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

DOI:
https://doi.org/10.1090/S0002-9939-04-07390-3

Published electronically:
April 8, 2004

MathSciNet review:
2054779

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Abstract | References | Similar Articles | Additional Information

Abstract: Given a connected linear algebraic group over an algebraically closed field of characteristic 0, we construct a pure Picard-Vessiot extension for , namely, a Picard-Vessiot extension , with differential Galois group , such that and are purely differentially transcendental over . The differential field is the quotient field of a -stable proper differential subring with the property that if is any differential field with field of constants and is a Picard-Vessiot extension with differential Galois group a connected subgroup of , then there is a differential homomorphism such that is generated over as a differential field by .

**1.**A. K. Bhandari and N. Sankaran,*Generic differential equations and Picard-Vessiot extensions*, Rend. Sem. Mat. Univ. Politec. Torino**52**, 4 (1994), 353-358. MR**96f:12007****2.**A. Borel,*Linear Algebraic Groups*, second enlarged edition, Graduate Texts in Mathematics, no. 126, Springer-Verlag, New York, 1991. MR**92d:20001****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*, 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*, Kluwer Academic Publishers, Dordrecht, 1999. MR**2001c:12006****6.**A. Magid,*Lectures on differential Galois theory*, University Lecture Series, vol. 7, American Mathematical Society, Providence, RI, 1994. MR**95j:12008****7.**E. Noether,*Gleichungen mit vorgeschriebener Gruppen*, Math. Ann.**78**(1918), 221-229.**8.**M. Van de Put and M. Singer,*Galois theory of linear differential equations*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 328, Springer-Verlag, Berlin, 2003. MR**2004c:12010****9.**T. A. Springer,*Linear Algebraic Groups*, second edition, Progress in Mathematics, vol. 9, Birkhäuser Boston, Boston, MA, 1998. MR**99h:20075**

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Additional Information

**Lourdes Juan**

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

Email:
ljuan@math.ttu.edu

DOI:
https://doi.org/10.1090/S0002-9939-04-07390-3

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