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Inventiones mathematicae

, Volume 103, Issue 1, pp 1–8 | Cite as

Hilbert's Tenth Problem for fields of rational functions over finite fields

  • Thanases Pheidas
Article

Summary

We prove that there is no algorithm to solve arbitrary polynomial equations over a field of rational functions in one letter with constants in a finite field of characteristic other than 2 and hence, Hilbert's Tenth Problem for any such field is undecidable.

Keywords

Rational Function Polynomial Equation Finite Field 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Cherlin, G.: Undecidability of rational function fields in nonzero characteristic, Logic colloquium '82, Lolli, G., Logo, G., Marcja A. (eds), pp. 85–95. North Holland 1984Google Scholar
  2. 2.
    Davis, M., Matijasevic, Yu., Robinson, J.: Diophantine equations: Positive Aspects of a negative solution. Proc. Symp. Pure Math.28, 323–378 (1976)Google Scholar
  3. 3.
    Denef, J.: The Diophantine problem for polynomial rings of positive characteristic, Logic Colloquium '78, Boffa, M., van Dalen, D., McAloon, K. (eds), pp. 131–145. North Holland 1979Google Scholar
  4. 4.
    Denef, J.: The Diophantine problem for polynomial rings and fields of rational functions. Trans. Am. Math. Soc.242, 391–399 (1978)Google Scholar
  5. 5.
    Denef, J.: Diophantine sets over algebraic integer rings II. Trans. Am. Math. Soc.257, 227–336 (1980)Google Scholar
  6. 6.
    Denef, J., Lipshitz, L.: Diophantine sets over some rings of algebraic integers. J. Lond. Math. Soc.18, 385–391 (1978).Google Scholar
  7. 7.
    Ersov, Yu.: Undecidability of certain fields. Dokl. Akad. Nauk SSSR161, 349–352 (1965)Google Scholar
  8. 8.
    Lang, S.: Algebra, Reading, MA: Addison Wesley 1971Google Scholar
  9. 9.
    Matijasevic, Yu.: Enumerable sets are Diophantine. Dokl. Akad. Nauk SSSR191, 279–282 (1970)Google Scholar
  10. 10.
    Penzin, Yu.: Undecidability of fields of rational functions over fields of characteristic 2. Algebra Logic12, 205–219 (1973)Google Scholar
  11. 11.
    Pheidas, T.: An undecidability result for power series rings of positive characteristic II. Proc. Am. Math. Soc.100, 526–530 (1987)Google Scholar
  12. 12.
    Pheidas, T.: Hilbert's Tenth Problem for a class of rings of algebraic integers. Proc. Am. Math. Soc.104, 611–620 (1988)Google Scholar
  13. 13.
    Robinson, J.: Definability and decision problems in arithmetic. J. Symb. Logic14, 98–114 (1949)Google Scholar
  14. 14.
    Rumely, R.: Undecidability and definability for the theory of global fields. Trans. Am. Math. Soc.262, 195–217 (1980)Google Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Thanases Pheidas
    • 1
  1. 1.Department of MathematicsFlorida International UniversityMiamiUSA

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