Journal of Mathematical Sciences

, Volume 209, Issue 6, pp 935–952 | Cite as

On Transcendental Functions Arising from Integrating Differential Equations in Finite Terms

  • M. D. MalykhEmail author

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.


Basic Function Rational Function Galois Theory Transcendental Function Rational Integral 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Moscow State University, Peoples’ Friendship University of RussiaMoscowRussia

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