Inventiones mathematicae

, Volume 51, Issue 2, pp 189–203 | Cite as

Ultraproducts and approximation in local rings I

  • Joseph Becker
  • J. Denef
  • L. Lipshitz
  • L. van den Dries
Article

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Artin, M.: On the solutions of analytic equations, Inventiones Math.5, 277–291 (1968)CrossRefGoogle Scholar
  2. 2.
    Artin, M.: Algebraic approximation of structures over complete local rings, Publ. Math. I.H.E.S.,36, 23–58 (1969)Google Scholar
  3. 3.
    Ax, J., Kochen, S.: Diophantine problems over local fileds I, II Amer. J. Math.,87, 605–648 (1965); III Ann. Math. (2),83, 437–456 (1966)Google Scholar
  4. 4.
    Becker, J.: A counterexample to Artin approximation with respect to subrings, Math. Ann.,230, 195–196 (1977)Google Scholar
  5. 5.
    Becker, J., Lipshitz, L.: Remarks on the elementary theories of formal and convergent power series, Fund. Math., (to appear)Google Scholar
  6. 6.
    Birch, B.J., McCann, K.: A criterion for thep-adic solubility of Diophantine equations, Quart. J. Math., Oxford, (2)18, 59–63 (1967)Google Scholar
  7. 7.
    Chang, C.C., Keisler, H.J.: Model Theory, North Holland, AmsterdamGoogle Scholar
  8. 8.
    Delon, F.: Résultats D'indécidabilité Dans Les Anneaux de Séries Formelles, (to appear)Google Scholar
  9. 9.
    Eršov, Yu.: On the elementary theory of maximal normed fields, Algebra i Logika,4, 31–69 (1965);5, 8–40 (1966);6, 31–37 (1967)Google Scholar
  10. 10.
    Greenberg, M.J.: Rational points in henselian discrete valuation rings, Publ. Math. I.H.E.S.31, 59–64 (1966)Google Scholar
  11. 11.
    Greenberg, M.J.: Strictly local solutions of diophantine equations, Pacific J. Math.,51, 143–153 (1974)Google Scholar
  12. 12.
    Lang, S.: Algebra, Addison-Wesley, 1965Google Scholar
  13. 13.
    Lascar, D.: Caractère effectif des théorèmes d'approximation de M. Artin, (to appear)Google Scholar
  14. 14.
    Luxemburg, W.A.J.: A General Theory of Monads. In: Applications of Model Theory to Algebra, Analysis and Probability, pp. 18–86, New York: Holt Rinehart and Winston, 1969Google Scholar
  15. 15.
    Nagata, M.: Local rings, Interscience, New York, 1962Google Scholar
  16. 16.
    Nerode, A.: A decision method forp-adic integral zeros of diophantine equations. Bull. A.M.S.69, 513–517 (1963)Google Scholar
  17. 17.
    Pfister, G., Popescu, D.: Die strenge Approximationseigenschaft lokaler Ringe, Inventiones Math.,30, 145–174 (1975)Google Scholar
  18. 18.
    Popescu, D.: A strong approximation theorem over discrete valuation rings, Rev. Roum. Math. Pures et Appl. Tome XX, No6, 659–692 (1975)Google Scholar
  19. 19.
    Robinson, A.: Elementary embeddings of fields of power series, Journal of Number Theory,2, 237–247 (1970)Google Scholar
  20. 20.
    van den Dries, L.: Model Theory of Fields, Thesis, Utrecht, June 1978Google Scholar

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • Joseph Becker
    • 1
  • J. Denef
    • 2
  • L. Lipshitz
    • 3
  • L. van den Dries
    • 4
  1. 1.Purdue UniversityWest LafayetteUSA
  2. 2.Princeton UniversityPrinceton
  3. 3.Institute for Advanced StudyPrincetonUSA
  4. 4.University of UtrechtUtrechtNetherlands

Personalised recommendations