NIP FOR THE ASYMPTOTIC COUPLE OF THE FIELD OF LOGARITHMIC TRANSSERIES

作者：Gehret Allen^*

来源：Journal of Symbolic Logic, 2017, 82(1): 35-61.

DOI：10.1017/jsl.2016.59

• asymptotic couples
• asymptotic integration
• logarithmic transseries
• independence property
• contraction groups

The derivation on the differential-valued field T-log of logarithmic transseries induces on its value group Gamma(log) a certain map psi. The structure Gamma = (G(log), psi) is a divisible asymptotic couple. In [7] we began a study of the first-order theory of (G(log), psi) where, among other things, we proved that the theory T-log = Th(G(log), psi) 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 T-log, which, in the presence of QE, is equivalent to a characterization of all simple extensions Gamma alpha of G. 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 [9].

• 出版日期2017-3