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On transcendental functions arising from integrating differential equations in finite terms. (English. Russian original) Zbl 1337.12001

J. Math. Sci., New York 209, No. 6, 935-952 (2015); translation from Zap. Nauchn. Semin. POMI 432, 196-223 (2015).
Let \(\mathfrak{D}\) be a simply connected domain of the complex plane \(\mathbb{C}\) and \(k\) be the field of meromorphic in \(\mathfrak{D}\) functions. The paper discusses announced earlier (see [M. D. Malykh, “On integrals of ordinary differential equations systems which are representable in finite terms”, Vestn. RUDN. Ser. Mat. Inform. Fiz. 2014, No. 3, 11–16 (2014)]) “version of Galois theory” for systems of nonlinear differential equations of the form \[ {{P}_{1}}=0,\ldots ,{{P}_{m}}=0,\tag{1} \] where \({{P}_{1}},\ldots ,{{P}_{m}}\in k[{{Y}_{1}},\ldots ,{{Y}_{n}},{{Y}_{1}}^{\prime },\ldots ,{{Y}_{n}}^{\prime }]\). The solutions of system (1) are taken from some sufficiently large algebraically closed extension \(K\) of \(k\) (for example, \(K\) is a differential closure of \(k\)). It is also assumed that the field of constants \(K\) and \(k\) coincide, an ideal \(\mathfrak{p}\) generated by \({{P}_{1}},\ldots ,{{P}_{m}}\) in \(K[{{Y}_{1}},\ldots ,{{Y}_{n}},{{Y}_{1}}^{\prime },\ldots ,{{Y}_{n}}^{\prime }]\) is prime, the system is totally consistent and closed.
The author distinguishes in the field of rational functions \(\operatorname{Re} (K[{{Y}_{1}},\ldots ,{{Y}_{n}},{{Y}_{1}}^{\prime },\ldots ,{{Y}_{n}}^{\prime }]/\mathfrak{p})\) subfield of functions that are constant on the solutions of the system (1). Such functions (other than constants) is called rational integrals of the system. He proved a theorem:
{ Theorem 6. } If a totally consistent closed system of differential equations allows \(m\) rational integrals, then its field of integrals is isomorphic to the field of rational functions on a hypersurface in the affine space of dimension \(m\) over the field \(\mathbb{C}\).
In turn coefficients of the functions representing the rational integrals generate some extension \({k}'\) of the field \(k\) in the field \(K\). It is proved that this field is finitely generated over \(k\) and it is a differential field. Moreover its group \(k\)- automorphisms is contained in the group \(\mathbb{C}\)-automorphisms of the field integrals. The thus obtained extensions \({k}'\) are offered as normal extensions for the new version of Galois theory.

MSC:

12H05 Differential algebra
34A34 Nonlinear ordinary differential equations and systems
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions

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