The removal of 𝜋 from some undecidable problems involving elementary functions
Miklós Laczkovich · Proceedings of the American Mathematical Society · 2002
We show that in the ring generated by the integers and the functions x , sin x n x, \ \sin x^{n} and sin ( x ⋅ sin x n ) \sin (x\cdot \sin x^{n}) ( n = 1 , 2 , … ) (n=1,2,\ldots ) defined on R \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 C \mathbf {C} is undecidable.