Characterizations of Symmetrically Hyperbolic Hook-Shaped Polynomials

Definitions and mathematical statement

For a nonzero polynomial $f\in{\mathbb{R}}[t]$, real-rooted means that every complex root of $f$ is real. Roots are counted with algebraic multiplicity whenever a root count is specified. If $f(t)=a_m t^m+a_{m-1}t^{m-1}+\cdots$, with $m\geq1$ and $a_m\ne0$, then its roots have sum zero exactly when $a_{m-1}=0$. A real polynomial $g$ is a sum of squares (SOS) if there are finitely many real polynomials $h_1,\ldots,h_r$ in the same variables such that $g=\sum_{j=1}^r h_j^2$.

The supplied record requests the following two characterizations. Several essential terms are not defined in the supplied excerpt, so these are source-formulation schemata requiring clarification, rather than fully specified conjectures.

First, retain the source notation ${\mathbb{R}}[t]_{n,0}$ and ${\mathbb{R}}[t]_{d,0}$, and write \[ q_n(t)=(t-1)^{n-1}(t+n-1). \] For every diagonal map \[ T:{\mathbb{R}}[t]_{n,0}\longrightarrow{\mathbb{R}}[t]_{d,0}, \] the requested assertion is \[ \begin{gathered} T(q_n)\text{ is real-rooted and has }d-1\text{ roots of one sign} \\[2pt] \Longleftrightarrow \\[2pt] T\text{ preserves zero-sum hyperbolicity}. \end{gathered} \] Preservation here has the universally quantified form: every zero-sum hyperbolic input in the source domain has a zero-sum hyperbolic image in the source codomain. The excerpt does not define those domains, the bases with respect to which $T$ is diagonal, or the intended meaning of zero-sum hyperbolicity. It also does not specify the admissible values of $n,d$, degree or nonvanishing conditions on images, or whether the root condition means exactly or at least $d-1$ roots, whether multiplicities count, and whether zero roots are allowed. The multiplicity convention above must therefore be checked against the intended source convention. The variable $t$ replaces the source's $x$ only as a dummy variable.

Second, for every hook-shaped symmetric polynomial $p$ in the intended source class, the requested assertion is \[ p\text{ is symmetrically hyperbolic} \quad\Longleftrightarrow\quad \Delta_{\mathbf 1,\mathbf 1}p\text{ is SOS}. \] Here $\mathbf 1=(1,\ldots,1)$ in the ambient variable space. The supplied excerpt does not define hook-shaped, symmetrically hyperbolic, or $\Delta_{u,v}p$, nor specify the variable count, degree, homogeneity, normalization, or other hypotheses on $p$. These terms and hypotheses must be recovered before this equivalence can be treated as a precise mathematical conjecture.

There is also a material discrepancy between the requested characterizations and the supplied original excerpt. The latter states a quintic example as "Theorem 9" rather than stating either characterization. Let $e_k$ denote the elementary symmetric polynomial of degree $k$ in five variables. The excerpt's final monomial is transcribed as $\texttt{e21 e3}$. If this means $e_1^2e_3$, its displayed polynomial is \[ p_{\mathrm{ex}}=4500e_5-220e_1e_4+7e_1^2e_3. \] This reading of the transcription requires confirmation. For the example, the excerpt asserts symmetric hyperbolicity, nonextendability of its associated operator, the SOS property of $\Delta_{\mathbf 1,\mathbf 1}p_{\mathrm{ex}}$, and failure of SOS hyperbolicity. It attributes the SOS verification and a verification that $\Delta_{\mathbf 1,u}p_{\mathrm{ex}}$ is not SOS for some $u\in{\mathbb{R}}^5$ to computer calculations. It further records, for the associated operator $T$, \[ T\left((t-1)^4(t+4)\right)=750(t-1)^2(t-2)^2(t+6), \] and states that an associated operator satisfying \[ T\left((t-1)^4(t+4)\right)=(t-a)^2(t-b)^2(t+2a+2b), \qquad a,b\in{\mathbb{R}}_+, \] is not extendable. The meanings of associated operator, extendable, SOS hyperbolicity, and the boundary convention for ${\mathbb{R}}_+$ are not supplied. These are attributed contextual assertions, not independently verified results or substitutes for the two requested characterizations.

Short English statement

The record requests two equivalences: a single test polynomial should characterize diagonal operators preserving zero-sum hyperbolicity, and an SOS condition should characterize symmetric hyperbolicity for hook-shaped symmetric polynomials. Essential definitions and parameter restrictions are missing. The supplied original excerpt instead discusses a quintic example, so the intended conjectures and their relationship to that excerpt require human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005480$, source problem code catalogue entry, titled Characterizations of Symmetrically Hyperbolic Hook-Shaped Polynomials; collection: Oberwolfach Reports Open Problems. No individual proposer is identified in the supplied record.
• Supplied report 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.
• 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.
• Supplied dated literature curation, checked 2026-08-22: the record labels the problem open and reports that the workshop report identifies an open problem, with evidence and low-degree or special-case results but no general characterization. This is supplied dated curation, not an independently verified assessment of the literature or current status.
• The registry labels the statement corrected_verified and says it was reconstructed and checked against the official report. This is a supplied verification claim; no source was independently visited or checked in preparing this draft, and the supplied original excerpt does not itself state the two requested equivalences.