Henson and Rubel's Theorem for Zilber's Pseudoexponentiation

Journal of Symbolic Logic (Impact Factor: 0.54). 03/2009; 77(2). DOI: 10.2178/jsl/1333566630
Source: arXiv


In 1984, C. W. Henson and L. A. Rubel [Trans. Am. Math. Soc. 282, 1–32 (1984; Zbl 0533.03015)] proved the following theorem: If p(x 1 ,...,x n ) is an exponential polynomial with coefficients in ℂ with no zeroes in ℂ, then p(x 1 ,...,x n )=e g(x 1 ,...,x n ) where g(x 1 ,...,x n ) is some exponential polynomial over ℂ. In this paper, I 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.

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