Henson and Rubel's theorem for Zilber's pseudoexponentiation

In 1984, Henson and Rubel proved the following theorem: If p(x1, ..., xn) is an exponential polynomial with coefficients in C with no zeroes in C, then p(x1, ..., xn) = e g(x1,...,xn) for some exponential polynomial g(x1, ..., xn) over C. In this paper, I will prove the analog of this theorem for Zilber’s Pseudoexponentiation directly from the axioms. Furthermore, this proof relies only on the existential closedness axiom without any reference to Schanuel’s conjecture. Macintyre, D’Aquino, and Terzo have also explored this problem and offer a semantic proof of the analog of this theorem for Zilber’s Pseudoexponentiation in [2].