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Solutions of the third Painlevé equation. I. (English) Zbl 0917.34004
A rigorous and systematic proof of the irreducibility of the third Painlevé equation is given by applying Umemura’s theory on algebraic differential equations to this equation. The proof consists of two parts: (1) to determine a necessary condition for the parameters of the existence of principal ideals invariant under the Hamiltonian vector field associated to the third Painlevé equation; (2) to determine the principal invariant ideals for a parameter where the principal invariant ideals exist. The method is released from complicated calculation, and has been applied to the proof of the irreducibility of other Painlevé equations.
##### MSC:
 34M55 Painlevé and other special equations; classification, hierarchies 34A34 Nonlinear ODE and systems, general 34A05 Methods of solution of ODE
##### Keywords:
third Painlevé equation; Umemura’s theory