Picard-Vessiot Extensions of Real Differential Fields
For a linear differential equation defined over a formally real differential field K with real closed field of constants k, Crespo, Hajto, and van der Put proved that there exists a unique formally real Picard-Vessiot extension up to K-differential automorphism. However, such an equation may have Pi...
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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/123456789/210288 |
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| Cite this: | Picard-Vessiot Extensions of Real Differential Fields / T. Crespo, Z. Hajto // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 27 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
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Crespo, T. Hajto, Z. 2025-12-05T09:21:52Z 2019 Picard-Vessiot Extensions of Real Differential Fields / T. Crespo, Z. Hajto // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 27 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 12H05; 13B05; 14P05; 12D15 arXiv: 1403.3226 https://nasplib.isofts.kiev.ua/handle/123456789/210288 https://doi.org/10.3842/SIGMA.2019.100 For a linear differential equation defined over a formally real differential field K with real closed field of constants k, Crespo, Hajto, and van der Put proved that there exists a unique formally real Picard-Vessiot extension up to K-differential automorphism. However, such an equation may have Picard-Vessiot extensions that are not formally real fields. The differential Galois group of a Picard-Vessiot extension for this equation has the structure of a linear algebraic group defined over k and is a k-form of the differential Galois group H of the equation over the differential field K(√-1). These facts lead us to consider two issues: determining the number of K-differential isomorphism classes of Picard-Vessiot extensions and describing the variation of the differential Galois group in the set of k-forms of H. We address these two issues in the cases when H is a special linear, a special orthogonal, or a symplectic linear algebraic group and conclude that there is no general behaviour. Both authors acknowledge support of grant MTM2015-66716-P (MINECO/FEDER, UE). The authors thank the anonymous referees for their valuable remarks and suggestions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Picard-Vessiot Extensions of Real Differential Fields Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Picard-Vessiot Extensions of Real Differential Fields |
| spellingShingle |
Picard-Vessiot Extensions of Real Differential Fields Crespo, T. Hajto, Z. |
| title_short |
Picard-Vessiot Extensions of Real Differential Fields |
| title_full |
Picard-Vessiot Extensions of Real Differential Fields |
| title_fullStr |
Picard-Vessiot Extensions of Real Differential Fields |
| title_full_unstemmed |
Picard-Vessiot Extensions of Real Differential Fields |
| title_sort |
picard-vessiot extensions of real differential fields |
| author |
Crespo, T. Hajto, Z. |
| author_facet |
Crespo, T. Hajto, Z. |
| publishDate |
2019 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
For a linear differential equation defined over a formally real differential field K with real closed field of constants k, Crespo, Hajto, and van der Put proved that there exists a unique formally real Picard-Vessiot extension up to K-differential automorphism. However, such an equation may have Picard-Vessiot extensions that are not formally real fields. The differential Galois group of a Picard-Vessiot extension for this equation has the structure of a linear algebraic group defined over k and is a k-form of the differential Galois group H of the equation over the differential field K(√-1). These facts lead us to consider two issues: determining the number of K-differential isomorphism classes of Picard-Vessiot extensions and describing the variation of the differential Galois group in the set of k-forms of H. We address these two issues in the cases when H is a special linear, a special orthogonal, or a symplectic linear algebraic group and conclude that there is no general behaviour.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/210288 |
| citation_txt |
Picard-Vessiot Extensions of Real Differential Fields / T. Crespo, Z. Hajto // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 27 назв. — англ. |
| work_keys_str_mv |
AT crespot picardvessiotextensionsofrealdifferentialfields AT hajtoz picardvessiotextensionsofrealdifferentialfields |
| first_indexed |
2025-12-07T21:25:02Z |
| last_indexed |
2025-12-07T21:25:02Z |
| _version_ |
1850886274574974976 |