Exceptional Reducibility Sets in Polynomial Specialization

Definitions and mathematical statement

Let $F(t,X)\in{\mathbb{Q}}[t,X]$ be irreducible. For a field $k$, a nonzero polynomial in $k[X]$ is called reducible if it is a product of two polynomials of positive degree in $X$. The supplied problem asks for a description, up to finitely many parameter values, of \[ R_F=\{a\in{\mathbb{Q}}:F(a,X)\text{ is reducible in }{\mathbb{Q}}[X]\}. \] Here a description up to a finite set means specifying a set $S_F\subseteq{\mathbb{Q}}$ such that \[ R_F\mathbin{\triangle}S_F =(R_F\setminus S_F)\cup(S_F\setminus R_F) \quad\text{is finite}. \] The source does not specify the required form or effectiveness of such a description.

There is a specialization qualification in this formulation. If $n=\deg_X F\geq1$, write $c(t)$ for the coefficient of $X^n$, and put \[ U_F=\{a\in{\mathbb{Q}}:c(a)\ne0\},\qquad R_F^{\mathrm{reg}}=\{a\in U_F:F(a,X)\text{ is reducible in }{\mathbb{Q}}[X]\}. \] The complement of $U_F$ is finite, so the requested description up to a finite set can be formulated using $R_F^{\mathrm{reg}}$, without deciding conventions for specializations whose degree drops. The source does not explicitly assume $\deg_XF\geq1$; if $F$ is independent of $X$, its specializations are constants or zero, and the intended treatment of this case is unspecified. Under the definition above, nonzero constants are not reducible, and the zero polynomial is excluded.

The problem also requests analogous descriptions over number fields and for integral parameters. For a number field $k$, let $\mathcal O_k$ denote its ring of integers. Natural field-reducibility analogues, for an irreducible $G(t,X)\in k[t,X]$, are \[ R_{G,k}=\{a\in k:G(a,X)\text{ is reducible in }k[X]\}, \qquad R_{G,\mathcal O_k}=\{a\in\mathcal O_k:G(a,X)\text{ is reducible in }k[X]\}. \] For $k={\mathbb{Q}}$, integral parameters give $R_{F,{\mathbb{Z}}}=R_F\cap{\mathbb{Z}}$. Seek descriptions of these sets modulo finite symmetric difference, with the same specialization qualification. These are explicit natural interpretations of the requested analogues: the source does not specify whether the number-field question concerns arbitrary irreducible polynomials over each field or base change of the original $F$, nor whether integral reducibility is meant over the fraction field or an integral coefficient ring. No precise class of admissible descriptions is supplied.

Short English statement

Describe, apart from finitely many exceptions, the rational parameter values at which an irreducible polynomial in two variables specializes to a reducible polynomial in $X$. Seek corresponding descriptions over number fields and for integral parameter values. The treatment of degenerate specializations, the precise integral formulation, and the required form of the description need clarification.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30004651$, title Exceptional Reducibility Sets in Polynomial Specialization, collection Oberwolfach Reports Open Problems, exact source problem code catalogue entry. The record gives proposed year $2021$ and names no proposer.
• Supplied report citation: Homotopic and Geometric Galois Theory (2022), Oberwolfach Reports, DOI 10.4171/owr/2021/12, pp. 663--744. https://doi.org/10.4171/owr/2021/12. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions. No problem-specific page or named contributor is supplied.
• König--Neftin, Reducible Fibers of Polynomial Maps. Supplied literature link: https://arxiv.org/abs/2001.03630.
• Behajaina--König--Neftin, DLS problem. Supplied literature link: https://arxiv.org/abs/2603.27728.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved and states that exceptional reducibility sets are explicitly described for substantial classes, including many decomposable polynomials, while exceptional low-degree solvable cases of the general arbitrary-polynomial Hilbert--Siegel problem remain open. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its status and cited claims require independent verification.
• The registry supplies the statement-status label corrected_verified and reports manual semantic, typographic, and LaTeX review. This records the supplied review claim and does not establish independent mathematical verification of this draft.