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A note on the axioms for Zilber’s pseudo-exponential fields. (English) Zbl 1345.03070

Summary: We show that Zilber’s conjecture that complex exponentiation is isomorphic to his pseudo-exponentiation follows from the a priori simpler conjecture that they are elementarily equivalent. An analysis of the first-order types in pseudo-exponentiation leads to a description of the elementary embeddings, and the result that pseudo-exponential fields are precisely the models of their common first-order theory which are atomic over exponential transcendence bases. We also show that the class of all pseudo-exponential fields is an example of a nonfinitary abstract elementary class, answering a question of Kesälä and Baldwin.

MSC:

03C60 Model-theoretic algebra
03C48 Abstract elementary classes and related topics
12L12 Model theory of fields
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References:

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