The Asymptotic Couple of the Field of Logarithmic Transseries

Nip for the Asymptotic Couple of the Field of Logarithmic Transseries

“…The structure Γ = (Γ log , ψ) is a divisible asymptotic couple. In [Geh14] 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 [Geh14] we began a study of the model-theoretic and algebraic properties of (Γ log , ψ), the asymptotic couple of the differential-valued field T log of logarithmic transseries. This paper is intended to be its sequel.…”

“…Here we give a complete survey of the space of 1-types over a model of the theory T log = Th(Γ log , ψ) and use that to show that T log has the Non-Independence Property (NIP), largely settling a question we raised in [Geh14,§8].…”

“…As usual, Z is the ring of integers, Q is the field of rational numbers, and R is the field of real numbers. In this paper, like its prequel [Geh14], we study asymptotic couples such as (Γ log , ψ) as independent objects of interest, completely removed from any differential-valued fields from which they may arise. A complete discussion of differential-valued fields such as T log and how they give rise to asymptotic couples is outside the scope of this paper.…”

“…In [Geh14] we gave a complete axiomatization for the first-order theory Th(Γ log , ψ) and proved that it is model complete; see Definition 2.6 below. This followed from exhibiting quantifier elimination for Th(Γ log , ψ) in a natural language L log which we recall in Section 3.6.…”