Turing Degrees of Differential-Field Embedding Images

Definitions and mathematical statement

A differential field with one derivation is a field $K$ equipped with a map $\delta_K:K\to K$ satisfying \[ \delta_K(a+b)=\delta_K(a)+\delta_K(b),\qquad \delta_K(ab)=a\delta_K(b)+b\delta_K(a) \quad(a,b\in K). \] For differential fields $(K,\delta_K)$ and $(L,\delta_L)$, a differential-field embedding is an injective field homomorphism $g:K\to L$ such that \[ g(\delta_K(a))=\delta_L(g(a))\qquad(a\in K). \] These definitions describe the single-derivation setting; the supplied fragment does not specify the number of derivations, the characteristic, or the hypotheses of Harrington's construction.

Given presentations in which elements have natural-number codes, an embedding is computable when its action on those codes is computable. Its image is \[ g(K)=\{g(a):a\in K\}\subseteq L. \] For a specified coding $\iota:L\to{\mathbb{N}}$, the degree of this image means \[ \deg_T\left(\{\iota(g(a)):a\in K\}\right), \] where $A\leq_T B$ means that membership in $A\subseteq{\mathbb{N}}$ is decidable by an algorithm with oracle $B$, and $\deg_T(A)$ is the equivalence class of $A$ under mutual Turing reducibility. The relevant presentations and coding must be specified to interpret this expression.

For comparison, a usual splitting-set convention for a computably presented field $F$, with an effective coding of $F[X]$, is \[ \operatorname{Split}(F)= \{\ulcorner f\urcorner: f\in F[X],\ \deg f\geq2, \ \exists u,v\in F[X]\ (\deg u,\deg v\geq1\ \land\ f=uv)\}. \] This convention is explanatory; the supplied record does not define its intended splitting set or identify the differential property that should replace polynomial reducibility.

Source question, with unresolved parameters. For the computable differential-field embedding $g$ attributed in the record to Harrington, determine the Turing degree of $g(K)$, and identify an appropriate differential analogue of the splitting set. The record does not specify $K$, the target field, their presentations, the construction or choices defining $g$, or the criterion that makes a differential analogue appropriate. It also does not state whether a single degree or a classification over a family of embeddings is sought. Consequently, this is a faithful statement of the extracted question with explicit gaps, not a fully specified mathematical problem.

Short English statement

The question asks how computationally complicated the image of Harrington's computable differential-field embedding is, and what should play the role of a splitting set for differential fields. The supplied fragment omits the embedding's construction, field hypotheses, presentations, and the intended differential analogue, so these parts require source clarification before a precise target can be fixed.

Sources

• Local UnsolvedMath registry, permanent ID $30001987$, title Turing Degrees of Differential-Field Embedding Images, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied record gives proposed year 2012 and no named proposer; it attributes the embedding to Harrington.
• Supplied report citation: Computability Theory (2012), Oberwolfach Reports, pp. 397--455, DOI 10.4171/owr/2012/08. Source URL and supplied literature-source link labelled "original Oberwolfach report": https://doi.org/10.4171/owr/2012/08.
• Literature assessment dated exactly 2026-08-21: supplied status open. The supplied assessment describes a historical fragment about reducibility of polynomial representations, says that it does not state a complete decidable theorem, and describes the underlying question as unresolved as extracted. This is supplied dated curation, not an independently verified literature assessment; the record requests independent verification of time-sensitive status and cited claims.
• The registry supplies statement status corrected_verified and reports that the mathematical content was checked against the official report, with notation, context, and question form normalized without changing the source-level claim. This draft preserves that verification claim as source metadata and does not independently confirm it.