Umemura, Hiroshi; Watanabe, Humihiko Solutions of the second and fourth Painlevé equations. I. (English) Zbl 0934.33029 Nagoya Math. J. 148, 151-198 (1997). 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)]. Cited in 1 ReviewCited in 34 Documents 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 Citations:Zbl 0714.12009; Zbl 0704.12007; Zbl 0707.34010; Zbl 0833.34005 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Algebraic Geometry and Commutative Algebra in Honor of Masayoshi NAGATA pp 771– (1987) [2] Nagoya Math. J. 117 pp 125– (1990) · Zbl 0688.34006 · doi:10.1017/S0027763000001835 [3] Editions du C. N. R. S., Paris (1972) [4] C. R. Acad. Sci. Paris 143 pp 1111– (1906) [5] Acta Math. 25 pp 1– (1900) [6] Bull. Soc. Math. France 28 pp 201– (1900) [7] (1987) [8] DOI: 10.1007/BF01458459 · Zbl 0589.58008 · doi:10.1007/BF01458459 [9] DOI: 10.14492/hokmj/1380892594 · Zbl 0833.34005 · doi:10.14492/hokmj/1380892594 [10] Diff. Eq. 1 pp 58– (1965) [12] Funk. Ekvac 40 pp 139– (1997) [14] Nagoya Math. J. 109 pp 63– (1988) · Zbl 0613.34030 · doi:10.1017/S0027763000002762 [15] Funk. Ekvac 28 pp 1– (1985) [16] Nagoya Math. J. 139 pp 37– (1995) · Zbl 0846.34002 · doi:10.1017/S0027763000005298 [17] Diff. Eq. 18 pp 317– (1982) [18] Diff. Eq. 7 pp 853– (1971) [19] Dokl. Akad. Nauk. BSSR. 33 pp 209– (1989) [20] Diff. Eq. 3 pp 395– (1967) [21] Vesti A. N. BSSR. Ser. Fiz-Tekh. Nauk. 3 pp 30– (1959) [22] Dokl. Akad. Nauk. BSSR. 33 pp 17– (1989) [23] Differential Algebra and Algebraic groups (1973) [24] Dokl. Akad. Nauk. BSSR. 32 pp 395– (1988) [25] Ordinary differential equations [26] Diff. Eq. 23 pp 506– (1987) [27] Diff. Eq. 27 pp 910– (1991) [28] 18 pp 537– (1982) [29] Diff. Eq. 14 pp 1510– (1978) [30] Acta Math. 33 pp 1– (1909) · JFM 40.0098.04 [31] Groupes et algèbres de Lie (1968) [32] Nagoya Math. J. 144 pp 59– (1996) · Zbl 0878.12002 · doi:10.1017/S0027763000006024 [33] Nagoya Math. J. 119 pp 1– (1990) · Zbl 0714.12009 · doi:10.1017/S0027763000003111 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.