= Painlevé determinateness theorem
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For a first-order <ordinary differential equation> rational in the dependent variable, with coefficients <meromorphic> on the chosen independent-variable surface, <analytic continuation> along an arc ending outside the fixed exceptional set has a definite limit in the <Riemann sphere>. Its movable singularities are <poles> or <algebraic branch points>. To see the limiting assertion, the cluster set of a continuous lifted arc is connected and compact. Any cluster value where the vector field is <holomorphic> permits uniform local continuation by the <Picard-Lindelöf theorem>. Otherwise the cluster set lies in finitely many coefficient poles, so it is a singleton. At a pole of the vector field, the inverse equation is <holomorphic>; its solution has a finite-order critical point, giving a convergent root expansion. The reciprocal dependent variable handles infinity. Points where both reduced numerator and denominator vanish belong to the fixed exceptional set and are excluded from this argument.
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