Linear Differential Systems with Small Coefficients: Various Types of Solvability and their Verification
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 the solvability of the Lie algebra of the differential Galois group of the system. However...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2019 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2019
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/210237 |
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| Cite this: | Linear Differential Systems with Small Coefficients: Various Types of Solvability and their Verification / M.A. Barkatou, R.R. Gontsov // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 23 назв. — англ. |
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Barkatou, M.A. Gontsov, R.R. 2025-12-04T13:07:22Z 2019 Linear Differential Systems with Small Coefficients: Various Types of Solvability and their Verification / M.A. Barkatou, R.R. Gontsov // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 23 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 34M03; 34M25; 34M35; 34M50 arXiv: 1901.09951 https://nasplib.isofts.kiev.ua/handle/123456789/210237 https://doi.org/10.3842/SIGMA.2019.058 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 the solvability of the Lie algebra of the differential Galois group of the system. However, the 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. The authors are grateful to Thomas Cluzeau for helpful discussions about the problem of simultaneous triangularizability of a set of matrices. They also thank the referee for his/her nice suggestions, which have refined the text. The work of R.G. was partially supported by the Russian Foundation for Basic Research (projects 16-51-1500005 and 17-01-00515). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Linear Differential Systems with Small Coefficients: Various Types of Solvability and their Verification Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
Linear Differential Systems with Small Coefficients: Various Types of Solvability and their Verification |
| spellingShingle |
Linear Differential Systems with Small Coefficients: Various Types of Solvability and their Verification Barkatou, M.A. Gontsov, R.R. |
| title_short |
Linear Differential Systems with Small Coefficients: Various Types of Solvability and their Verification |
| title_full |
Linear Differential Systems with Small Coefficients: Various Types of Solvability and their Verification |
| title_fullStr |
Linear Differential Systems with Small Coefficients: Various Types of Solvability and their Verification |
| title_full_unstemmed |
Linear Differential Systems with Small Coefficients: Various Types of Solvability and their Verification |
| title_sort |
linear differential systems with small coefficients: various types of solvability and their verification |
| author |
Barkatou, M.A. Gontsov, R.R. |
| author_facet |
Barkatou, M.A. Gontsov, R.R. |
| publishDate |
2019 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
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Інститут математики НАН України |
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Article |
| description |
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 the solvability of the Lie algebra of the differential Galois group of the system. However, the 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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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/210237 |
| citation_txt |
Linear Differential Systems with Small Coefficients: Various Types of Solvability and their Verification / M.A. Barkatou, R.R. Gontsov // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 23 назв. — англ. |
| work_keys_str_mv |
AT barkatouma lineardifferentialsystemswithsmallcoefficientsvarioustypesofsolvabilityandtheirverification AT gontsovrr lineardifferentialsystemswithsmallcoefficientsvarioustypesofsolvabilityandtheirverification |
| first_indexed |
2025-12-07T21:24:52Z |
| last_indexed |
2025-12-07T21:24:52Z |
| _version_ |
1850886263575412736 |