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Second proof of the irreducibility of the first differential equation of painlevé

Published online by Cambridge University Press:  22 January 2016

Hiroshi Umemura
Hiroshi Umemura*
Affiliation:
Department of Mathematics, Kumamoto University, Kumamoto, 860, Japan
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In our paper [U2], we proved the irreducibility of the first differential equation y″ = 6y2 + x of Painlevé. In that paper we explained the origin of the problem and the importance of giving a rigorous proof. We can say that our method in [U2] is algebraic and finite dimensional in contrast to a prediction of Painlevé who expected a proof depending on the infinite dimensional differential Galois theory. Even nowadays the latter remains to be established. It seems that Painlevé needed an armament with the general theory (the infinite dimensional differential Galois theory) in the controversy with R. Liouville on the mathematical foundation of the proof of the irreducibility of the first differential equation (1902-03).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 117 , March 1990 , pp. 125 - 171
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1990

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