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General solutions depending algebraically on arbitrary constants

Published online by Cambridge University Press:  22 January 2016

Keiji Nishioka
Keiji Nishioka*
Affiliation:
Takabatake-cho, 184-632 Nara 630, Japan
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In his famous lectures [7] Painlevé investigates general solutions of algebraic differential equations which depend algebraically on some of arbitrary constants. Although his discussions are beyond our understanding, the rigorous and accurate interpretation to make his intuition true would be possible. Successful accomplishments have been done by some authors, for example, Kimura [1], Umemura [8, 9]. From differential algebraic viewpoint in [5] the author introduces the notion of rational dependence on arbitrary constants of general solutions of algebraic differential equations, and in [6] clarifies the relation between it and the notion of strong normality. Here we aim at generalizing to higher order case the result in [4] that in the first order case solutions of equations depend algebraically on those of equations free from moving singularities which are determined uniquely as the closest ones to the given. Part of our result can be seen in [7].

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 113 , March 1989 , pp. 1 - 6
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1989

References

[1] Kimura, T., On conditions for ordinary differential equations of the first order to be reducible to a Riccati equation by a rational transformation, Funkcial. Ekvac, 9 (1966), 251–259.Google Scholar
[2] Kolchin, E. R., Differential Algebra and Algebraic Groups, Academic Press, New York, 1973.Google Scholar
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[4] Nishioka, K., A theorem of Painlevé on parametric singularities of algebraic differential equations of the first order, Osaka J. Math., 18 (1981),Google Scholar
[5] Nishioka, K., A note on the transcendency of Panleve’s first transcendent, Nagoya Math. J., 109 (1988), 63-67.CrossRefGoogle Scholar
[6] Nishioka, K., Differential algebraic function fields depending rationally on arbitrary constants, Nagoya Math. J., 113 (1989), 173–179.Google Scholar
[7] Painlevé, P., Leçon de Stockholm, OEuvres de P. Painlevé I, 199–818, Éditions du C.N.R.S., Paris, 1972.Google Scholar
[8] Umemura, H., Birational automorphism groups and differential equations, to appear.Google Scholar
[9] Umemura, H., On the irreducibility of the first differential equation of Painlevé, in preprint.Google Scholar