la découverte par P. Painlevé des équations qui portent son nom 6 .…”

Section: La Naissance Des éQuations De Painlevé Leur Redécouverte à Strasbourg Vers 1970 Et Les Débuts De La Collaboration Franco-japonaiunclassified

Hiroshi Umemura et les mathématiques françaises

Ramis¹

2021

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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Section: La Naissance Des éQuations De Painlevé Leur Redécouverte à Strasbourg Vers 1970 Et Les Débuts De La Collaboration Franco-japonaiunclassified

Hiroshi Umemura et les mathématiques françaises

Ramis¹

2021

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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“…The spirit of this result and of our proof appears in the proof of [3,Theorem 3] in a particular case. It is extended here to all constructions and this is useful for reduced form algorithms; see [4,Lemma 5.1], where this result was alluded to, without a full proof for lack of space. Theorem 3.9.…”

Section: Gauge Transformation To a Reduced Form With Local Conditionsmentioning

confidence: 99%

“…The algorithm in [4]. In the latter reference, we showed how one can compute the Galois-Lie algebra of an (absolutely) irreducible linear differential system and hence (a good part of) its differential Galois group.…”

Section: Introductionmentioning

confidence: 99%

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Reduced Forms of Linear Differential Systems and the Intrinsic Galois-Lie Algebra of Katz

Barkatou

Cluzeau²,

Vizio

et al. 2020

SIGMA

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Generalizing the main result of [Aparicio-Monforte A., Compoint E., Weil J.-A., J. Pure Appl. Algebra 217 (2013), 1504-1516], we prove that a linear differential system is in reduced form in the sense of Kolchin and Kovacic if and only if any differential module in an algebraic construction admits a constant basis. Then we derive an explicit version of this statement. We finally deduce some properties of the Lie algebra of Katz's intrinsic Galois group.

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“…Under the assumptions of the theorem, this type of solvability is equivalent to solvability by exponentials of integrals.2 Under the assumptions of the theorem, this type of solvability is equivalent to solvability by integrals and radicals 3. Under the assumptions of the theorem, this type of solvability is equivalent to solvability by radicals.…”

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confidence: 99%

Linear Differential Systems with Small Coefficients: Various Types of Solvability and their Verification

Barkatou

Gontsov²

2019

SIGMA

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We study the problem of solvability of linear differential systems with small coefficients in the Liouvillian sense (or, by generalized quadratures). For a general system, this problem is equivalent to that of solvability of the Lie algebra of the differential Galois group of the system. However, dependence of this Lie algebra on the system coefficients remains unknown. We show that for the particular class of systems with non-resonant irregular singular points that have sufficiently small coefficient matrices, the problem is reduced to that of solvability of the explicit Lie algebra generated by the coefficient matrices. This extends the corresponding Ilyashenko-Khovanskii theorem obtained for linear differential systems with Fuchsian singular points. We also give some examples illustrating the practical verification of the presented criteria of solvability by using general procedures implemented in Maple.

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Computing the Lie Algebra of the Differential Galois Group of a Linear Differential System

Cited by 14 publications

References 17 publications

Hiroshi Umemura et les mathématiques françaises

Hiroshi Umemura et les mathématiques françaises

Reduced Forms of Linear Differential Systems and the Intrinsic Galois-Lie Algebra of Katz

Linear Differential Systems with Small Coefficients: Various Types of Solvability and their Verification

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