Hilbert's Tenth Problem in Generic Rational Subrings

Definitions and mathematical statement

For a subring $R\subseteq{\mathbb{Q}}$ and a specified permitted coefficient ring $C\subseteq R$, a set $D\subseteq R^n$ is Diophantine over $R$ with coefficients in $C$ if there are integers $m\geq0$, $s\geq1$, and polynomials $p_1,\ldots,p_s\in C[X_1,\ldots,X_n,Y_1,\ldots,Y_m]$ such that \[ D=\{\mathbf x\in R^n:\exists\mathbf y\in R^m\;\bigwedge_{i=1}^{s}p_i(\mathbf x,\mathbf y)=0\}. \] A Diophantine model of $({\mathbb{Z}},+,\cdot)$ in $R$, relative to this coefficient convention, consists of an integer $n\geq1$, a Diophantine set $D\subseteq R^n$, and a bijection $\iota:{\mathbb{Z}}\to D$ such that the sets \[ \{(\iota(a),\iota(b),\iota(a+b)):a,b\in{\mathbb{Z}}\},\qquad \{(\iota(a),\iota(b),\iota(ab)):a,b\in{\mathbb{Z}}\} \] are Diophantine subsets of $R^{3n}$ under the same convention.

The supplied record asks two existence questions:

1. Does there exist a subring $R\subseteq{\mathbb{Q}}$ that is HTP-generic and admits a Diophantine model of $({\mathbb{Z}},+,\cdot)$?
2. Does there exist a polynomial $f$ such that $\mu(B(f))>0$, where the source calls $B(f)$ a boundary set?

The record does not define HTP-genericity, specify the permitted coefficients or parameters for the Diophantine model, or state whether subrings must contain $1$. It also does not specify the coefficient domain or number of variables of $f$, the underlying space and topology defining $B(f)$, or the measure $\mu$. Consequently, the second question remains schematic, and the first depends on unresolved source conventions. No particular topology, probability measure, or interpretation of HTP-genericity is adopted here. The record does not specify that the witnesses in the two questions must be related.

Short English statement

The source asks whether an HTP-generic subring of ${\mathbb{Q}}$ can contain a Diophantine model of integer arithmetic, and whether some polynomial has a boundary set of positive measure. Essential definitions and conventions are absent from the supplied record, so a fully specified formulation requires source clarification. The supplied open status is a dated curation label.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30003247$, title: Hilbert's Tenth Problem in Generic Rational Subrings. Exact source problem code: catalogue entry. The supplied record lists proposed year 2016 and identifies no individual proposer.
• Supplied report citation: Definability and Decidability Problems in Number Theory (2017), Oberwolfach Reports, pp. 2793--2866, DOI 10.4171/owr/2016/49. https://doi.org/10.4171/owr/2016/49. The supplied background describes the report as documenting workshop discussions and open problems.
• Supplied literature source, labeled Oberwolfach Reports 2017: https://ems.press/content/serial-article-files/46652.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, describes the HTP-generic subring and positive-boundary-measure questions as conjectural, and reports finding no demonstrable resolution of the measure assertion. This is supplied dated curation, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.
• The registry supplies the statement status corrected_verified and reports a manual semantic, typographic, and LaTeX review claiming self-containment and fidelity. These are supplied review assertions; the technical definitions needed for self-containment are absent from the supplied record.