Painlevé equation PII and strongly normal extensions

• Autorzy: El-Hadi Miri S.
• Języki publikacji: EN

Abstrakty

EN

The aim of this paper is to show that if F is a differential field and y is a PII transcendent such that tr.deg. F=2, then every constant in F is in F. We also show that in this case, F is not contained in any strongly normal extension.

Słowa kluczowe

EN

Painlevé equation PII; differential algebra; strongly normal extensions

PL

równanie Painlevéa; równanie różniczkowe; algebra różniczkowa

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2016
• Tom: Vol. 49, nr 4
• Strony: 386--397
• Opis fizyczny: Bibliogr. 21 poz.

Twórcy

autor

El-Hadi Miri S.

• Laboratoire D’analyse Non Linéaire et Mathématiques Appliquées BP 119 Tlemcen 13000
• Faculté de Technologie, Département Gee BP 230 Tlemcen 13000, Université de Tlemcen, Algerie

Bibliografia

• [1] L. Brenig, A. Goriely, Painlevé analysis and normal forms, Computer Algebra and Differential Equations 193 (1994).
• [2] G. Casale, Le groupoïde de Galois de la P1 et son irréductibilité, Comment. Math. Helv. 83 (2008), 471–519.
• [3] G. Casale, J. Weil, Galoisian Methods for Testing Irreducibility of Order Two Nonlinear Differential Equations, 2015. arXiv preprint arXiv:1504.08134
• [4] R. Conte, The Painlevé Property: One Century Later, Springer, 1999.
• [5] E. L. Ince, Ordinary Differential Equations, Dover Publications Inc., 1956.
• [6] K. Iwasaki, H. Kimura, S. Shimomura, M. Yoshida, From Gauss to Painlevé, Vieweg, 1991.
• [7] I. Kaplansky, An Introduction to Differential Algebra, Hermann, 1957.
• [8] E. R. Kolchin, Differential Algebra and Algebraic Groups, Academic Press, New York, 1973.
• [9] E. R. Kolchin, Selected Works of Ellis Kolchin with Commentary, American Mathematical Society, Providence, RI, 1999.
• [10] J. Kovacic, Differential Galois Theory, Talk presented in the KSDA, 2001. Downloaded from Pr. Kovacic personal web page: http://mysite.verizon.net/jkovacic
• [11] J. Kovacic, The differential Galois theory of strongly normal extensions, Trans. Amer. Math. Soc. 355(11) (2003), 4475–4522.
• [12] B. Malgrange, On non linear differential Galois theory, Chinese Ann. Math. Ser. B. 23(2) (2002), 219–226.
• [13] J. Nagloo, P. Anand, On algebraic relations between solutions of a generic Painlevé equation, Journal Reine Angew. Math. (Crelles Journal), 2011.
• [14] K. Nishioka, A note on the transcendency of Painlevé’s first transcendent, Nagoya Math. J. 109 (1988), 63–67.
• [15] K. Nishioka, Algebraic function fields depending rationally on arbitrary constants, Nagoya Math. J. 113 (1989), 173–179.
• [16] M. F. Singer, Introduction to the Galois Theory of Linear Differential Equations, Algebraic Theory of Differential Equations, 2008. arXiv preprint arXiv:0712.4124.
• [17] M. van der Put, M. F. Singer, Galois Theory of Linear Differential Equations, Springer, Heidelberg, 2003.
• [18] H. Umemura, Second proof of the irreducibility of the first differential equation of Painlevé, Nagoya Math. J. 117 (1990), 125–171.
• [19] H. Umemura, Differential Galois theory of infinite dimension, Nagoya Math. J. 144 (1996), 59–135.
• [20] H. Umemura, Solutions of the scond and fourth Painlevé equations, I, Nagoya Math. J. 148 (1997), 151–198.
• [21] J. Weil, De l’importance d’être constant, Talk presented at the ”Journées MEDICIS”, Luminy, 1992.

Uwagi

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę.