Schanuel Nullstellensatz for Zilber fields
We characterize the unsolvable exponential polynomials over the exponential fields introduced by Zilber, and deduce Picard's Little Theorem for such fields.
Strong initial segments of models of IΔ₀
McAloon showed that if 𝓐 is a nonstandard model of IΔ₀, then some initial segment of 𝓐 is a nonstandard model of PA. Sommer and D'Aquino characterized, in terms of the Wainer functions, the elements that can belong to such an initial segment. The characterization used work of Ketonen and Solovay, and Paris. Here we give conditions on a model 𝓐 of IΔ₀ guaranteeing that there is an n-elementary initial segment that is a nonstandard model of PA. We also characterize the elements that can be included....
Real closed exponential fields
Ressayre considered real closed exponential fields and “exponential” integer parts, i.e., integer parts that respect the exponential function. In 1993, he outlined a proof that every real closed exponential field has an exponential integer part. In the present paper, we give a detailed account of Ressayre’s construction and then analyze the complexity. Ressayre’s construction is canonical once we fix the real closed exponential field R, a residue field section k, and a well ordering ≺ on R. The...
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