# zbMATH — the first resource for mathematics

Picard-Vessiot theory for real fields. (English) Zbl 1298.12004
Let $$F$$ be a differential field of characteristic $$0$$ with field of constants $$C$$ and (1) $$L(y)=0$$ be a linear differential equation over $$F$$. In the case where the field $$C$$ is not algebraically closed a question of the existence of Picard-Vessiot extension of $$F$$ for the equation (1) is poorly understood see M. P. Epstein [Ann. Math. (2) 62, 528–547 (1955; Zbl 0065.27001)], but it is of particular interest, for example, in the study of equations with parameters [P. J. Cassidy and M. F. Singer, in: Differential equations and quantum groups. Andrey A. Bolibrukh memorial volume. IRMA Lect. Math. Theor. Phys. 9, 113–155 (2007; Zbl 1230.12003)] or solution of the equation by real Liouville functions see O. A. Gel’fond and A. G. Khovanskii [Funkts. Anal. Prilozh. 14, No. 2, 52–53 (1980; Zbl 0468.30006)].
In this paper the authors show on concrete examples why it is difficult to investigate the case where $$C$$ is not algebraically closed. Then they prove the existence theorem and main theorem of Galois theory when $$F$$ is real and $$C$$ is real closed.

##### MSC:
 12H05 Differential algebra 34A30 Linear ordinary differential equations and systems, general 37C10 Dynamics induced by flows and semiflows
Full Text:
##### References:
 [1] M. Audin, Exemples de hamiltoniens non intégrables en mécanique analytique réelle, Annales de la Faculté des Sciences de Toulouse 12 (2003), 1–23. · Zbl 1042.37039 · doi:10.5802/afst.1042 [2] J. Bochnak, M. Coste and M.-F. Roy, Real Algebraic Geometry, Springer Verlag, Berlin, 1998. [3] A. Borel, Algebraic groups and Galois theory in the works of Ellis Kolchin in Selected works of Ellis Kolchin with Commentary (H. Bass, A. Buium and P. J. Cassidy, eds.), American Mathematical Society, Providence, RI, 1999, pp. 505–525. [4] T. Crespo and Z. Hajto, Algebraic Groups and Differential Galois Theory, Graduate Studies in Mathematics, Vol. 122, American Mathematical Society, Providence, RI, 2011. · Zbl 1215.12001 [5] T. Crespo, Z. Hajto and E. Sowa, Constrained extensions of real type, Comptes Rendus Mathématique. Académie des Sciences. Paris 350 (2012), 235–237. · Zbl 1285.12003 · doi:10.1016/j.crma.2012.03.006 [6] O. A. Gel’fond and A. G. Khovanskii, Real Liouville functions, Funktsional’nyi Analiz i Ego Prilozheniya 14 (1980), 52–53. [7] H. Gillet, S. Gorchinskiy and A. Ovchinnikov, Parameterized Picard-Vessiot extensions and Atiyah extensions, arXiv:1110.3526. · Zbl 1328.12010 [8] E. Kolchin, Differential Algebra and Algebraic Groups, Academic Press, New York, 1973. · Zbl 0264.12102 [9] E. Kolchin, Existence theorems connected with Picard-Vessiot theory of homogeneous linear ordinary differential equations, Bulletin of the American Mathematical Society 54 (1948), 927–932. · Zbl 0037.06601 · doi:10.1090/S0002-9904-1948-09099-1 [10] E. Kolchin, Algebraic matric groups and the Picard-Vessiot theory of homogeneous linear ordinary differential equations, Annals of Mathematics 49 (1948), 1–42. · Zbl 0037.18701 · doi:10.2307/1969111 [11] E. R. Kolchin, Constrained extensions of differential fields, Advances in Mathematics 12 (1974), 141–170. · Zbl 0279.12103 · doi:10.1016/S0001-8708(74)80001-0 [12] J. J. Kovacic, The differential Galois theory of strongly normal extensions, Transactions of the American Mathematical Society 355 (2003), 4475–4522. · Zbl 1036.12005 · doi:10.1090/S0002-9947-03-03306-3 [13] A. Magid, Lectures on Differential Galois Theory, University Lecture Series, Vol. 7, American Mathematical Society, Providence, RI, 1997. · Zbl 0855.12001 [14] J. J. Morales-Ruiz, Differential Galois Theory and Non-integrability of Hamiltonian Systems, Progress in Mathematics, Vol. 179, Birkhäuser, Basel, 1999. · Zbl 0934.12003 [15] A. Pillay, Differential Galois theory I, Illinois Journal of Mathematics 42 (1998), 678–699. · Zbl 0916.03028 [16] M. van der Put and M. Singer, Galois Theory of Linear Differential equations, Grundlehren der Mathematischen Wissenschaften, Vol. 328, Springer, Berlin, 2003. · Zbl 1036.12008 [17] A. Seidenberg, Contribution to the Picard-Vessiot theory of homogeneous linear differential equations, American Journal of Mathematics 78 (1956), 808–817. · Zbl 0072.26502 · doi:10.2307/2372470 [18] J.-P. Serre, Galois Cohomology, Springer, Berlin, 1997. [19] M. F. Singer, A class of differential fields with minimal differential closures, Proceedings of the American Mathematical Society 69 (1978), 319–322. · Zbl 0396.03032 · doi:10.1090/S0002-9939-1978-0465851-4 [20] E. Sowa, Picard-Vessiot extensions for real fields, Proceedings of the American Mathematical Society 139 (2011), 2407–2413. · Zbl 1252.12006 · doi:10.1090/S0002-9939-2010-10700-1 [21] H. Umemura, Galois theory of algebraic and differential equations, Nagoya Mathematical Journal 144 (1996), 1–58. · Zbl 0885.12004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.