A generic Picard-Vessiot extension for GL(n) and the inverse differential Galois problem.
| dc.contributor.advisor | Magid, Andy, | en_US |
| dc.contributor.author | Juan, Lourdes. | en_US |
| dc.date.accessioned | 2013-08-16T12:30:52Z | |
| dc.date.available | 2013-08-16T12:30:52Z | |
| dc.date.issued | 2000 | en_US |
| dc.description.abstract | Let F be a differential field with field of constants the algebraically dosed field C. Let Yij be differential indeterminates and let R = F{ Yij}[X11, ..., Xnn] be a differential ring with derivation D (Xij) = k=1n YikXkj. We show that the quotient field Q = F⟨ Yij⟩ ( Xij) of R is a Picard-Vessiot extension of F(Yij) for GL n(C). Moreover, we show that Q is a generic extension in the following sense: If E ⊃ F is a Picard-Vessiot extension with differential Galois group G = GLn( C) then E is isomorphic to F( Xij) as a G-module and as an F-module. Under this isomorphism DE, the derivation of E, goes to a G-equivariant derivation which has a form similar to D with the coefficients Yij specialized to elements fij from F. The differential subfield C⟨ fij⟩ (Xij) ⊂ F(Xij) is shown to be a Picard-Vessiot extension of C⟨ fij⟩ with group GLn(C). From this, one can retrieve the Picard-Vessiot extension F( Xij) ⊃ F by extension of scalars from C⟨ fij⟩ to F. | en_US |
| dc.description.abstract | Conversely, if given F we specialize the Y ij to fij in F so that the corresponding extension C⟨ f ij⟩ (X) ⊃ C⟨ fij⟩ has no new constants we obtain a solution to the inverse differential Galois problem for GLn(C). In the second part of this dissertation we show necessary and sufficient conditions for such a specialization to exist. | en_US |
| dc.format.extent | vii, 67 leaves ; | en_US |
| dc.identifier.uri | http://hdl.handle.net/11244/5942 | |
| dc.note | Major Professor: Andy Magid. | en_US |
| dc.note | Source: Dissertation Abstracts International, Volume: 61-02, Section: B, page: 0884. | en_US |
| dc.subject | Inverse Galois theory. | en_US |
| dc.subject | Differential algebra. | en_US |
| dc.subject | Mathematics. | en_US |
| dc.subject | Differential-algebraic equations. | en_US |
| dc.thesis.degree | Ph.D. | en_US |
| dc.thesis.degreeDiscipline | Department of Mathematics | en_US |
| dc.title | A generic Picard-Vessiot extension for GL(n) and the inverse differential Galois problem. | en_US |
| dc.type | Thesis | en_US |
| ou.group | College of Arts and Sciences::Department of Mathematics | |
| ou.identifier | (UMI)AAI9962971 | en_US |
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