Reduction of Transcendental Decision Problems over the Reals

A special class of univariate transcendental decision problems called "trigonometric extension" is studied in this paper. Roughly speaking, a trigonometric extension is a ring of univariate analytic functions obtained by adjoining trigonometric functions to a ring consisting of functions having only finitely many real zeros. It is shown that in this case, the decision problem can be reduced to looking for solutions in a bounded domain. Based on the reduction, several new decidability results are established when Schanuel's Conjecture is assumed. Furthermore, for the theory of multivariate trigonometric extension, it is proved that, although a small fragment of the theory can be reduced to the univariate case, the general theory is undecidable.