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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
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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

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