Advertisement

Differential Galois Theory

  • Frits Beukers
Chapter

Abstract

Perhaps the easiest description of differential Galois theory is that it is about algebraic dependence relations between solutions of linear differential equations. To clarify this statement, let us consider three examples. First consider the differential equation
$$z(1-z){y}''+(\frac{1}{2}-\frac{7}{6}z){y}'+\frac{11}{3600}y = 0$$
(1.1)

Keywords

Algebraic Group Linear Differential Equation Galois Group Linear Algebraic Group Algebraic Subgroup 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Babbitt, D.G., Varadarajan, V.S. (1983) Formal reduction theory of meromorphic differential equations: a group theoretic view, Pacific J.Math. 109(1983), 1–80.MathSciNetMATHCrossRefGoogle Scholar
  2. Barnes (1907), The asymptotic expansion of integral functions defined by generalized hypergeometric series, Proc. London Math. Soc (2)5(1907) 59–116.MATHCrossRefGoogle Scholar
  3. Beukers, F., Brownawell, W.D., Heckman, G. (1988) Siegel normality, Annals of Math. 127 (1988) 279–308.MathSciNetMATHCrossRefGoogle Scholar
  4. Beukers, F., Heckman, G. (1989) Monodromy for the hypergeometric function n F n−1, Inv.Math. 95 (1989) 325–354.MathSciNetMATHCrossRefGoogle Scholar
  5. Beukers, F., Peters, C.A.M. (1984) A family of K3-surfaces and ζ(3), J. reine angew. Math. 351 42–54.MathSciNetMATHGoogle Scholar
  6. Deligne, P. (1987) Categories tannakiennes, in Grothendieck Festschrift, part. II, Birkhäuser 1991.Google Scholar
  7. Duval, A., Mitschi, C. (1989) Matrices de Stokes et groupe de Galois des équations hypergéometriques confluentes généralisées, Pacific J. Math. 138 (1989), 25–56.MathSciNetMATHCrossRefGoogle Scholar
  8. Erdélyi, A. et al. (1953) Higher transcendental functions (Bateman manuscript project) Vol II, Ch.7 McGraw-Hill, New York 1953.Google Scholar
  9. Fano, G. (1900) Über lineare homogene Differentialgleichungen, Math.Annalen 53 493–590.MathSciNetMATHCrossRefGoogle Scholar
  10. Freudenthal, H., de Vries, H. (1969) Linear Lie groups, Academic Press New York 1969.MATHGoogle Scholar
  11. Gray, J. (1986) Linear differential equations and group theory from Riemann to Poincaré Birkhäuser, Boston 1986.MATHGoogle Scholar
  12. Hille, E. (1976) Ordinary differential equations in the complex domain Wiley, New York 1976.MATHGoogle Scholar
  13. Humphreys, J.E. (1972) Introduction to Lie algebras and representation theory Springer Verlag, New York 1972.MATHCrossRefGoogle Scholar
  14. Humphreys, J. E. (1975) Linear algebraic groups Springer Verlag, New York 1975.MATHCrossRefGoogle Scholar
  15. Ince, E.L. (1926) Ordinary differential equations Dover Publication (reprinted) New York 1956.Google Scholar
  16. Kaplansky, I. (1957) Differential algebras Hermann, Paris 1957.Google Scholar
  17. Katz, N.M. (1970) Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin, Publ. Math IHES 39 175–232.MATHGoogle Scholar
  18. Katz, N.M. (1987) On the calculation of some differential Galois groups, Inv.Math. 87 (1987) 13–61.MATHCrossRefGoogle Scholar
  19. Katz, N.M. (1990) Exponential sums and differential equations Ann. of Math. Study 124 Princeton University Press, Princeton 1990.MATHGoogle Scholar
  20. Katz, N.M., Pink, R. (1987) A note on pseudo-CM representations and differential Galois groups, Duke Math.J. 54 (1987) 57–65.MathSciNetMATHCrossRefGoogle Scholar
  21. Klein, F. (1933) Vorlesungen über die hypergeometrische Funktion Springer Verlag, New York 1933.Google Scholar
  22. Kolchin, E.R (1948) Algebraic matrix groups and the Picard-Vessiot theory of homogeneous linear ordinary differential equations, Annals of Math. 49 (1948) 1–42.MathSciNetMATHCrossRefGoogle Scholar
  23. Kolchin, E.R (1948) Existence theorems connected with the Picard-Vessiot theory of homogeneous linear ordinary differential equations, Bull. Amer. Math. Soc. 54 (1948) 927–932.MathSciNetMATHCrossRefGoogle Scholar
  24. Kolchin, E.R. (1973) Differential algebra and algebraic groups, Academic Press, New York 1973.MATHGoogle Scholar
  25. Kolchin, E.R. (1985) Differential algebraic groups, Academic Press, New York 1985.MATHGoogle Scholar
  26. Kostant, B. (1958) A characterisation of the classical groups, Duke Math. J. 25 (1958) 107–123.MathSciNetMATHCrossRefGoogle Scholar
  27. Levelt, A.H.M. (1961) Hypergeometric functions, thesis University of Amsterdam 1961.Google Scholar
  28. Levelt, A.H.M. (1975) Jordan decomposition for a class of singular differential operators, Arkiv Math. 13 (1975) 1–27.MathSciNetMATHCrossRefGoogle Scholar
  29. Levelt, A.H.M. (1990) Differential Galois theory and tensor products, Indag. Math. 1 (new series) (1990) 439–449.MathSciNetMATHCrossRefGoogle Scholar
  30. Martinet, J. Ramis, J.P. (1989) Théorie de Galois différentielle in Computer Algebra and Differential Equations, ed. E. Tournier, Academic Press, New York 1989.Google Scholar
  31. Meijer, C.S. (1946) On the G-function I, Indag.Math. 8 (1946) 124–134.Google Scholar
  32. Picard, E. (1898) Traité d’analyse Vol III, Chapter 17, Gauthier-Villars, Paris 1898. Reprinted as Analogies entre la théorie des équations différentielles linéaires et la théorie des équations algébriques, Gauthier-Villars Paris, 1936.Google Scholar
  33. Pommaret, J.F. (1983) Differential Galois theory, Gordon and Breach New York, 1983.MATHGoogle Scholar
  34. Poole, E.G.C. Introduction to the theory of linear differential equations Oxford 1936, reprinted in Dover Publ.Google Scholar
  35. Ritt, J.F. (1932) Differential equations from the algebraic standpoint AMS Coll. Publ. XIV, New York 1932.Google Scholar
  36. Schwartz, H.A. (1873) Über diejenigen Fälle in welchen die Gaussische hypergeometrische Reihe einer algebraische Funktion ihres vierten Elementes darstellt, Crelle J. 75 (1873) 292–335.CrossRefGoogle Scholar
  37. Sibuya, Y. (1977) Stokes phenomena, Bull.American Math. Soc. 83 (1977), 1075–1077.MathSciNetMATHCrossRefGoogle Scholar
  38. Singer, M. (1989) An outline of differential Galois theory in Computer Algebra and Differential Equations, ed. E. Tournier, Academic Press, New York 1989.Google Scholar
  39. Springer, T.A. (1981) Linear algebraic groups, Progress in math. 9, Birkhäuser, Boston 1986.MATHGoogle Scholar
  40. Turrittin, H.L. (1955) Convergent solutions of ordinary linear homogeneous differential equations in the neighbourhood of an irregular singular point, Acta Math. 93 (1955) 27–66.MathSciNetMATHCrossRefGoogle Scholar
  41. Vessiot, E. (1892) Sur les intégrations des équations différentielles linéaires, Ann. Sci. École Norm. Sup.(3) 9 (1892), 197–280.MathSciNetMATHGoogle Scholar
  42. Wasow, W. (1965) Asymptotic expansions for ordinary differential equations Wiley, New York 1965.MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Frits Beukers

There are no affiliations available

Personalised recommendations