Some Consequences of Schanuel's Conjecture in Exponential Rings

On freely generated E-subrings

“…Here we prove our main result using the ideas and the techniques from [10] together with the versions of Schanuel's Conjecture for real and complex numbers not definable without parameters in o-minimal expansions of the real exponential field proved in the last section.…”

“…In this paper we generalize the result from [10] to any real number not definable without parameters in the real exponential field, without using Schanuel's Conjecture: Theorem 1.1. Let τ be a real number not definable without parameters in the real exponential field, then the E-subring of R generated by τ is isomorphic to the free E-ring on τ .…”

“…In particular if Schanuel's Conjecture is true, then π is not definable without parameters in the real exponential field, so Theorem 1.1 implies the result for π in [10] (although the technique is similar).…”

“…The second author proved in [10] that the E-subring generated by π , modulo Schanuel's Conjecture, is isomorphic to the free exponential ring on π . Recall that:…”

From Schanuel’s Conjecture to Shapiro’s Conjecture

“…Schanuel's Conjecture has played a crucial role in exponential algebra (see [10], [16]), and in the model theory of exponential fields (see [11], [17], [12]).…”