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Real Liouvillian Extensions of Partial Differential Fields
Symmetry, integrability and geometry: methods and applications, Tome 17 (2021) Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper, we establish Galois theory for partial differential systems defined over formally real differential fields with a real closed field of constants and over formally $p$-adic differential fields with a $p$-adically closed field of constants. For an integrable partial differential system defined over such a field, we prove that there exists a formally real (resp. formally $p$-adic) Picard–Vessiot extension. Moreover, we obtain a uniqueness result for this Picard–Vessiot extension. We give an adequate definition of the Galois differential group and obtain a Galois fundamental theorem in this setting. We apply the obtained Galois correspondence to characterise formally real Liouvillian extensions of real partial differential fields with a real closed field of constants by means of split solvable linear algebraic groups. We present some examples of real dynamical systems and indicate some possibilities of further development of algebraic methods in real dynamical systems.
Keywords: real and $p$-adic Picard–Vessiot theory, gradient dynamical systems, integrability.
Mots-clés : real Liouvillan extension, split solvable algebraic group 
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     author = {Teresa Crespo and Zbigniew Hajto and Rouzbeh Mohseni},
     title = {Real {Liouvillian} {Extensions} of {Partial} {Differential} {Fields}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2021},
     volume = {17},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2021_17_a94/}
}
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Teresa Crespo; Zbigniew Hajto; Rouzbeh Mohseni. Real Liouvillian Extensions of Partial Differential Fields. Symmetry, integrability and geometry: methods and applications, Tome 17 (2021). http://geodesic.mathdoc.fr/item/SIGMA_2021_17_a94/

[1] Bochnak J., Coste M., Roy M.-F., Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 36, Springer-Verlag, Berlin, 1998 | DOI

[2] Borel A., Linear algebraic groups, Graduate Texts in Mathematics, 126, 2nd ed., Springer-Verlag, New York, 1991 | DOI

[3] Colding T. H., Minicozzi II W.P., “Arnold–Thom gradient conjecture for the arrival time”, Comm. Pure Appl. Math., 72 (2019), 1548–1577, arXiv: 1712.05381 | DOI

[4] Colding T. H., Minicozzi II W.P., “Analytical properties for degenerate equations”, Geometric analysis, Progr. Math., 333, Birkhäuser/Springer, Cham, 2020, 57–70, arXiv: 1804.08999 | DOI

[5] Crespo T., Hajto Z., Algebraic groups and differential Galois theory, Graduate Studies in Mathematics, 122, Amer. Math. Soc., Providence, RI, 2011 | DOI

[6] Crespo T., Hajto Z., “Picard–Vessiot theory and the Jacobian problem”, Israel J. Math., 186 (2011), 401–406 | DOI

[7] Crespo T., Hajto Z., “Real Liouville extensions”, Comm. Algebra, 43 (2015), 2089–2093, arXiv: 1206.2283 | DOI

[8] Crespo T., Hajto Z., Sowa-Adamus E., “Galois correspondence theorem for Picard–Vessiot extensions”, Arnold Math. J., 2 (2016), 21–27, arXiv: 1502.08026 | DOI

[9] Crespo T., Hajto Z., van der Put M., “Real and $p$-adic Picard–Vessiot fields”, Math. Ann., 365 (2016), 93–103, arXiv: 1307.2388 | DOI

[10] Dubrovin B. A., Fomenko A. T., Novikov S. P., Modern geometry – methods and applications, v. I, Graduate Texts in Mathematics, 93, The geometry of surfaces, transformation groups, and fields, 2nd ed., Springer-Verlag, New York, 1992 | DOI

[11] Gel'fond O. A., Khovanskii A. G., “Real Liouville functions”, Funct. Anal. Appl., 14 (1980), 122–123 | DOI

[12] Gillet H., Gorchinskiy S., Ovchinnikov A., “Parameterized Picard–Vessiot extensions and Atiyah extensions”, Adv. Math., 238 (2013), 322–411, arXiv: 1110.3526 | DOI

[13] 243–283 | DOI

[14] Hajto Z., Mohseni R., Tame topology and non-integrability of dynamical systems, arXiv: 2008.12074

[15] Kamensky M., Pillay A., “Interpretations and differential Galois extensions”, Int. Math. Res. Not., 2016 (2016), 7390–7413 | DOI

[16] Khovanskii A. G., Fewnomials, Translations of Mathematical Monographs, 88, Amer. Math. Soc., Providence, RI, 1991 | DOI

[17] Kolchin E. R., “Algebraic matric groups and the Picard–Vessiot theory of homogeneous linear ordinary differential equations”, Ann. of Math., 49 (1948), 1–42 | DOI

[18] Kolchin E. R., “Picard–Vessiot theory of partial differential fields”, Proc. Amer. Math. Soc., 3 (1952), 596–603 | DOI

[19] Kurdyka K., Mostowski T., Parusiński A., “Proof of the gradient conjecture of R Thom”, Ann. of Math., 152 (2000), 763–792, arXiv: math.AG/9906212 | DOI

[20] Łojasiewicz S., “On semi-analytic and subanalytic geometry”, Panoramas of Mathematics (Warsaw, 1992/1994), Banach Center Publ., 34, Polish Acad. Sci. Inst. Math., Warsaw, 1995, 89–104

[21] Maciejewski A. J., Przybylska M., “Differential Galois theory and integrability”, Int. J. Geom. Methods Mod. Phys., 6 (2009), 1357–1390, arXiv: 0912.1046 | DOI

[22] Prestel A., Lectures on formally real fields, Lecture Notes in Math., 1093, Springer-Verlag, Berlin, 1984 | DOI

[23] Prestel A., Roquette P., Formally $p$-adic fields, Lecture Notes in Math., 1050, Springer-Verlag, Berlin, 1984 | DOI

[24] van der Put M., Singer M. F., Galois theory of linear differential equations, Grundlehren der mathematischen Wissenschaften, 328, Springer-Verlag, Berlin, 2003 | DOI