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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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2019 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2019
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/210288 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | 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. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862732006501122048 |
|---|---|
| author | Crespo, T. Hajto, Z. |
| author_facet | Crespo, T. Hajto, Z. |
| 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. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| 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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| first_indexed | 2025-12-07T21:25:02Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-210288 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T21:25:02Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | 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 |
| spellingShingle | Picard-Vessiot Extensions of Real Differential Fields Crespo, T. Hajto, Z. |
| title | 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_short | Picard-Vessiot Extensions of Real Differential Fields |
| title_sort | picard-vessiot extensions of real differential fields |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210288 |
| work_keys_str_mv | AT crespot picardvessiotextensionsofrealdifferentialfields AT hajtoz picardvessiotextensionsofrealdifferentialfields |