Splitting Sets Under Differential-Transcendental Extensions

Definitions and mathematical statement

A computable differential field is a field with a computable presentation in which the field operations and the distinguished derivation are computable. In the ordinary differential-field convention, a derivation $\delta$ is an additive map satisfying $\delta(ab)=a\delta(b)+b\delta(a)$. The supplied record does not explicitly specify the characteristic or the number of derivations; the notation below unpacks the ordinary, single-derivation interpretation without imposing characteristic zero.

Let $(K,\delta)$ be a computable differential field contained in a larger computable differential field $(L,\delta)$, with the derivation on $L$ restricting to that on $K$. An element $z\in L$ is differentially transcendental over $K$ if, for every integer $n\geq 0$ and every nonzero polynomial $P\in K[X_0,\ldots,X_n]$, \[ P(z,\delta z,\ldots,\delta^n z)\neq 0. \] Write \[ K\langle z\rangle=K(z,\delta z,\delta^2z,\ldots) \] for the smallest differential subfield of $L$ containing $K$ and $z$.

The supplied question asks whether, for every such $K$, $L$, and $z$, \[ T_{K\langle z\rangle}\leq_T T_K. \] For subsets $A,B\subseteq{\mathbb{N}}$, the relation $A\leq_T B$ means that an oracle Turing machine with oracle $B$ computes the characteristic function of $A$.

Unresolved qualification. The supplied record does not define $T_F$, its effective coding, or its dependence on a presentation of the differential field $F$. Thus the displayed question preserves the source notation but is not yet a fully specified computability statement. The title mentions splitting sets, but this alone does not identify $T_F$. The record also does not state whether the inclusion $K\subseteq L$ must be computable or how $K\langle z\rangle$ is to be presented effectively. No uniform reduction across all instances is explicitly requested.

Short English statement

The question asks whether adjoining a differentially transcendental element to a computable differential field preserves computability of the associated set $T_F$ relative to its value for the original field. The meaning and coding of $T_F$ are missing, so a precise formulation requires further source clarification. The supplied curation labels the problem open and reports that a relevant proof remains to be checked; these are supplied assessments, not independently verified conclusions.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001988$, exact source problem code catalogue entry, title Splitting Sets Under Differential-Transcendental Extensions; collection: Oberwolfach Reports Open Problems. The record gives the proposed year as 2012 and supplies no individual proposer.
• Computability Theory (2012), Oberwolfach Reports, pp. 397--455, DOI 10.4171/owr/2012/08. Supplied source URL and literature reference labeled "original Oberwolfach report": https://doi.org/10.4171/owr/2012/08.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open and states that the report presents the statement as a conjecture whose relevant proof remains to be checked. This is supplied dated curation, not independently verified literature assessment; the record itself requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports comparison with the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. This is supplied verification metadata; no independent source inspection or mathematical verification is asserted here.