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Solutions of the second and fourth Painlevé equations. I. (English) Zbl 0934.33029

Summary: A rigorous proof of the irreducibility of the second and fourth Painlevé equations is given by applying H. Umemura’s theory on algebraic differential equations to the two equations [cf. Nagoya Math. J. 119, 1-80 (1990; Zbl 0714.12009); Algebraic geometry and commutative algebra, in Honor of Masayoshi Nagata, Vol. II, 771-789 (1988; Zbl 0704.12007); Nagoya Math. J. 117, 125-171 (1990; Zbl 0707.34010)]. The proof consists of two parts: to determine a necessary condition for the parameters of the existence of principal ideals invariant under the Hamiltonian vector field; to determine the principal invariant ideals for a parameter where the principal invariant ideals exist. Our method is released from complicated calculation, and applicable to the proof of the irreducibility of the third, fifth and sixth equation [e.g. Hokkaido Math. J. 24, No. 2, 231-267 (1995; Zbl 0833.34005)].

MSC:

34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
33E30 Other functions coming from differential, difference and integral equations
34A34 Nonlinear ordinary differential equations and systems
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