Real Liouvillian Extensions of Partial Differential Fields
In this paper, we establish Galois theory for partial differential systems defined over formally real differential fields with a real closed field of constants and over formally -adic differential fields with a -adically closed field of constants. For an integrable partial differential system define...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2021 |
| Автори: | , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2021
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/211432 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Real Liouvillian Extensions of Partial Differential Fields. Teresa Crespo, Zbigniew Hajto and Rouzbeh Mohseni. SIGMA 17 (2021), 095, 16 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862646184000094208 |
|---|---|
| author | Crespo, Teresa Hajto, Zbigniew Mohseni, Rouzbeh |
| author_facet | Crespo, Teresa Hajto, Zbigniew Mohseni, Rouzbeh |
| citation_txt | Real Liouvillian Extensions of Partial Differential Fields. Teresa Crespo, Zbigniew Hajto and Rouzbeh Mohseni. SIGMA 17 (2021), 095, 16 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | In this paper, we establish Galois theory for partial differential systems defined over formally real differential fields with a real closed field of constants and over formally -adic differential fields with a -adically closed field of constants. For an integrable partial differential system defined over such a field, we prove that there exists a formally real (resp. formally -adic) Picard-Vessiot extension. Moreover, we obtain a uniqueness result for this Picard-Vessiot extension. We give an adequate definition of the Galois differential group and obtain a Galois fundamental theorem in this setting. We apply the obtained Galois correspondence to characterise formally real Liouvillian extensions of real partial differential fields with a real closed field of constants by means of split solvable linear algebraic groups. We present some examples of real dynamical systems and indicate some possibilities for further development of algebraic methods in real dynamical systems.
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| first_indexed | 2026-03-15T10:41:54Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211432 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-15T10:41:54Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Crespo, Teresa Hajto, Zbigniew Mohseni, Rouzbeh 2026-01-02T08:31:49Z 2021 Real Liouvillian Extensions of Partial Differential Fields. Teresa Crespo, Zbigniew Hajto and Rouzbeh Mohseni. SIGMA 17 (2021), 095, 16 pages 1815-0659 2020 Mathematics Subject Classification: 12H05; 37J35; 12D15; 14P05 arXiv:2104.09548 https://nasplib.isofts.kiev.ua/handle/123456789/211432 https://doi.org/10.3842/SIGMA.2021.095 In this paper, we establish Galois theory for partial differential systems defined over formally real differential fields with a real closed field of constants and over formally -adic differential fields with a -adically closed field of constants. For an integrable partial differential system defined over such a field, we prove that there exists a formally real (resp. formally -adic) Picard-Vessiot extension. Moreover, we obtain a uniqueness result for this Picard-Vessiot extension. We give an adequate definition of the Galois differential group and obtain a Galois fundamental theorem in this setting. We apply the obtained Galois correspondence to characterise formally real Liouvillian extensions of real partial differential fields with a real closed field of constants by means of split solvable linear algebraic groups. We present some examples of real dynamical systems and indicate some possibilities for further development of algebraic methods in real dynamical systems. We are very thankful to the anonymous referees for their valuable comments, which helped us to improve significantly the presentation of our results. R. Mohseni acknowledges the support of the Polish Ministry of Science and Higher Education. T. Crespo and Z. Hajto acknowledge support of grant PID2019-107297GB-I00 (MICINN). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Real Liouvillian Extensions of Partial Differential Fields Article published earlier |
| spellingShingle | Real Liouvillian Extensions of Partial Differential Fields Crespo, Teresa Hajto, Zbigniew Mohseni, Rouzbeh |
| title | Real Liouvillian Extensions of Partial Differential Fields |
| title_full | Real Liouvillian Extensions of Partial Differential Fields |
| title_fullStr | Real Liouvillian Extensions of Partial Differential Fields |
| title_full_unstemmed | Real Liouvillian Extensions of Partial Differential Fields |
| title_short | Real Liouvillian Extensions of Partial Differential Fields |
| title_sort | real liouvillian extensions of partial differential fields |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211432 |
| work_keys_str_mv | AT crespoteresa realliouvillianextensionsofpartialdifferentialfields AT hajtozbigniew realliouvillianextensionsofpartialdifferentialfields AT mohsenirouzbeh realliouvillianextensionsofpartialdifferentialfields |