Henson and Rubel's theorem for Zilber's pseudoexponentiation

Abstract

In 1984, Henson and Rubel [2] proved the following theorem: If p(x₁,…,x_n) is an exponential polynomial with coefficients in ℂ with no zeroes in ℂ, then p(x₁,…,x_n)= e^{g(x₁,…,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.