Power Series Solutions of Non-linear q-Difference Equations and the Newton–Puiseux Polygon

• J. Cano ; P. Fortuny Ayuso ^[1]
  1. [1] Universidad de Oviedo

     Universidad de Oviedo

     Oviedo, España

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 21, Nº 4, 2022
• Idioma: inglés
• Enlaces
  • Texto completo (pdf)
• Resumen

  • Adapting the Newton–Puiseux Polygon process to nonlinear q-difference equations of any order and degree, we compute their power series solutions, study the properties of the set of exponents of the solutions and give a bound for their q-Gevrey order in terms of the order of the original equation.

• Referencias bibliográficas
  • 1. Adams, C.R.: On the linear ordinary q-difference equation. Ann. Math. 30(1), 195–205 (1928–1929)
  • 2. Adams, C.R.: Linear q-difference equations. Bull. Am. Math. Soc. 31, 361–400 (1931)
  • 3. Barbe, P., Cano, J., Ayuso, P. Fortuny., McCormick, W. C.: q-algebraic equations, their power series solutions, and the asymptotic behavior...
  • 4. Bézivin, Jean-Paul.: Convergence des solutions formelles de certaines équations fonctionnelles. Aequationes Math. 44(1), 84–99 (1992)
  • 5. Bézivin, Jean-Paul.: Sur les équations fonctionnelles aux q-différences. Aequationes Math. 43(2–3), 159–176 (1992)
  • 6. Bézivin, Jean-Paul., Boutabaa, Abdelbaki: Sur les équations fonctionelles p-adiques aux q-différences. Collect. Math. 43(2), 125–140 (1992)
  • 7. Cano, J.: On the series defined by differential equations, with an extension of the Puiseux Polygon construction to these equations. Analysis...
  • 8. Cano, J., Fortuny Ayuso, P.: Power series solutions of non-linear q-difference equations and the Newton–Puiseux polygon (2012) arXiv: 1209.0295v1
  • 9. Cano, J., Fortuny Ayuso, P.: The space of generalized formal power series solution of an ordinary differential equation. Astérisque 323,...
  • 10. Christensen, C.: Newton’s method for resolving affected equations. Coll. Math. J. 27(5), 330–340 (1996)
  • 11. Della Dora, J., Richard-Jung, F.: About the newton polygon algorithm for non linear ordinary differential equations. In: ISSAC, pp. 298–304...
  • 12. Enriques, F.: Lezioni Sulla Teoria Geometrica delle Equazioni e delle Funzioni Algebriche, Libro Quarto. Zanichelli, Bologna (1915)
  • 13. Fine, H.B.: On the functions defined by differential equations, with an extension of the Puiseux Polygon construction to these equations....
  • 14. Forsyth, A.R.: Theory of Differential Equations: Part II. Cambridge University Press, Cambridge (1900)
  • 15. Gérard, R.: Sur le théorème de maillet. Funkcial. Ekvac. 34, 117–125 (1991)
  • 16. Grigor’ev, DYu., Singer, M.: Solving ordinary differential equations in terms of series with real exponents. Trans. AMS 327(1), 329–351...
  • 17. Hahn, H.: Über die nichtarchimedischen Größensysteme, Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Wien. Mathematisch...
  • 18. Hironaka, H.: Characteristic Polyhedra of Singularities. J. Math. Kyoto Univ. 7, 251–293 (1967)
  • 19. Ince, E.L.: Ordinary Differential Equations. Dover, New York (1956)
  • 20. Li, X., Zhang, C.: Existence of analytic solutions to analytic nonlinear q-difference equations. J. Math. Anal. Appl. 2, 412–417 (2011)
  • 21. Mahler, K.: On formal power series as integrals of algebraic differential equations. Lincei–Rend. Sc. Fis. Mat. et Nat L, 76–89 (1971)
  • 22. Maillet, E.: Sur les séries divergentes et les équations différentielles. Ann. Sci. École Norm. Sup. 20, 487–518 (1903)
  • 23. Malgrange, B.: Sur le théoreme de maillet. Asymptot. Anal. 2, 1–4 (1989)
  • 24. Puiseux, V.A.: Recherches sur les fonctions algébriques. J. Math. Pures Appl. 15, 365–480 (1850)
  • 25. Puiseux, V.A.: Recherches sur les fonctions algébriques. J. Math. Pures Appl. 16, 228–240 (1851)
  • 26. Ramis, J.-P.: About the growth of entire functions solutions linear algebraic q-difference equations. Ann. Fac. Sci. Toulouse, 6 série...
  • 27. Ribenboim, P.: Noetherian rings of generalized power series. J. Pure Appl. Algebra 79, 293–312 (1992)
  • 28. Sauloy, J.: La filtration canonique par les pentes d’un module aux q-différences et le gradué associé. Ann. Inst. Fourier (Grenoble) 54(1),...
  • 29. Seidenberg, A.: Reduction of singularities of the differential equation Ady = Bdx. Am. J. Math. 90, 248–269 (1968)
  • 30. van der Hoeven, J.: Operators on generalized power series. J. Univ. Ill. 45(4), 1161–1190 (2001)
  • 31. van der Hoeven, J.: Transseries and Real Differential Algebra, LNM, vol. 1888. Springer, Berlin (2006)
  • 32. Zhang, C.: Sur un théorème de maillet-malgrange pour les équations q-différences–différentielles. Asymptot. Anal. 17, 309–314 (1998)