Extendability of Operators Associated with Hook-Shaped Polynomials

Definitions and mathematical statement

The supplied record describes this assertion as a speculative conjecture, identified as Conjecture 3. For a polynomial $p$, write $T_p$ for the operator referred to in the record as its associated operator. This notation does not define that operator: its construction, domain, and codomain are absent from the supplied text.

The intended conjecture is the following conditional statement, universally quantified over the intended class of polynomials: \[ \begin{gathered} p is hook-shaped and symmetrically hyperbolic \ \Longrightarrow
\left(T_p is extendable\right)  \Longleftrightarrow  \left(p is weakly sum-of-squares hyperbolic\right). \end{gathered} Here the entire equivalence is the conclusion of the implication.

The intended polynomial class and the meanings of "hook-shaped," "symmetrically hyperbolic," "extendable," and "weakly sum-of-squares hyperbolic" are not defined in the supplied record. In particular, the coefficient field, number of variables, degree restrictions, any homogeneity or hyperbolicity-direction assumptions, and the permitted operator extensions are unspecified. Consequently, this display preserves the supplied assertion but is only a formulation schema; a precise, self-contained conjecture requires the missing definitions and hypotheses from the cited source.

Short English statement

The supplied conjecture says that, for every hook-shaped, symmetrically hyperbolic polynomial in the intended class, its associated operator is extendable exactly when the polynomial is weakly sum-of-squares hyperbolic. The record omits the specialized definitions and operator construction needed to make this statement precise.

Sources

• Local UnsolvedMath registry: permanent ID $30005481$, source problem code catalogue entry, title "Extendability of Operators Associated with Hook-Shaped Polynomials," in the Oberwolfach Reports Open Problems collection. The record supplies the year 2023 and names no proposer.
• Supplied source citation: New Directions in Real Algebraic Geometry (2023), Oberwolfach Reports, pp. 813--881, DOI 10.4171/owr/2023/15. https://doi.org/10.4171/owr/2023/15. The supplied original-statement field mentions a speculative Conjecture 3 but does not reproduce its mathematical content.
• Supplied literature source, "OWR 2023 report": https://ems.press/content/serial-article-files/47009?nt=1.
• Supplied literature source, "Symmetric Hyperbolic Polynomials": https://arxiv.org/abs/2308.09653.
• Supplied literature source, "Journal of Pure and Applied Algebra": https://doi.org/10.1016/j.jpaa.2025.107869.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, reports that the Oberwolfach report states the conjecture as open, and says that the available degree-three and degree-four classifications and computational evidence do not prove it in general. This is supplied dated curation, not an independently verified literature assessment; the record expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says it was reconstructed and checked against the official report. These are supplied provenance claims, not independent verification performed for this draft.