Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Pure Picard-Vessiot extensions with generic properties

Author: Lourdes Juan
Journal: Proc. Amer. Math. Soc. 132 (2004), 2549-2556
MSC (2000): Primary 12H05; Secondary 12F12, 20G15
Published electronically: April 8, 2004
MathSciNet review: 2054779
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Given a connected linear algebraic group $G$ over an algebraically closed field $C$ of characteristic 0, we construct a pure Picard-Vessiot extension for $G$, namely, a Picard-Vessiot extension $\mathcal E\supset \mathcal F$, with differential Galois group $G$, such that $\mathcal E$ and $\mathcal F$ are purely differentially transcendental over $C$. The differential field $\mathcal E$ is the quotient field of a $G$-stable proper differential subring $\mathcal R$ with the property that if $F$ is any differential field with field of constants $C$ and $E\supset F$ is a Picard-Vessiot extension with differential Galois group a connected subgroup $H$ of $G$, then there is a differential homomorphism $\phi:\mathcal R\rightarrow E$ such that $E$ is generated over $F$ as a differential field by $\phi(\mathcal R)$.

References [Enhancements On Off] (What's this?)

  • 1. A. K. Bhandari and N. Sankaran, Generic differential equations and Picard-Vessiot extensions, Rend. Sem. Mat. Univ. Politec. Torino 52 (1994), no. 4, 353–358. MR 1345605
  • 2. Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012
  • 3. L. Goldman, Specialization and Picard-Vessiot theory, Trans. Amer. Math. Soc. 85 (1957), 327-356.MR 19:384b
  • 4. L. Juan, Principal differential ideals and a generic inverse differential Galois problem for GL$_n$, Comm. Algebra 30, 12 (2002), 6071-6103.
  • 5. M. V. Kondratieva, A. B. Levin, A. V. Mikhalev, and E. V. Pankratiev, Differential and difference dimension polynomials, Mathematics and its Applications, vol. 461, Kluwer Academic Publishers, Dordrecht, 1999. MR 1676955
  • 6. Andy R. Magid, Lectures on differential Galois theory, University Lecture Series, vol. 7, American Mathematical Society, Providence, RI, 1994. MR 1301076
  • 7. E. Noether, Gleichungen mit vorgeschriebener Gruppen, Math. Ann. 78 (1918), 221-229.
  • 8. Marius van der Put and Michael F. Singer, Galois theory of linear differential equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 328, Springer-Verlag, Berlin, 2003. MR 1960772
  • 9. T. A. Springer, Linear algebraic groups, 2nd ed., Progress in Mathematics, vol. 9, Birkhäuser Boston, Inc., Boston, MA, 1998. MR 1642713

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 12H05, 12F12, 20G15

Retrieve articles in all journals with MSC (2000): 12H05, 12F12, 20G15

Additional Information

Lourdes Juan
Affiliation: Department of Mathematics and Statistics, Texas Tech University, Box 41042, Lubbock, Texas 79409-1042

Received by editor(s): August 26, 2002
Received by editor(s) in revised form: June 2, 2003
Published electronically: April 8, 2004
Additional Notes: The author was supported in part by NSA grant No. MDA904-02-1-0084
Communicated by: Lance W. Small
Article copyright: © Copyright 2004 American Mathematical Society