  Assuming Schanuel's conjecture, we prove that any polynomial exponential
equation in one variable must have a solution that is transcendental over a
given finitely generated field. With the help of some recent results in
Diophantine geometry, we obtain the result by proving (unconditionally) that
certain polynomial exponential equations have only finitely many rational
solutions.
  This answers affirmatively a question of David Marker, who asked, and proved
in the case of algebraic coefficients, whether at least the one-variable case
of Zilber's strong exponential-algebraic closedness conjecture can be reduced
to Schanuel's conjecture.
