We give an axiomatization of the class ECF of exponentially closed fields, which includes the pseudo-exponential fields previously introduced by the second author, and show that it is superstable over its interpretation of arithmetic. Furthermore, ECF is exactly the elementary class of the pseudo-exponential fields if and only if the Diophantine conjecture CIT on atypical intersections of tori with subvarieties is true.

A careful exposition of Zilber's quasiminimal excellent classes and their categoricity is given, leading to two new results: the L ω₁ ,ω (Q)-definability assumption may be dropped, and each class is determined by its model of dimension $\aleph _{0}$.

In 1984, Henson and Rubel [2] proved the following theorem: If p(x₁, ..., x n ) is an exponential polynomial with coefficients in with no zeroes in ℂ, then $p({x_1},...,{x_n}) = {e^{g({x_{1......}}{x_n})}}$ where g(x₁......x n ) is some exponential polynomial over ℂ. In this paper, I will prove the analog of this theorem for Zilber's Pseudoexponential fields directly from the axioms. Furthermore, this proof relies only on the existential closedness axiom without any reference to Schanuel's conjecture.

We construct a new class of 1 categorical structures, disproving Zilber's conjecture, and study some of their properties.