On Transcendental Functions Arising from Integrating Differential Equations in Finite Terms

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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  1. 1.

    J. M. Borwein and R. E. Crandall, “Closed forms: what they are and why we care,” Notices Amer. Math. Soc., 60, No. 1, 50–65 (2013).

    MATH  MathSciNet  Article  Google Scholar 

  2. 2.

    M. F. Singer, “Liouvillian first integral of differential equations,” Trans. Amer. Math. Soc., 333, No. 2, 673–688 (1992).

    MATH  MathSciNet  Article  Google Scholar 

  3. 3.

    G. Casale, “Liouvillian first integrals of differential equations,” Banach Center Publ., 94, 153–161 (2011).

    MathSciNet  Article  Google Scholar 

  4. 4.

    J. Moses, “Symbolic integration,” AI Technical Reports (1967).

  5. 5.

    E. S. Cheb-Terrab, L. G. S. Duarte, and L. A. C. P. da Mota, “Computer algebra solving of first order ODEs using symmetry methods,” Comput. Phys. Comm., 101, 254–267 (1997).

    MATH  MathSciNet  Article  Google Scholar 

  6. 6.

    E. S. Cheb-Terrab and A. D. Roche, “Symmetries and first order ODE patterns,” Comput. Phys. Comm., 113, 239–260 (1998).

    MATH  Article  Google Scholar 

  7. 7.

    E. S. Cheb-Terrab and A. D. Roche, “Integrating factors for second order ODEs,” J. Symbolic Comput., 27, No. 5, 501–519 (1999).

    MATH  MathSciNet  Article  Google Scholar 

  8. 8.

    E. S. Cheb-Terrab and T. Kolokolnikov, “First order ODEs, symmetries and linear transformations,” European J. Appl. Math., 14, No. 2, 231–246 (2000).

    MathSciNet  Article  Google Scholar 

  9. 9.

    D. D. Mordukhai-Boltovskoi, Commentary to Euclid, in: Euclid, Elements [in Russian], Books 1–6, Moscow (1955).

  10. 10.

    N. A. Kudryashov, Analytic Theory of Nonlinear Differential Equations [in Russian], Moscow (2004).

  11. 11.

    A. S. Fokas, A. R. Its, A. A. Kapaev, and V. Yu. Novokshenov, Painlevé Transcendents: The Riemann–Hilbert Approach, Amer. Math. Soc. (2006).

  12. 12.

    S. L. Sobolevsky, Movable Singularities of Solutions of Ordinary Differential Equations [in Russian], Minsk (2006).

  13. 13.

    V. V. Golubev, Vorlesungen über Differentialgleichungen im Komplexen, VEB Deutscher Verlag der Wissenschaften, Berlin (1958).

  14. 14.

    L. Schlesinger, Einführung in die Theorie der gewöhnlichen Differentialgleichungen, VWV, Leipzig (1922).

  15. 15.

    N. N. Parfentiev, “A review on the work by Prof. Schlesinger from Giessen,” Izvestiya Fiz.-Mat. Obshchestva pri Imperat. Kazan. Universitete, Ser. 2, XVIII, 4 (1912).

  16. 16.

    P. Painlevé, Leçons sur la theorie analytique des equations differentielles, Paris (1897); Œuvres, T. 1, Paris (1971).

  17. 17.

    P. Painlevé, “Memoire sur les equations differentielles du premier ordre,” in: Œuvres, T. 2, Paris (1974), pp. 237–461.

  18. 18.

    P. Painlevé, Appendix to P. Boutroux’s book, in: Œuvres, T. 2, Paris (1974), pp. 767–813.

  19. 19.

    A. N. Bogolyubov and M. D. Malykh, Transcendental functions introduced by integrating differential equations, in: Dynamics of Complex Systems. XXI Century [in Russian], No. 3 (2010).

  20. 20.

    H. Umemura, “Birational automorphism groups and differential equations,” Nagoya Math. J., 119, 1–80 (1990).

    MATH  MathSciNet  Google Scholar 

  21. 21.

    L. Königsberger, Lehrbuch der Theorie der Differentialgleichungen mit einer unabhägigen Variabeln, Tuebner, Leipzig (1889).

  22. 22.

    L. Königsberger, Die Principien der Mechanik, Tuebner, Leipzig (1901).

  23. 23.

    A. M. Vinogradov and I. S. Krasil’shchik (eds.), Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, Amer. Math. Soc. (1999).

  24. 24.

    R. Hartshorne, Algebraic Geometry, Springer (1977).

  25. 25.

    O. Zariski and P. Samuel, Commutative Algebra, D. van Nostrand Company (1958).

  26. 26.

    D. Maclagan and B. Sturmfels, Introduction to Tropical Geometry, http://homepages.warwick.ac.uk/staff/D.Maclagan/papers/TropicalBook.html.

  27. 27.

    C. L. Siegel and J. K. Moser, Lectures on Celestial Mechanics (1971).

  28. 28.

    J. Liouville, “Memoire sur l’integration d’une classe de fonctions transcendantes,” J. Reine Angew. Math., 13, 93–118 (1835).

    MATH  MathSciNet  Article  Google Scholar 

  29. 29.

    J. F. Ritt, Integration in Finite Terms: Liouville’s Theory of Elementary Methods, Columbia Univ. Press, New York (1949).

    Google Scholar 

  30. 30.

    A. Khovanskii, Topological Galois Theory, Springer (2014).

  31. 31.

    M. Bronstein, Symbolic Integration I: Transcendental Functions, Springer, Berlin (1999).

    Google Scholar 

  32. 32.

    N. G. Chebotarev, The Theory of Algebraic Functions [in Russian], Moscow (2013).

  33. 33.

    G. Castelnuovo and F. Enriques, “Die algebraischen Flachen vom Gesichtspunkte der birationalen Transformationen aus,” in: Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, III C 6b, Teubner, Leipzig, 1903–1932.

  34. 34.

    F. Enriques, Le superficie algebriche, Bologna (1946).

  35. 35.

    Yu. G. Prokhorov, “The Cremona group and its subgroups,” A talk in the Moscow Mathematical Society, March 26, 2013.

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Correspondence to M. D. Malykh.

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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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  • Basic Function
  • Rational Function
  • Galois Theory
  • Transcendental Function
  • Rational Integral