Abstract:
An algebraic technique is presented that does not use results of model theory and makes it possible to construct a general Galois theory of arbitrary nonlinear systems of partial differential equations. The algebraic technique is based on the search for prime differential ideals of special form in tensor products of differential rings. The main results demonstrating the work of the technique obtained are the theorem on the constructedness of the differential closure and the general theorem on the Galois correspondence for normal extensions.
Bibliography: 14 titles.
\Bibitem{Tru10}
\by D.~V.~Trushin
\paper Splitting fields and general differential Galois theory
\jour Sb. Math.
\yr 2010
\vol 201
\issue 9
\pages 1323--1353
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Linking options:
https://www.mathnet.ru/eng/sm7581
https://doi.org/10.1070/SM2010v201n09ABEH004114
https://www.mathnet.ru/eng/sm/v201/i9/p77
This publication is cited in the following 10 articles:
Arreche C.E., Zhang Y., “Computing Differential Galois Groups of Second-Order Linear Q-Difference Equations”, Adv. Appl. Math., 132 (2022), 102273
Lucia Di Vizio, “Action of an endomorphism on (the solutions of) a linear differential equation”, Publications mathématiques de Besançon. Algèbre et théorie des nombres, 2019, no. 1, 21
Arreche C.E., “Computation of the Difference-Differential Galois Group and Differential Relations Among Solutions For a Second-Order Linear Difference Equation”, Commun. Contemp. Math., 19:6 (2017), 1650056
Arreche C.E., “On the computation of the parameterized differential Galois group for a second-order linear differential equation with differential parameters”, J. Symbolic Comput., 75:SI (2016), 25–55
A. Minchenko, A. Ovchinnikov, M. F. Singer, “Unipotent differential algebraic groups as parameterized differential Galois groups”, J. Inst. Math. Jussieu, 13:4 (2014), 671–700
C. E. Arreche, “Computation of the unipotent radical of the differential Galois group for a parameterized second-order linear differential equation”, Adv. in Appl. Math., 57 (2014), 44–59
Minchenko A., Ovchinnikov A., “Extensions of differential representations of $\mathbf{SL}_2$ and tori”, J. Inst. Math. Jussieu, 12:1 (2013), 199–224
H. Gillet, S. Gorchinskiy, A. Ovchinnikov, “Parameterized Picard–Vessiot extensions and Atiyah extensions”, Adv. in Math., 238 (2013), 322–411
C. E. Arreche, “A Galois-theoretic proof of the differential transcendence of the incomplete Gamma function”, J. Algebra, 389 (2013), 119–127
Mariya Bessonov, Alexey Ovchinnikov, Maxwell Shapiro, Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation, 2013, 45
Citation in format AMSBIB
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