Real-Rootedness of Generalized Chebyshev Polynomials

Definitions and mathematical statement

A polynomial $p(x)$ is real-rooted if every $z\in{\mathbb{C}}$ satisfying $p(z)=0$ belongs to ${\mathbb{R}}$.

The supplied record asks for a characterization of the triangulations $L$ for which the generalized Tchebyshev polynomial $T_n^L(x)$ is real-rooted for every $n$. The record does not define the admissible triangulations, the construction of $T_n^L$, or the range of $n$. Thus, let $\mathcal A$ and $I$ denote, provisionally, the admissible class and full index set intended by that construction. The requested characterization is of \[ \left\{L\in\mathcal A:\ \forall n\in I\ \forall z\in{\mathbb{C}},\quad T_n^L(z)=0\Longrightarrow z\in{\mathbb{R}}\right\}. \] This is a conditional formulation: specifying $\mathcal A$, $I$, and the polynomial construction from an authoritative source is necessary for a complete mathematical statement. No particular interpretation of these missing definitions is asserted here.

The supplied literature assessment dated 2026-08-21 reports real-rootedness for important subclasses, including all valid subdivisions of a two-dimensional simplex, and a counterexample arising from a simple three-dimensional subdivision. It reports that a full characterization remains open. These are supplied curation claims, not independently verified results; the meaning of "valid" and the counterexample construction are not provided.

Short English statement

Characterize the triangulations $L$ whose associated generalized Tchebyshev polynomials $T_n^L(x)$ have only real roots for every index $n$. The supplied record omits the polynomial construction, admissibility conditions on $L$, and index range, so the formulation remains conditional on those definitions.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002774$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2015 and identifies no individual proposer.
• Geometric and Algebraic Combinatorics (2015), Oberwolfach Reports, pp. 285--368, DOI 10.4171/owr/2015/5: https://doi.org/10.4171/owr/2015/5. This is the supplied workshop-report provenance and also a supplied literature-review source; no exact problem page is given.
• Hetyei--Nevo, JCTA, supplied literature-source label, DOI 10.1016/j.jcta.2015.08.004: https://doi.org/10.1016/j.jcta.2015.08.004. No article title or precise result locator is supplied.
• Literature assessment checked at 2026-08-21: supplied dated curation, not independently verified. It labels the problem open, reports real-rootedness for important subclasses including all valid subdivisions of a two-dimensional simplex, reports a simple three-dimensional subdivision counterexample, and states that a full characterization remains open. The record explicitly calls for independent verification of time-sensitive status and cited claims.