On Transcendental Functions Arising from Integrating Differential Equations in Finite Terms

In this paper, we discuss a version of Galois theory for systems of ordinary differential equations in which there is no fixed list of allowed transcendental operations. We prove a theorem saying that the field of integrals of a system of differential equations is equivalent to the field of rational...

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Bibliographic Details
Published in:Journal of mathematical sciences (New York, N.Y.) N.Y.), 2015-09, Vol.209 (6), p.935-952
Main Author: Malykh, M. D.
Format: Article
Language:English
Subjects:
Differential equations
Mathematics
Mathematics and Statistics
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Summary:In this paper, we discuss a version of Galois theory for systems of ordinary differential equations in which there is no fixed list of allowed transcendental operations. We prove a theorem saying that the field of integrals of a system of differential equations is equivalent to the field of rational functions on a hypersurface having a continuous group of birational automorphisms whose dimension coincides with the number of algebraically independent transcendentals introduced by integrating the system. The suggested construction is a development of the algebraic ideas presented by Paul Painlevé in his Stockholm lectures.
ISSN:1072-3374
1573-8795
DOI:10.1007/s10958-015-2539-6