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An extension of Painlev\'e's determinateness theorem

In this paper we state and proof an extended version of Painlevé's determinateness theorem. We study the C 0 −continuability along analytic arcs of solutions of some Cauchy's problems, which will be 'multiple-valued', i.e. defined on Riemann domains spread over C 2. The theory of ordinary differential equations in the complex domain is a classical subject; in this paper we pursue the goal of extending the well known Painlevé's determinateness theorem (see e.g. [HIL], theorem 3.3.1): informally speaking, we shall study some aspects of the C 0 −continuability along analytic arcs of solutions of one-dimensional (nonautonomous) Cauchy's problems, which will be defined by finitely 'multiple-valued' functions on C 2. Branch points will be supposed to lie on algebraic curves; of course we shall formalize 'multiple valuedness' by the use of Riemann domains spread over C 2 (see e.g. [GRO]). It seems worthwhile to remark a difference between theorem 3.3.1 of [HIL] and theorem 6: they both assure the existence of some limits of analytical continuations of solutions of some Cauchy's problems, the former dealing with domains of the form (S × C) (for some algebraic Riemann surface S), the latter concerning some proper Riemann domain (R, p). We start by recalling some definitions and statements which we shall use in the following; for further details, the reader is referred to [HIL] or [INC].