Mathematische Annalen

, Volume 267, Issue 2, pp 213–238

Power series solutions of algebraic differential equations

  • J. Denef
  • L. Lipshitz
Article

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References

  1. 1.
    Aberth, O.: The failure in computable analysis of a classical existence theorem for differential equations. Proc. AMS30, 151–156 (1971)Google Scholar
  2. 2.
    Artin, M.: On the solutions of analytic equations. Invent. Math.5, 277–291 (1968)Google Scholar
  3. 3.
    Artin, M.: Algebraic approximation of structures over complete local rings. Publ. Math. I.H.E.S.36, 23–58 (1969)Google Scholar
  4. 4.
    Ax, J., Kochen, S.: Diophantine problems over local fields III. Ann. Math.83, 437–456 (1966)Google Scholar
  5. 5.
    Becker, J., Denef, J., Lipshitz, L.: Further remarks on the elementary theory of formal power series rings. In: Model theory of algebra and arithmetic. Proceedings Karpacz, Poland (1979). Lecture Notes in Mathematics, Vol. 834. Berlin, Heidelberg, New York: Springer 1979Google Scholar
  6. 6.
    Becker, J., Denef, J., Lipshitz, L., van den Dries, L.: Ultraproducts and approximation in local rings I.Invent. Math.51, 189–203 (1979)Google Scholar
  7. 7.
    Birch, B.J., McCann, K.: A criterion for thep-adic solubility of Diophantine equations. Quart. J. Math. Oxford18, 54–63 (1967)Google Scholar
  8. 8.
    Blum, L.: Ph. D. Thesis, M.I.T. (1968)Google Scholar
  9. 9.
    Denef, J.: The diophantine problem for polynomial rings and fields of rational functions. Trans. A.M.S.242, 391–399 (1978)Google Scholar
  10. 10.
    Denef, J., Lipshitz, L.: Ultraproducts and approximation in local rings II. Math. Ann.253, 1–28 (1980)Google Scholar
  11. 11.
    Greenberg, M.: Rational points in henselian discrete valuation rings. Publ. Math. I.H.E.S.31, 59–64 (1966)Google Scholar
  12. 12.
    Hurwitz, A.: Sur le développement des fonctions satisfaisant à une équation differentielle algébrique. Ann. Ecole Norm. Sup.6, 327–332 (1889)Google Scholar
  13. 13.
    Kaplansky, I.: An introduction to differential algebra. Actualités Scientifiques et Industrielles No. 1251. Paris: Hermann 1957Google Scholar
  14. 14.
    Lang, S.: Algebra. New York: Addison Wesley 1965Google Scholar
  15. 15.
    Mahler, K.: Lectures on transcendental numbers. Lecture notes in Mathematics, Vol. 546. Berlin, Heidelberg, New York: Springer 1976Google Scholar
  16. 16.
    Maillet, E.: Sur les series divergentes et les équations differentielles. Ann. Sci. Ecole Norm. Sup. Ser. 3,20, 487–518 (1903)Google Scholar
  17. 17.
    Malgrange, B.: Sur les points singuliers des équations differentielles. L'Enseignement Math.20, 147–176 (1974)Google Scholar
  18. 18.
    Matijaseviĉ, Yu.: Enumerable sets are diophantine. Dokl. Akad. Nauk SSSR191, 279–282 (1970)=Sov. Math. Dokl.11, 354–357 (1970)Google Scholar
  19. 19.
    Mumford, D.: Introduction to algebraic geometry. Mimeographed notesGoogle Scholar
  20. 20.
    Popken J.: Über arithmetische Eigenschaften analytisches Functionen. Amsterdam: North-Holland 1935Google Scholar
  21. 21.
    Pour-el, M., Richards, I.: A computable ordinary differential equation which possesses no computable solution. Ann. Math. Logic17, 61–90 (1979)Google Scholar
  22. 22.
    Pour-el, M., Richards, I.: The wave equation with computable initial data such that its unique solution is not computable. Adv. Math.39, 215–239 (1981)Google Scholar
  23. 23.
    Ritt, J.F.: Differential algebra. AMS colloquium publications XXXIII (1950)Google Scholar
  24. 24.
    Robinson, A.: On the concept of a differentially closed field. Bull. Res. Counc. Israel8F, 113–128 (1959)Google Scholar
  25. 25.
    Rubel, L.: An elimination theorem for systems of algebraic differential equations. Houston J. of Math.8, 289–295 (1982)Google Scholar
  26. 26.
    Seidenberg, A.: An elimination theory for differential algebra, Cal. U. Publ. Math. (New Ser)3, 31–36 (1956)Google Scholar
  27. 27.
    Sibuya, Y., Sperber, S.: Arithmetic properties of power series solutions of algebraic differential equations. Ann. Math.113, 111–157 (1981)Google Scholar
  28. 28.
    Singer, M.: The model theory of ordered differential fields. J. Symb. Logic43, 82–91 (1978)Google Scholar
  29. 29.
    Wood, C.: The model theory of differential fields revisited. Israel J. of Math.25, 331–352 (1976)Google Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • J. Denef
    • 1
    • 2
  • L. Lipshitz
    • 1
    • 2
  1. 1.Department of MathematicsUniversity of LeuvenHeverleeBelgium
  2. 2.Department of MathematicsPurdue UniversityWest LafayetteUSA

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