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Preprint Number 849

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849. Allen Gehret
NIP for the Asymptotic Couple of the Field of Logarithmic Transseries
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Submission date: 22 March 2015.

Abstract:

The derivation on the differential-valued field T_log of logarithmic transseries induces on its value group Γ_log a certain map ψ. The structure Γ = (Γ_log , ψ) is a divisible asymptotic couple. In [gehret] we began a study of the first-order theory of (Γ_log , ψ) where, among other things, we proved that the theory T_log = Th(Γ_log , ψ) has a universal axiomatization, is model complete and admits elimination of quantifiers (QE) in a natural first-order language. In that paper we posed the question whether T_log has NIP (i.e., the Non-Independence Property). In this paper, we answer that question in the affirmative: T_log does have NIP. Our method of proof relies on a complete survey of the 1-types of T_log, which, in the presence of QE, is equivalent to a characterization of all simple extensions Γ<α> of Γ. We also show that T_log does not have the Steinitz exchange property and we weigh in on the relationship between models of T_log and the so-called precontraction groups of [kuhlmann1].

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Full text arXiv 1503.06496: pdf, ps.


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