×

Solvable-by-finite groups as differential Galois groups. (English) Zbl 1066.12003

Let \(C\) be an algebraically closed field of characteristic zero and let \(K=C(x)\) be the field of rational functions over \(C\) with the derivation \(d/dx\). Let \(G\) be an affine algebraic group defined over \(C\). The inverse problem of differential Galois theory asks whether there is a Picard-Vessiot extension \(E \supset K\) whose differential Galois group is isomorphic to \(G\). The authors consider the (stronger, in case \(G\) is not connected) problem of finding a principal homogeneous space \(V\) for \(G\) and a derivation on the function field \(K(V)\) such that \(E=K(V)\), with the given action of \(G\), is such a Picard-Vessiot extension. They prove this when the identity component of \(G\) is solvable, and their proof is constructive up to finding finite Galois extensions of \(C(x)\).
In fact their proof, which is explained with careful attention to exposition, potentially covers a much wider class of groups (and has been extended to all groups by Julia Hartmann [On the inverse problem in differential Galois theory. Thesis Heidelberg (2002; Zbl 1063.12005)]). Because of this attention to background, this paper is an excellent introduction to this approach to the inverse problem.

MSC:

12H05 Differential algebra

Citations:

Zbl 1063.12005
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML Link

References:

[1] Bertrand, D.). - Review of “Lectures on differential Galois theory” by A. Magid, Bull. AMS, 33 (1996), 289-294. Springer Verlag, 1991.
[2] Borel, A.). - Linear Algebraic Groups, second edition, Springer Verlag, 1991. · Zbl 0726.20030
[3] Borel, A.), Serre, J.-P.). - Théorèmes de finitude en cohomologie galoisienne, Comment. Math. Helv., 39 (1964-1965), 111-164. · Zbl 0143.05901
[4] Chevalley, C.). - Théorie des groupes de Lie, Vol.II, Groupes algébriques, Hermann, Paris, 1951. · Zbl 0054.01303
[5] Deligne, P.). - Catégories tannakiennes, The Grothendieck Festschrift2, p. 111-195, Progress in Math., 87 (1990), Birkhäuser. · Zbl 0727.14010
[6] Hartmann, J.). - On the Inverse Problem in Differential Galois Theory, Preprint, Universität Heidelberg, 2002. · Zbl 1063.12005
[7] Hochschild, G.). - Basic Theory of Algebraic Groups and Lie Algebras, , 75, Springer-Verlag, 1981. · Zbl 0589.20025
[8] Kolchin, E.R.). - Differential Algebra and Algebraic Groups, Academic Press, 1976. · Zbl 0264.12102
[9] Kolchin, E.R.). - Algebraic groups and algebraic dependence, Amer. J. Math., 90 (1968), 1151-1164. · Zbl 0169.36701
[10] Kovacic, J.). - The Inverse Problem in the Galois Theory of Differential Fields, Annals of Mathematics, 89 (1969), 583-608. · Zbl 0188.33801
[11] Kovacic, J.). - On the Inverse Problem in the Galois Theory of Differential Fields, Annals of Mathematics, 93 (1971), 269-284. · Zbl 0214.06004
[12] Lang, S.). - Algebra, Third Edition, Addison-Wesley, 1993. · Zbl 0848.13001
[13] Magid, A.). - Lectures on Differential Galois theory, University Lecture Series, American Mathematical Society, Second edition, 1994. · Zbl 0855.12001
[14] Mitschi, C.), Singer, M.F.). - Connected Linear Groups as Differential Galois Groups, Journal of Algebra, 184 (1996), 333-361. · Zbl 0867.12004
[15] Van Der Put, M.). - Galois Theory of differential equations, algebraic groups and Lie algebras, Journal of Symbolic Computation, 28 (1999), 441-472. · Zbl 0997.12008
[16] Van Der Put, M.). - Recent work in differential Galois theory, Séminaire Bourbaki, volume 1997/98 (exposé 849), Astérisque, Société Mathématique de France, Paris, 1998. · Zbl 0931.12008
[17] Van Der Put, M.), Singer, M.F.).- Galois Theory of Differential Equations, to appear. · Zbl 0997.12008
[18] Ritt, J.F.). - Differential Algebra, , Vol. 33, Am. Math.Soc., 1950. · Zbl 0037.18402
[19] Ramis, J.P.). - About the Inverse Problem in Differential Galois Theory: the Differential Abhyankar Conjecture, unpublished manuscript, 1996. · Zbl 0860.12003
[20] Ramis, J.P.). - About the inverse problem in differential Galois theory: the differential Abhyankar conjecture, in The Stokes phenomenon and Hilbert’s 16th Problem, World Scientific Publ. (Editors B.L.J. Braaksma, G.K. Immink, M. van der Put), Singapore1996. · Zbl 0860.12003
[21] Serre, J.P.).- Cohomologie Galoisienne, cinquième édition, , 5, Springer-Verlag, 1994. · Zbl 0812.12002
[22] Singer, M.F.). - Moduli of linear differential equations on the Riemann sphere with fixed Galois groups, Pacific J. Math., 106, No. 2 (1993), 343-395. · Zbl 0778.12007
[23] Tretkoff, C.), Tretkoff, M.). - Solution of the inverse problem of differential Galois theory in the classical case, Amer. J. Math., 1979, 1327-1332. · Zbl 0423.12021
[24] Volklein, H.). - Groups as Galois Groups, Cambridge Studies in Advanced Mathematics, Vol. 53, Cambridge University Press, 1996. · Zbl 0868.12003
[25] Wehfritz, B.A.F.). - Infinite Linear Groups, Springer-Verlag, 1973. · Zbl 0261.20038
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.