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NIP FOR THE ASYMPTOTIC COUPLE OF THE FIELD OF LOGARITHMIC TRANSSERIES

Published online by Cambridge University Press:  21 March 2017

ALLEN GEHRET
ALLEN GEHRET*
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN URBANA, IL 61801, USAE-mail: agehret2@illinois.edu
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Abstract

The derivation on the differential-valued field Tlog of logarithmic transseries induces on its value group ${{\rm{\Gamma }}_{{\rm{log}}}}$ a certain map ψ. The structure ${\rm{\Gamma }} = \left( {{{\rm{\Gamma }}_{{\rm{log}}}},\psi } \right)$ is a divisible asymptotic couple. In [7] we began a study of the first-order theory of $\left( {{{\rm{\Gamma }}_{{\rm{log}}}},\psi } \right)$ where, among other things, we proved that the theory $T_{{\rm{log}}} = Th\left( {{\rm{\Gamma }}_{{\rm{log}}} ,\psi } \right)$ 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 Tlog has NIP (i.e., the Non-Independence Property). In this paper, we answer that question in the affirmative: Tlog does have NIP. Our method of proof relies on a complete survey of the 1-types of Tlog, which, in the presence of QE, is equivalent to a characterization of all simple extensions ${\rm{\Gamma }}\left\langle \alpha \right\rangle$ of ${\rm{\Gamma }}$. We also show that Tlog does not have the Steinitz exchange property and we weigh in on the relationship between models of Tlog and the so-called precontraction groups of [9].

Keywords

03C64asymptotic couplesasymptotic integrationlogarithmic transseriesindependence propertycontraction groups
Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 1 , March 2017 , pp. 35 - 61
Copyright
Copyright © The Association for Symbolic Logic 2017 

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