Transseries and Todorov-Vernaeve's asymptotic fields

We study the relationship between fields of transseries and residue fields of convex subrings of non-standard extensions of the real numbers. This was motivated by a question of Todorov and Vernaeve, answered in this paper. In this note we answer a question by Todorov and Vernaeve (see, e.g., [35]) concerning the relationship between the field of logarithmic-exponential series from [14] and the residue field of a certain convex subring of a non-standard extension of the real numbers, introduced in [34] in connection with a non-standard approach to Colombeau’s theory of generalized functions. The answer to this question can almost immediately be deduced from well-known (but non-trivial) results about o-minimal structures. It should therefore certainly be familiar to logicians working in this area, but perhaps less so to those in non-standard analysis, and hence may be worth recording. We begin by explaining the question of Todorov-Vernaeve. Let ∗R be a nonstandard extension of R. Given X ⊆ R we denote the non-standard extension of X by ∗X, and given also a map f : X → R, by abuse of notation we denote the non-standard extension of f to a map ∗X → ∗Rn by the same symbol f . Let O be a convex subring of ∗R. Then O is a valuation ring of ∗R, with maximal ideal o := {x ∈ ∗R : x = 0, or x 6= 0 and x−1 / ∈ O}. We denote the residue field O/o of O by Ô, with natural surjective morphism x 7→ x̂ := x+ o : O → Ô. The ordering of ∗R induces an ordering of Ô making Ô an ordered field and x 7→ x̂ order-preserving. By standard facts from real algebra [26], Ô is real closed. Residue fields of convex subrings of ∗R are called “asymptotic fields” in [34] (although this terminology is already used with a different meaning elsewhere [2]). The collection of convex subrings of ∗R is linearly ordered by inclusion, and the smallest convex subring of ∗R is Rfin = {x ∈ ∗R : |x| 6 n for some n}, with maximal ideal Rinf = { x ∈ ∗R : |x| 6 1 n for all n > 0 } . The inclusions R→ Rfin → O give rise to a field embedding R→ Ô, by which we identify R with a subfield of Ô. In the case O = Rfin we have Ô = R, and x̂ is the standard part of x ∈ Rfin, also denoted in the following by st(x).