Exponential rings and Schanuel's Conjecture

In last years Schanuel's Conjecture (SC) has played a fundamental role in the Theory of Transcendental Numbers and on in decidability issues. Macintyre and Wilkie proved the decidability of the real exponential field, modulo (SC), solving in this way a problem left opened by A. Tarski. Moreover, Macintyre proved that the exponential subring of R generated by 1 is free on no generators. In this line of research we obtained that in the exponential ring (C, e^x), there are not further relations except i2 = -1 and e^{i\pi} = -1 modulo SC. Assuming Schanuel's Conjecture we proved that the E-subring of R generated by \pi is isomorphic to the free E-ring on \pi. These results have consequences in decidability issues both on (C, e^x) and (R, e^x). Moreover, we generalize the previous results obtaining, without assuming Schanuel's conjecture, that the E-subring generated by a real number not definable in the real exponential field is freely generated. We also obtain a similar result for the complex exponential field.