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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The removal of $\pi$ from some undecidable problems involving elementary functions


Author: M. Laczkovich
Journal: Proc. Amer. Math. Soc. 131 (2003), 2235-2240
MSC (2000): Primary 03B25, 03D40; Secondary 26A09
Published electronically: October 18, 2002
MathSciNet review: 1963772
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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

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06753-9
PII: S 0002-9939(02)06753-9
Keywords: Undecidable problems, rings of elementary functions
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.
Article copyright: © Copyright 2002 American Mathematical Society