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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The removal of $\pi$ from some undecidable problems involving elementary functions
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by M. Laczkovich PDF
Proc. Amer. Math. Soc. 131 (2003), 2235-2240 Request permission

Abstract:

We show that in the ring generated by the integers and the functions $x, \ \sin x^{n}$ and $\sin (x\cdot \sin x^{n})$ $(n=1,2,\ldots )$ defined on $\mathbf {R}$ it is undecidable whether or not a function has a positive value or has a root. We also prove that the existential theory of the exponential field $\mathbf {C}$ is undecidable.
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Additional Information
  • M. Laczkovich
  • Affiliation: Department of Analysis, Eötvös Loránd University, Budapest, Pázmány Péter sétány 1/C 1117, Hungary
  • Email: laczko@renyi.hu
  • Received by editor(s): February 7, 2002
  • Received by editor(s) in revised form: February 22, 2002
  • Published electronically: October 18, 2002
  • Additional Notes: This research was partially supported by the Hungarian National Foundation for Scientific Research Grant No. T032042
  • Communicated by: Carl G. Jockusch, Jr.
  • © Copyright 2002 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 131 (2003), 2235-2240
  • MSC (2000): Primary 03B25, 03D40; Secondary 26A09
  • DOI: https://doi.org/10.1090/S0002-9939-02-06753-9
  • MathSciNet review: 1963772