From Schanuel's Conjecture to Shapiro's Conjecture

Schanuel's Conjecture is a famous, and very powerful, conjecture about algebraic relations between complex numbers and their exponentials. Shapiro's Conjecture gives a necessary and sufficient conditin for two complex exponential polynomials to have infinitely many common zeros (this is a rare event). Montgomery gave an argument to show that this was not likely to yield to conventional approximation arguments from complex analysis. I sketch a proof that Schanuel implies Shapiro.