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

SIGMA 15 (2019), 058, 15 pages      arXiv:1901.09951      https://doi.org/10.3842/SIGMA.2019.058
Contribution to the Special Issue on Algebraic Methods in Dynamical Systems

Linear Differential Systems with Small Coefficients: Various Types of Solvability and their Verification

Moulay A. Barkatou a and Renat R. Gontsov bc
a) Laboratoire XLIM (CNRS UMR 72 52), Département Mathématiques-Informatique, Université de Limoges, Faculté des Sciences et Techniques, 123 avenue Albert Thomas, F-87060 LIMOGES Cedex, France
b) Institute for Information Transmission Problems RAS, Bolshoy Karetny per. 19, build. 1, Moscow 127051, Russia
c) Moscow Power Engineering Institute, Krasnokazarmennaya 14, Moscow 111250, Russia

Received January 30, 2019, in final form July 31, 2019; Published online August 09, 2019

Abstract
We study the problem of solvability of linear differential systems with small coefficients in the Liouvillian sense (or, by generalized quadratures). For a general system, this problem is equivalent to that of solvability of the Lie algebra of the differential Galois group of the system. However, dependence of this Lie algebra on the system coefficients remains unknown. We show that for the particular class of systems with non-resonant irregular singular points that have sufficiently small coefficient matrices, the problem is reduced to that of solvability of the explicit Lie algebra generated by the coefficient matrices. This extends the corresponding Ilyashenko-Khovanskii theorem obtained for linear differential systems with Fuchsian singular points. We also give some examples illustrating the practical verification of the presented criteria of solvability by using general procedures implemented in Maple.

Key words: linear differential system; non-resonant irregular singularity; formal exponents; solvability by generalized quadratures; triangularizability of a set of matrices.

pdf (373 kb)   tex (20 kb)  

References

  1. Barkatou M.A., An algorithm to compute the exponential part of a formal fundamental matrix solution of a linear differential system, Appl. Algebra Engrg. Comm. Comput. 8 (1997), 1-23.
  2. Barkatou M.A., Cluzeau T., On simultaneous triangularization of a set of matrices, in preparation.
  3. Barkatou M.A., Cluzeau T., Weil J.A., Di Vizio L., Computing the Lie algebra of the differential Galois group of a linear differential system, in Proceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2016, 63-70.
  4. Barkatou M.A., Pflügel E., ISOLDE: a Maple package for solving systems of linear ODEs (1996), available at http://sourceforge.net/projects/isolde/.
  5. Compoint E., Singer M.F., Computing Galois groups of completely reducible differential equations, J. Symbolic Comput. 28 (1999), 473-494.
  6. Feng R., Hrushovski's algorithm for computing the Galois group of a linear differential equation, Adv. in Appl. Math. 65 (2015), 1-37, arXiv:1312.5029.
  7. Gontsov R.R., On the dimension of the subspace of Liouvillian solutions of a Fuchsian system, Math. Notes 102 (2017), 149-155.
  8. Gontsov R.R., Vyugin I.V., Solvability of linear differential systems with small exponents in the Liouvillian sense, Arnold Math. J. 1 (2015), 445-471.
  9. Hrushovski E., Computing the Galois group of a linear differential equation, in Differential Galois Theory (Bęedlewo, 2001), Banach Center Publ., Vol. 58, Polish Acad. Sci. Inst. Math., Warsaw, 2002, 97-138.
  10. Humphreys J.E., Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York - Berlin, 1972.
  11. Kaplansky I., An introduction to differential algebra, Actualités Sci. Ind., Vol. 1251, Hermann, Paris, 1957.
  12. Khovanskii A.G., On solvability and unsolvability of equations in explicit form, Russian Math. Surveys 59 (2004), 661-736.
  13. Khovanskii A.G., Topological Galois theory: solvability and unsolvability of equations in finite terms, Springer Monographs in Mathematics, Springer, Heidelberg, 2014.
  14. Kimura T., On Riemann's equations which are solvable by quadratures, Funkcial. Ekvac. 12 (1970), 269-281.
  15. Kolchin E.R., Algebraic matric groups and the Picard-Vessiot theory of homogeneous linear ordinary differential equations, Ann. of Math. 49 (1948), 1-42.
  16. Maciejewski A., Moulin-Ollagnier J., Nowicki A., Simple quadratic derivations in two variables, Comm. Algebra 29 (2001), 5095-5113.
  17. Morales Ruiz J.J., Differential Galois theory and non-integrability of Hamiltonian systems, Progress in Mathematics, Vol. 179, Birkhäuser Verlag, Basel, 1999.
  18. van der Hoeven J., Around the numeric-symbolic computation of differential Galois groups, J. Symbolic Comput. 42 (2007), 236-264.
  19. Vidunas R., Differential equations of order two with one singular point, J. Symbolic Comput. 28 (1999), 495-520.
  20. Vyugin I.V., Gontsov R.R., On the question of solubility of Fuchsian systems by quadratures, Russian Math. Surveys 67 (2012), 585-587.
  21. Wasow W., Asymptotic expansions for ordinary differential equations, Pure and Applied Mathematics, Vol. 14, Interscience Publishers John Wiley & Sons, Inc., New York - London - Sydney, 1965.
  22. Żoładek H., Polynomial Riccati equations with algebraic solutions, in Differential Galois Theory (Będlewo, 2001), Banach Center Publ., Vol. 58, Polish Acad. Sci. Inst. Math., Warsaw, 2002, 219-231.
  23. Żoładek H., The monodromy group, Monografie Matematyczne, Vol. 67, Birkhäuser Verlag, Basel, 2006.

Previous article  Next article  Contents of Volume 15 (2019)