Hostname: page-component-76fb5796d-r6qrq Total loading time: 0 Render date: 2024-04-28T15:23:41.657Z Has data issue: false hasContentIssue false
  • English
  • Français

Differential algebraic function fields depending rationally on arbitrary constants

Published online by Cambridge University Press:  22 January 2016

Keiji Nishioka
Keiji Nishioka*
Affiliation:
Takabatake-cho 184-632 Nara 630, Japan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The general solution of an algebraic differential equation depends on the initial conditions, though it is in general too difficult to make explicit the shape of the relationship. Painlevé studied in [8] algebraic differentia] equations of second order with the general solutions depending rationally on the initial conditions and the solvability of such equations. Giving the precise definition of the notion “rational dependence on the initial conditions”, Umemura [10] revived and generalized rigorously the discussion of Painlevé in the language of modern algebraic geometry. The theorem of Umemura is as follows; Let K be a differential field extension of complex number field C generated by a finite number of meromorphic functions on some domain in C. Let y be the general solution of a given algebraic differential equation over K. Suppose that y depends rationally on the initial conditions. Then it is contained in the terminal Km of a finite chain of differential field extensions: K = K0 ⊂ K1 ⊂… ⊂Km such that each Ki is strongly normal over Ki−1.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 113 , March 1989 , pp. 173 - 179
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1989

References

[1] Białynicki-Birula, A., On Galois theory of fields with operators, Amer. J. Math., 84 (1962), 89–109.Google Scholar
[2] Kolchin, E.R., Existence theorems connected with the Picard Vessiot theory of homogeneous linear ordinary differential equations, Bull. Amer. Math. Soc., 54 (1948), 927–932.CrossRefGoogle Scholar
[3] Kolchin, E. R., Differential Algebra and Algebraic Groups, Academic Press, New York, 1973.Google Scholar
[4] Matsuda, M., First order algebraic differential equations, A differential algebraic approach, Lecture Notes in Math., 804, Springer, Berlin, 1980.Google Scholar
[5] Nishioka, K., Algebraic differential equations of Clairaut type from the differential-algebraic standpoint, J. Math. Soc. Japan, 31 (1979), 553–559.Google Scholar
[6] Nishioka, K., A note on the transcendency of Painlevé’s first transcendent, Nagoya Math. J., 109 (1988), 63–67.Google Scholar
[7] Nishioka, K., General solutions depending algebraically on arbitrary constants, Nagoya Math. J., 113 (1989), 1–6.Google Scholar
[8] Painlevé, P., Leçon de Stockholm, OEuvres de P. Painlevé I, 199–818, Editions du C.N.R.S., Paris, 1972.Google Scholar
[9] Ritt, J. F., Differential algebra, Amer. Math. Soc. Colloq. Publ. Vol. 33, New York, 1950.Google Scholar
[10] Umemura, H., Birational automorphism groups and differential equations, to appear in the Proc. Franco-Japanese colloquium on differential equations, Strasbourg, 1985.Google Scholar