Algebraic Independence of Painlevé First Transcendents

MR2108680 Keiji NISHIOKA Keiji NISHIOKA Algebraic Independence of Painlevé First Transcendents Funkcialaj Ekvacioj. Serio Internacia 47 2004 351--360 http://fe.math.kobe-u.ac.jp/FE/FullPapers/47-2/47_351.pdf http://www.ams.org/mathscinet-getitem?mr=MR2108680 It will be proved that Painlevé first transcendents and their first derivatives are algebraically independent over the rational function field with complex coefficients, by the use of the irreducibility. A particular case indicates that the group of differential automorphisms of the differential field generated by a Painlevé first transcendent over the rational function field is trivial. Painlevé first transcendent, Differential automorphism, Irreducibility. 12H05, 34M15. 47-351 2004 Algebraic Independence of Painlevé First Transcendents Keiji NISHIOKA Keiji NISHIOKA

1 Nishioka, K. A note on the transcendency of Painlevé's first transcendent Nagoya Math. J. 109 1988 63-67 MR0931951

2 Umemura, H. On the irreducibility of the first differential equation of Painlevé Algebraic Geometry and Commutative Algebra in honor of Masayoshi NAGATA, Kinokuniya, Tokoyo 1987 771-789 MR0977782