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, Henson and Rubel [2] 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) = eg(x1,…, xn)
where g(x1,…, xn) 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.

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    • "Using the classification of strong extensions, we show that the Schanuel nullstellensatz, studied by D'Aquino, Terzo, and Macintyre [2] and also Shkop [13], is strictly weaker than the property of being strongly exponentially-algebraically closed (SEAC), and that SEAC fields are not model-complete. Zilber's pseudo-exponential fields are identified in Construction 7.1 as the strong exponential-algebraic closures of fields of rational functions in a set of indeterminates. "
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    ABSTRACT: The algebra of exponential fields and their extensions is developed. The focus is on ELA-fields, which are algebraically closed with a surjective exponential map. In this context, finitely presented extensions are defined, it is shown that finitely generated strong extensions are finitely presented, and these extensions are classified. An algebraic construction is given of Zilber's pseudo-exponential fields. As applications of the general results and methods of the paper, it is shown that Zilber's fields are not model-complete, answering a question of Macintyre, and a precise statement is given explaining how Schanuel's conjecture answers all transcendence questions about exponentials and logarithms. Connections with the Kontsevich-Zagier, Grothendieck, and Andr\'e transcendence conjectures on periods are discussed, and finally some open problems are suggested.
    Preview · Article · Dec 2009 · Algebra and Number Theory
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    • "Recall the following fact: (See [3]) "
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    ABSTRACT: In this article, I will prove that assuming Schanuel's conjecture, an exponential polynomial with algebraic coefficients can have only finitely many algebraic roots. Furthermore, this proof demonstrates that there are no unexpected algebraic roots of any such exponential polynomial. This implies a special case of Shapiro's conjecture: if p(x) and q(x) are two exponential polynomials with algebraic coefficients, each involving only one iteration of the exponential map, and they have common factors only of the form exp(g) for some exponential polynomial g, then p and q have only finitely many common zeros.
    Preview · Article · Oct 2009 · Communications in Algebra