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.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 432, 2015, pp. 196–223.
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Malykh, M.D. On Transcendental Functions Arising from Integrating Differential Equations in Finite Terms. J Math Sci 209, 935–952 (2015). https://doi.org/10.1007/s10958-015-2539-6
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DOI: https://doi.org/10.1007/s10958-015-2539-6