Nagoya Mathematical Journal

On a class of numbers generated by differential equations related with algebraic groups

Hiroshi Umemura

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Nagoya Math. J. Volume 133 (1994), 1-55.

First available in Project Euclid: 14 June 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 12H05: Differential algebra [See also 13Nxx]
Secondary: 12F99: None of the above, but in this section


Umemura, Hiroshi. On a class of numbers generated by differential equations related with algebraic groups. Nagoya Mathematical Journal 133 (1994), 1--55.

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  • [B] Borel, A., Linear algebraic groups, Benjamin, New York, 1969.
  • [D] Dieudonne,J., Calcul infinitesimal, Collection Methodes, Hermann,Paris, 1968. a
  • [G] Gauss,C. F., Disquisitionesgeneralescircaserieminfinitam1 jx f a l)( 1)a(al)(a2)( 1) Q8 2) 3,' 1. 2. (l)x x1. 2. 3. ( 1) (r 2) e t c 'p a r s prior, Gesamelte Werke Bd. Ill, 125-163, Gttingen, 1863-1933.
  • [H] Hartshorne, R., Algebraic geometry, Graduate. Texts, in Math., 52, Springer- Verlag, Berlin-Heidelberg-New York, 1977.
  • [K] Kolchin, E., Differential algebra and the algebraic groups, Academic Press, New York and London, 1973.
  • [L] Liouville, J., Sur la classification des transcendentes et sur impossibilite d'exprimer les racines de certaines equations en fonction finie explicite des coefficient, Journal de Math., 2 (1837), 56-104 3 (1838), 523-546 4 (1839), 423-456.
  • [Ml] Mumford, D., Tata Lectures on Theta I, Progress in Math. Vol. 28, Birkhauser, Boston-Basel-Stuttgart, 1983.
  • [M2] Mumford, The red book of varieties and schemes, Lee. Notes, in Math., Vol. 1358, Springer-Verlag, Berlin-Heidelberg-New York, 1988.
  • [PjPain] eve, P., Leons de Stockholm, Oeuvres, Edition du C. N. R. S., Paris, 1972.
  • [Rl] Rosenlicht, M., Some basic theorems on algebraic groups, Amer. J Math., 78 (1956), 401-443.
  • [R2] Rosenlicht, Liouville's theorem on functions with elementary integrals, Pacific J. Math., 24(1968), 153-161.
  • [S] Shafarevich, I. R., Basic algebraic geometry, Grundl. Math. Wiss., Bd. 213, Springer-Verlag, Berlin-Heidelberg-New York, 1974.
  • [Ul] Umemura, H., Resolution of algebraic equations by theta constants, in D. Mum- ford Tata Lectures on Theta II, Progress in Math. Vol. 43, Birkhauser, Boston-Basel-Stuttgart, 1985.
  • [U2] Umemura, On the irreducibility of the first differential equation of Painleve, Algebraic geometry and Commutative algebra in Honor of Masayoshi NAGATA, Kinokuniya 1987, Tokyo, 101-119.
  • [U3] Umemura, Second proof of the irreducibility of the first differential equation of Pain- leve, Nagoya Math. J., 117 (1990), 125-171.
  • [U4] Umemura, Birational automorphism groups and differential equations, Nagoya Math.J., 119(1990), 1-80.
  • [W] Weyl, H.,Die Idee der Riemannschen Flache, Teubner, 1913.
  • [WW] Whittacker, E. T. and Watson, G. N., A course of modern analysis, Cambridge University Press, 1902.
  • [Z] Zariski, O. and Samuel, P., Commutative algebra, Van Nostrand, Princeton, 1985. Appendix This part is added to help the reader who is not familiar with algebraic geometry. Our aim here is not review rigorous definitions but to explain the termi-