Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 095, 16 pages      arXiv:2104.09548      https://doi.org/10.3842/SIGMA.2021.095

Real Liouvillian Extensions of Partial Differential Fields

Teresa Crespo a, Zbigniew Hajto b and Rouzbeh Mohseni b
a) Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain
b) Faculty of Mathematics and Computer Science, Jagiellonian University, ul. L ojasiewicza 6, 30-348 Kraków, Poland

Received February 28, 2021, in final form October 25, 2021; Published online October 29, 2021

Abstract
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.

Key words: real Liouvillan extension; real and $p$-adic Picard-Vessiot theory; split solvable algebraic group; gradient dynamical systems; integrability.

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References

  1. Bochnak J., Coste M., Roy M.-F., Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Vol. 36, Springer-Verlag, Berlin, 1998.
  2. Borel A., Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, Vol. 126, Springer-Verlag, New York, 1991.
  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.
  4. Colding T.H., Minicozzi II W.P., Analytical properties for degenerate equations, in Geometric analysis, Progr. Math., Vol. 333, Birkhäuser/Springer, Cham, 2020, 57-70, arXiv:1804.08999.
  5. Crespo T., Hajto Z., Algebraic groups and differential Galois theory, Graduate Studies in Mathematics, Vol. 122, Amer. Math. Soc., Providence, RI, 2011.
  6. Crespo T., Hajto Z., Picard-Vessiot theory and the Jacobian problem, Israel J. Math. 186 (2011), 401-406.
  7. Crespo T., Hajto Z., Real Liouville extensions, Comm. Algebra 43 (2015), 2089-2093, arXiv:1206.2283.
  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.
  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.
  10. Dubrovin B.A., Fomenko A.T., Novikov S.P., Modern geometry - methods and applications. Part I. The geometry of surfaces, transformation groups, 2nd ed., and fields, Graduate Texts in Mathematics, Vol. 93, Springer-Verlag, New York, 1992.
  11. Gel'fond O.A., Khovanskii A.G., Real Liouville functions, Funct. Anal. Appl. 14 (1980), 122-123.
  12. Gillet H., Gorchinskiy S., Ovchinnikov A., Parameterized Picard-Vessiot extensions and Atiyah extensions, Adv. Math. 238 (2013), 322-411, arXiv:1110.3526.
  13. Grothendieck A., Esquisse d'un programme, in Geometric Galois Actions, 1, Editors L. Schneps, P. Lochak, London Math. Soc. Lecture Note Ser., Vol. 242, Cambridge University Press, Cambridge, 1997, 5-48, English translation on pp. 243-283.
  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.
  16. Khovanskii A.G., Fewnomials, Translations of Mathematical Monographs, Vol. 88, Amer. Math. Soc., Providence, RI, 1991.
  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.
  18. Kolchin E.R., Picard-Vessiot theory of partial differential fields, Proc. Amer. Math. Soc. 3 (1952), 596-603.
  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.
  20. Łojasiewicz S., On semi-analytic and subanalytic geometry, in Panoramas of Mathematics (Warsaw, 1992/1994), Banach Center Publ., Vol. 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.
  22. Prestel A., Lectures on formally real fields, Lecture Notes in Math., Vol. 1093, Springer-Verlag, Berlin, 1984.
  23. Prestel A., Roquette P., Formally $p$-adic fields, Lecture Notes in Math., Vol. 1050, Springer-Verlag, Berlin, 1984.
  24. van der Put M., Singer M.F., Galois theory of linear differential equations, Grundlehren der mathematischen Wissenschaften, Vol. 328, Springer-Verlag, Berlin, 2003.

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