Log-Concavity of Cohen–Macaulay H-Vectors

Definitions and mathematical statement

Use the usual convention that a standard graded algebra is an algebra over a field $k$: \[ R=\bigoplus_{n\geq 0}R_n,\qquad R_0=k,\qquad R=k[R_1],\qquad \dim_k R_1<\infty. \] The supplied source does not specify the field $k$ or any restriction on its characteristic; the intended range of ground fields remains unresolved.

Assume that $R$ is an integral domain and is Cohen--Macaulay, meaning that every local ring $R_{\mathfrak p}$ has depth equal to its Krull dimension. Set $d=\dim R$. Define the $h$-vector $(h_0,\ldots,h_s)$ by the Hilbert series identity \[ \sum_{n\geq 0}\dim_k(R_n)t^n =\frac{h_0+h_1t+\cdots+h_st^s}{(1-t)^d}, \qquad h_s\neq 0. \] Let $X=\operatorname{Proj}(R)$, the projective scheme associated with the grading. The hypothesis that $X$ is normal means that each local ring $\mathcal O_{X,x}$, for $x\in X$, is an integrally closed domain.

Question. Must every such $R$ with normal $\operatorname{Proj}(R)$ have a log-concave $h$-vector, that is, \[ h_i^2\geq h_{i-1}h_{i+1} \qquad\text{for every integer }i\text{ with }1\leq i\leq s-1? \] For $s\leq 1$, this condition is vacuous. This formulation uses the usual field-based meaning of standard graded; it does not resolve the omitted ground-field qualifications. The supplied background concerns polytope normality and does not specify an additional hypothesis or an equivalent formulation of this question.

Short English statement

Does every standard graded Cohen--Macaulay domain whose associated projective scheme is normal have a log-concave Hilbert-series numerator? The source leaves the ground field unspecified. Its supplied open-status label and unrelated polytope literature assessment require independent review.

Sources

• Local registry: data/unsolvedmath/problems.json, supplied record with permanent UnsolvedMath ID $30003601$, title "Log-Concavity of Cohen--Macaulay H-Vectors," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2017.
• Supplied report citation: Mini-Workshop: Lattice Polytopes: Methods, Advances, Applications (2018), Oberwolfach Reports, pp. 2659--2701, DOI 10.4171/owr/2017/44. Supplied source URL: https://doi.org/10.4171/owr/2017/44. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No problem-specific page or named contributor is supplied.
• Supplied literature source: "Hofscheier--Kasprzyk, smooth polytopes and IDP," https://arxiv.org/abs/2512.20680.
• Supplied literature source: "Bruns, questions on normal polytopes," https://www.home.uni-osnabrueck.de/wbruns/brunsw/pdf-article/2-quest-WB.pdf.
• Supplied literature source: "original OWR report," https://doi.org/10.4171/OWR/2017/44.
• Literature assessment dated exactly 2026-08-21: this is supplied dated curation, not independently verified. The record labels the problem open. Its assessment states that the existence of a very ample nonnormal simple polytope would give a counterexample to Oda's conjecture, that no such example is known, and that the general smooth-polytope normality conjecture remains open. These are attributed statements from the supplied assessment; their accuracy and relevance to the present $h$-vector question are unresolved. The background itself calls this a dated literature triage derived from supplied OWR curation and requests independent verification of time-sensitive status and cited claims.