Closed asymptotic couples

Abstract

The derivation of a Hardy field induces on its value group a certain function ψ. If a Hardy field extends the real field and is closed under powers, then its value group is also a vector space over . Such "ordered vector spaces with ψ-function" are called H-couples. We define closed H-couples and show that every H-couple can be embedded into a closed one. The key fact is that closed H-couples have an elimination theory: solvability of an arbitrary system of equations and inequalities (built up from vector space operations, the function ψ, parameters, and the unknowns to be solved for) is equivalent to an effective condition on the parameters of the system. The H-couple of a maximal Hardy field is closed, and this is also the case for the H-couple of the field of logarithmic-exponential series over . We analyze in detail finitely generated extensions of a given H-couple.

Nip for the Asymptotic Couple of the Field of Logarithmic Transseries

“…At the moment this observation isn't very fruitful for models of T 0 since most of the action happens around this set anyway. However, for other asymptotic couples, such as the so-called closed asymptotic couples of [AvdD00], this can be useful in further relating the roles of s and ψ.…”

“…A closed asymptotic couple is a divisible H-asymptotic couple with asymptotic integration such that (Γ < ) = Ψ (see [AvdD00]). There they use essentially the same trick with (B, )-shifts, except they consider iterates of ψ instead of iterates of s. However, by Corollary 7.3, one can see that this is essentially the same notion for elements α 0.…”

The asymptotic couple of the field of logarithmic transseries

“…We determine here the elementary (i.e., first-order) theory of (Γ log , ψ), provide a quantifier elimination result in a natural language, and show that the induced structure on the set Ψ is just its structure as an ordered subset of Γ log (so Ψ is stably embedded in (Γ log , ψ)). This paper is in the spirit of [AvdD00], which proves a quantifier elimination result for so-called closed asymptotic couples. That paper did for the asymptotic couple of the field T of logarithmic-exponential transseries what is done here for the asymptotic couple of T log .…”

“…Our choice of primitives here yields a universal theory with QE. This makes some things simpler than in [AvdD00], and has various other benefits, as we shall see.…”

“…For an explicit construction of T, see [vdDMM01]. The main difficulty in getting QE, compared to [AvdD00], was to find the right extra primitives, and to establish a new Embedding Lemma 4.12. Our choice of primitives here yields a universal theory with QE.…”