Elementary Equivalence of Differential Generalized Series Fields

Definitions and mathematical statement

For a field $k$ and an ordered abelian group $G$, the Hahn field $k((G))$ consists of formal sums \[ f=\sum_{g\in G}a_g t^g,\qquad a_g\in k, \] whose support $\{g:a_g\ne0\}$ is well ordered. Addition is coefficientwise and multiplication is convolution. Its canonical valuation is \[ v_G(f)=\min\operatorname{supp}(f)\quad(f\ne0),\qquad v_G(0)=\infty, \] and its valuation ring is $\mathcal O_G=\{f:v_G(f)\ge0\}$.

An asymptotic couple $(G,\psi)$ consists of an ordered abelian group $G$ and a map $\psi:G\setminus\{0\}\to G$ satisfying, for nonzero $a,b\in G$, \[ \begin{aligned} \psi(a+b)&\ge\min\{\psi(a),\psi(b)\} &&\text{if }a+b\ne0,\\ \psi(na)&=\psi(a) &&\text{for }n\in{\mathbb{Z}}\setminus\{0\},\\ a+\psi(a)&>\psi(b) &&\text{if }a>0. \end{aligned} \] Any additional restrictions on the asymptotic couples required by the cited construction remain unspecified in the supplied record.

A derivation on a field $K$ is an additive map $D:K\to K$ satisfying $D(fh)=fD(h)+hD(f)$. Let $D$ and $D'$ denote the particular derivations on $k((G))$ and $k'((G'))$ referred to in the report. The supplied record does not give their construction, their action on coefficients or monomials, or the hypotheses under which they exist. Consequently these symbols denote missing construction data; the question must not be interpreted as quantifying over arbitrary derivations.

For a fixed first-order language $L$, write $A\equiv_L B$ when $A$ and $B$ satisfy exactly the same $L$-sentences. With $L_{\mathrm{ring}}$ the language of fields, and $L_{\mathrm{ac}}$, $L_{\mathrm{dv}}$ the intended languages of asymptotic couples and differential valued fields, respectively, the supplied corrected question asks whether, for every pair of inputs to the report's derivation construction, \[ \left. \begin{gathered} k\equiv_{L_{\mathrm{ring}}}k',\\ (G,\psi)\equiv_{L_{\mathrm{ac}}}(G',\psi') \end{gathered} \right\} \quad\Longrightarrow\quad \left(k((G)),D,\mathcal O_G\right) \equiv_{L_{\mathrm{dv}}} \left(k'((G')),D',\mathcal O_{G'}\right). \] Here the displayed structures use a valuation-ring predicate to represent the valuation. This is an explicit conventional presentation, not a verified specification of the report's language; any intended value-group sort or additional named structure must be supplied. No characteristic, coefficient-field derivation, or further compatibility assumptions are stated in the record. Thus this is a conditional formulation requiring the missing construction and language specifications before it becomes a fully determined mathematical question.

The supplied original wording instead compares $k((G))$ with $k'((G))$, using the same exponent group. The registry labels the version with $G'$ as corrected and verified; that supplied editorial label is not an independent verification here.

Short English statement

Does elementary equivalence of the coefficient fields and asymptotic couples imply elementary equivalence of the resulting Hahn fields with the report's specified derivations and valuations? The derivation construction, its hypotheses, and the precise languages are missing from the supplied record, and the original and corrected wordings use different exponent-group formulations.

Sources

• Local UnsolvedMath registry: supplied complete record for permanent ID $30003316$, source problem code catalogue entry, titled Elementary Equivalence of Differential Generalized Series Fields, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied.
• Supplied source citation: Mini-Workshop: Surreal Numbers, Surreal Analysis, Hahn Fields and Derivations (2017), Oberwolfach Reports, pp. 3313--3372, DOI 10.4171/owr/2016/60. Original OWR report: https://doi.org/10.4171/owr/2016/60. The record separately gives proposed year 2016; the differing year designations are retained without reconciliation.
• Supplied literature source, labeled "series derivations": https://doi.org/10.1016/j.jalgebra.2011.11.024. No authors, title, or specific construction passage are supplied for this reference.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the defining derivation construction is available while elementary equivalence of the resulting differential valued generalized-series fields remains explicitly open. This is supplied dated curation, not independently verified literature assessment. The background expressly requests independent verification of time-sensitive status and cited claims.
• The registry supplies the statement status corrected_verified and reports that the official passage was checked when preparing its corrected formulation. These are source-record assertions; no source visit or independent verification is claimed in this draft.