The removal of 𝜋 from some undecidable problems involving elementary functions

Laczkovich, M.

doi:10.1090/S0002-9939-02-06753-9

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