Differential algebraic function fields depending rationally on arbitrary constants

Abstract

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.