Positivity of Kac-Type Polynomials for Concealed-Canonical Algebras

Definitions and mathematical statement

Let $C$ range over the almost concealed-canonical algebras intended in the cited source, and let $d$ range over all dimension vectors allowed for $C$ in that source. Write $P_{C,d}(T)$ for the associated Kac-type polynomial named in the supplied record. The supplied dated literature assessment reports polynomiality and integrality; taking that reported integrality as part of the intended setup, write \[ P_{C,d}(T)=\sum_{j\geq 0}a_{C,d,j}T^j,\qquad a_{C,d,j}\in{\mathbb{Z}}, \] where only finitely many coefficients are nonzero. The question is whether \[ \forall C\text{ almost concealed-canonical},\quad \forall d\text{ a dimension vector for }C,\quad \forall j\in{\mathbb{Z}}_{\geq 0},\qquad a_{C,d,j}\geq 0. \] Equivalently, is $P_{C,d}(T)\in{\mathbb{Z}}_{\geq 0}[T]$ for every such pair $(C,d)$, where ${\mathbb{Z}}_{\geq 0}[T]$ denotes the polynomials with nonnegative integer coefficients?

The supplied record does not define an almost concealed-canonical algebra or the construction of $P_{C,d}(T)$, and it does not specify the base field, the indexing and admissibility conventions for dimension vectors, or any boundary restrictions. These conventions must be recovered from the cited source before this can be regarded as a fully specified mathematical statement; the displayed quantifiers retain the source's intended domains without selecting additional hypotheses.

Short English statement

For every almost concealed-canonical algebra $C$ and every dimension vector $d$ admitted by the source, must every coefficient of its polynomial $P_{C,d}(T)$ be nonnegative? The polynomial construction and algebraic conventions require clarification from the cited source. The supplied record labels the problem open; this draft does not independently verify that status.

Sources

• Local registry: supplied complete source record for permanent UnsolvedMath ID $30003424$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The recorded proposed year is 2017; no individual proposer is supplied.
• Supplied report citation: Representation Theory of Quivers and Finite Dimensional Algebras (2018), Oberwolfach Reports, pp. 591--681, DOI 10.4171/owr/2017/12. https://doi.org/10.4171/owr/2017/12. The supplied background describes Oberwolfach Reports as records of mathematical workshops and their open-problem discussions.
• Plamondon--Schiffmann, Kac polynomials for canonical algebras, supplied literature reference: https://arxiv.org/abs/1602.01207.
• Oberwolfach primary report, supplied literature reference: https://ems.press/content/serial-article-files/46676?nt=1.
• Literature assessment dated exactly 2026-08-21: supplied dated curation, not independently verified. It reports that the cited work establishes polynomiality and integrality for the relevant Kac-type polynomials, while nonnegativity of all coefficients for almost concealed-canonical algebras remains a conjecture in later primary literature. The supplied background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and records a manual semantic, typographic, and LaTeX review claiming that the clean statement is self-contained and preserves the target. These are supplied review claims; they do not resolve the missing definitions identified here.