Picard-Vessiot extensions for real fields

Sowa, Elżbieta

doi:10.1090/S0002-9939-2010-10700-1

Picard-Vessiot extensions for real fields
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by Elżbieta Sowa

Proc. Amer. Math. Soc. 139 (2011), 2407-2413

DOI: https://doi.org/10.1090/S0002-9939-2010-10700-1

Published electronically: December 9, 2010

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Abstract:

We define a notion of Picard-Vessiot extension for a homogeneous linear differential equation $\mathcal {L}=0$ defined over a real differential field $K$ with a real closed field of constants $C_{K}$. When $\mathcal {L}$ has differential Galois group $GL_{n}$ over the complexification of $K$, we prove that a Picard-Vessiot extension for $\mathcal {L}$ exists over $K$.

References

M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802
A. Buium, P. J. Cassidy, Differential algebraic geometry and differential algebraic groups: From algebraic differential equations to Diophantine geometry, in Selected works of Ellis Kolchin with commentary, H. Bass, A. Buium and P.J. Cassidy, eds., American Mathematical Society, Providence, RI, 1999, pp. 567-636.
Jacek Bochnak, Michel Coste, and Marie-Françoise Roy, Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 36, Springer-Verlag, Berlin, 1998. Translated from the 1987 French original; Revised by the authors. MR 1659509, DOI 10.1007/978-3-662-03718-8
T. Crespo, Z. Hajto, Introduction to Differential Galois Theory, Cracow University of Technology Publishers, 2007.
Marvin P. Epstein, An existence theorem in the algebraic study of homogeneous linear ordinary differential equations, Proc. Amer. Math. Soc. 6 (1955), 33–41. MR 67102, DOI 10.1090/S0002-9939-1955-0067102-6
Marvin P. Epstein, On the theory of Picard-Vessiot extensions, Ann. of Math. (2) 62 (1955), 528–547. MR 72868, DOI 10.2307/1970078
Irving Kaplansky, An introduction to differential algebra, Publ. Inst. Math. Univ. Nancago, No. 5, Hermann, Paris, 1957. MR 0093654
E. R. Kolchin, Differential algebra and algebraic groups, Pure and Applied Mathematics, Vol. 54, Academic Press, New York-London, 1973. MR 0568864
Andy R. Magid, Lectures on differential Galois theory, University Lecture Series, vol. 7, American Mathematical Society, Providence, RI, 1994. MR 1301076, DOI 10.1090/ulect/007
Juan J. Morales Ruiz, Differential Galois theory and non-integrability of Hamiltonian systems, Progress in Mathematics, vol. 179, Birkhäuser Verlag, Basel, 1999. MR 1713573, DOI 10.1007/978-3-0348-8718-2
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, DOI 10.1007/978-3-642-55750-7
Joseph Fels Ritt, Differential algebra, Dover Publications, Inc., New York, 1966. MR 0201431
A. Seidenberg, Contribution to the Picard-Vessiot theory of homogeneous linear differential equations, Amer. J. Math. 78 (1956), 808–818. MR 81897, DOI 10.2307/2372470