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Computing in Transcendental Extensions

  • A. C. Norman
Part of the Computing Supplementum book series (COMPUTING, volume 4)

Abstract

Performing the rational operations in a field extended by a transcendental element is equivalent to performing arithmetic in the field of rational functions over the field. The computational difficulty associated with such extensions is in verifying that proposed extensions are transcendental. When the extensions being considered are functions, and where a differentiation operator can be defined for them, structure theorems can be used to determine the character of the extension and to exhibit a relationship between the adjoined element and existing quantities in case the adjoined element is not transcendental.

Keywords

Structure Theorem Transcendental Function Rational Operation Diophantine Approximation Constant 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]
    Caviness, B. F.: Methods for Symbolic Computation with Transcendental Functions. MAXIMIN 1977, 16–43.Google Scholar
  2. [2]
    Kaplansky, I.: An Introduction to Differential Algebra. Paris: Hermann 1957.MATHGoogle Scholar
  3. [3]
    Lang, S.: Transcendental Numbers and Diophantine Approximations. Bull. AMS 77/5, 635–677.Google Scholar
  4. [4]
    Rothstein, M., Caviness, B. F.: A Structure Theorem for Exponential and Primitive Functions. SIAM J. Computing 8/3, 357–367 (1979).MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Wien 1982

Authors and Affiliations

  • A. C. Norman
    • 1
  1. 1.Computer LaboratoryUniversity of CambridgeCambridgeUK

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