The inverse problem of differential Galois theory over the field R(z)
We describe a Picard-Vessiot theory for differential fields with non algebraically closed fields of constants. As a technique for constructing and classifying Picard-Vessiot extensions, we develop a Galois descent theory. We utilize this theory to prove that every linear algebraic group $G$ over $\m...
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| Format: | Conference item |
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2008
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| _version_ | 1826256986673512448 |
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| author | Dyckerhoff, T |
| author_facet | Dyckerhoff, T |
| author_sort | Dyckerhoff, T |
| collection | OXFORD |
| description | We describe a Picard-Vessiot theory for differential fields with non algebraically closed fields of constants. As a technique for constructing and classifying Picard-Vessiot extensions, we develop a Galois descent theory. We utilize this theory to prove that every linear algebraic group $G$ over $\mathbb{R}$ occurs as a differential Galois group over $\mathbb{R}(z)$. The main ingredient of the proof is the Riemann-Hilbert correspondence for regular singular differential equations over $\mathbb{C}(z)$. |
| first_indexed | 2024-03-06T18:10:59Z |
| format | Conference item |
| id | oxford-uuid:0302e011-7c1f-45cc-bb3b-4eedea81b0d5 |
| institution | University of Oxford |
| last_indexed | 2024-03-06T18:10:59Z |
| publishDate | 2008 |
| record_format | dspace |
| spelling | oxford-uuid:0302e011-7c1f-45cc-bb3b-4eedea81b0d52022-03-26T08:43:51ZThe inverse problem of differential Galois theory over the field R(z)Conference itemhttp://purl.org/coar/resource_type/c_5794uuid:0302e011-7c1f-45cc-bb3b-4eedea81b0d5Symplectic Elements at Oxford2008Dyckerhoff, TWe describe a Picard-Vessiot theory for differential fields with non algebraically closed fields of constants. As a technique for constructing and classifying Picard-Vessiot extensions, we develop a Galois descent theory. We utilize this theory to prove that every linear algebraic group $G$ over $\mathbb{R}$ occurs as a differential Galois group over $\mathbb{R}(z)$. The main ingredient of the proof is the Riemann-Hilbert correspondence for regular singular differential equations over $\mathbb{C}(z)$. |
| spellingShingle | Dyckerhoff, T The inverse problem of differential Galois theory over the field R(z) |
| title | The inverse problem of differential Galois theory over the field R(z) |
| title_full | The inverse problem of differential Galois theory over the field R(z) |
| title_fullStr | The inverse problem of differential Galois theory over the field R(z) |
| title_full_unstemmed | The inverse problem of differential Galois theory over the field R(z) |
| title_short | The inverse problem of differential Galois theory over the field R(z) |
| title_sort | inverse problem of differential galois theory over the field r z |
| work_keys_str_mv | AT dyckerhofft theinverseproblemofdifferentialgaloistheoryoverthefieldrz AT dyckerhofft inverseproblemofdifferentialgaloistheoryoverthefieldrz |