Degree Definability and Existential Theory of Rational Function Fields

Definitions and mathematical statement

Let $R={\mathbb{C}}[t]$, where $t$ is an indeterminate, and let $\mathcal L_{\mathrm{ring}}=\{0,1,+,-,\cdot\}$. For $f\in R\setminus\{0\}$, let $\deg(f)$ be its polynomial degree; in particular, every nonzero constant has degree $0$. Define \[ E^{\times}=\{(f,g)\in(R\setminus\{0\})^2:\deg(f)=\deg(g)\}. \] The source asks about a relation $E$ on all of $R$, but does not specify the degree of the zero polynomial. Thus its restriction to nonzero arguments is unambiguous, while its values on pairs involving zero require a convention. For example, adopting $\deg(0)=-\infty$ would give $E=E^{\times}\cup\{(0,0)\}$; this convention is not supplied by the record.

For a relation $S\subseteq R^2$, definability in the ring language with parameters means that there exist a finite tuple $\mathbf a\in R^r$ and an $\mathcal L_{\mathrm{ring}}$-formula $\varphi(x,y,\mathbf u)$ such that \[ \forall f,g\in R,\qquad (f,g)\in S\ \Longleftrightarrow\ R\models\varphi(f,g,\mathbf a). \] Under the usual convention allowing coefficients from $R$, being Diophantine means that there exist integers $m\geq0$, $s\geq1$, and polynomials $P_1,\ldots,P_s\in R[X,Y,Z_1,\ldots,Z_m]$ such that \[ \forall f,g\in R,\qquad (f,g)\in S\ \Longleftrightarrow\ \exists\mathbf z\in R^m\ \bigwedge_{i=1}^{s}P_i(f,g,\mathbf z)=0. \] These finitely many coefficients constitute parameters. The source does not separately clarify whether its phrase "with parameters" also qualifies the Diophantine alternative.

Degree relation question. Is the degree-equivalence relation $E$ on ${\mathbb{C}}[t]$ Diophantine? Is it definable in $\mathcal L_{\mathrm{ring}}$ with parameters? The intended relation agrees with $E^{\times}$ on nonzero arguments; its treatment of zero, and the parameter convention for the Diophantine question, remain qualifications requiring review.

Existential theory question. Fix a prime $p$, let ${\mathbb{F}}_p$ be the field with $p$ elements, and let $K_p={\mathbb{F}}_p(t)$ be its rational function field. Define \[ \operatorname{Th}_{\exists}(K_p)=\left\{\exists x_1\cdots\exists x_n\,\psi(x_1,\ldots,x_n):\ n\geq0,\ \psi\text{ is quantifier-free in }\mathcal L_{\mathrm{ring}},\ K_p\models\exists x_1\cdots\exists x_n\,\psi\right\}. \] Is $\operatorname{Th}_{\exists}(K_p)$ decidable? Equivalently, for this fixed $p$, does an algorithm halt on every existential $\mathcal L_{\mathrm{ring}}$-sentence and correctly determine its truth in $K_p$? Here the language has no additional named parameters, in particular no constant symbol naming $t$. The question is posed for each fixed prime; a single algorithm uniform in $p$ is not explicitly requested.

Short English statement

The record asks whether equality of polynomial degrees over ${\mathbb{C}}[t]$ is Diophantine or definable with parameters, and whether the parameter-free existential theory of ${\mathbb{F}}_p(t)$ is decidable for a fixed prime $p$. The treatment of the zero polynomial and the parameters allowed in the Diophantine alternative need clarification. The supplied open-status assessment is dated curation, not independent verification.

Sources

• Supplied local problem registry record: permanent UnsolvedMath ID $30003259$, source problem code catalogue entry, title Degree Definability and Existential Theory of Rational Function Fields; collection: Oberwolfach Reports Open Problems. No individual proposer is identified. The supplied proposed year is 2016.
• Supplied source 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 report is described as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source, labelled "Oberwolfach Reports 2017": https://ems.press/content/serial-article-files/46652.
• Literature assessment checked at 2026-08-21: the supplied curation labels the problem open and states that the existential theory of ${\mathbb{F}}_p(t)$ in the pure ring language remains open, and that decidability results with a named parameter do not resolve the parameter-free question. This is supplied dated curation derived from the OWR material, not independently verified here; its time-sensitive status and cited claims require independent verification. No separate literature assessment of the degree-equivalence question is supplied.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review. These are supplied editorial assertions; this statement-only draft still requires human mathematical review.