Article
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
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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 exponentiallyalgebraically closed (SEAC), and that SEAC fields are not modelcomplete. Zilber's pseudoexponential fields are identified in Construction 7.1 as the strong exponentialalgebraic 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 ELAfields, 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 pseudoexponential fields. As applications of the general results and methods of the paper, it is shown that Zilber's fields are not modelcomplete, 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 KontsevichZagier, Grothendieck, and Andr\'e transcendence conjectures on periods are discussed, and finally some open problems are suggested.Algebra and Number Theory 12/2009; 7(4). DOI:10.2140/ant.2013.7.943 · 0.68 Impact Factor 
 "Recall the following fact: (See [3]) "
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