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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On Generic differential $ \operatorname{SO}_n$-extensions


Authors: Lourdes Juan and Arne Ledet
Journal: Proc. Amer. Math. Soc. 136 (2008), 1145-1153
MSC (2000): Primary 12H05; Secondary 12F12, 20G15
Published electronically: December 27, 2007
MathSciNet review: 2367088
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Abstract: Let $ \mathcal C$ be an algebraically closed field with trivial derivation and let $ \mathcal F$ denote the differential rational field $ \mathcal C\langle Y_{ij}\rangle$, with $ Y_{ij}$, $ 1\leq i\leq n-1$, $ 1\leq j\leq n$, $ i\leq j$, differentially independent indeterminates over $ \mathcal C$. We show that there is a Picard-Vessiot extension $ \mathcal E\supset \mathcal F$ for a matrix equation $ X'=X\mathcal A(Y_{ij})$, with differential Galois group $ \operatorname{SO}_n$, with the property that if $ F$ is any differential field with field of constants $ \mathcal C$, then there is a Picard-Vessiot extension $ E\supset F$ with differential Galois group $ H\leq\operatorname{SO}_n$ if and only if there are $ f_{ij}\in F$ with $ \mathcal A(f_{ij})$ well defined and the equation $ X'=X\mathcal A(f_{ij})$ giving rise to the extension $ E\supset F$.


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Additional Information

Lourdes Juan
Affiliation: Department of Mathematics, Texas Tech University, MS 1042, Lubbock, Texas 79409
Email: lourdes.juan@ttu.edu

Arne Ledet
Affiliation: Department of Mathematics, Texas Tech University, MS 1042, Lubbock, Texas 79409
Email: arne.ledet@ttu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-07-09314-8
PII: S 0002-9939(07)09314-8
Received by editor(s): July 5, 2006
Published electronically: December 27, 2007
Communicated by: Martin Lorenz
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.