Positivity and Closure of Flat Moment Matrices

Definitions and mathematical statement

The supplied record leaves the domain of positivity and the precise meaning of flatness implicit. The following gives a conventional real, unconstrained interpretation; these conventions require confirmation against the source.

Fix integers $n\geq 1$ and $m\geq 0$. For $\alpha\in{\mathbb{Z}}_{\geq 0}^{n}$, put $|\alpha|=\alpha_1+\cdots+\alpha_n$ and $x^\alpha=x_1^{\alpha_1}\cdots x_n^{\alpha_n}$. Write ${\mathbb{R}}[x]_{\leq m}$ for the real polynomials in $n$ variables of total degree at most $m$. A real truncated moment sequence is an array \[ \beta^{(m)}=(\beta_\alpha)_{|\alpha|\leq m},\qquad \beta_\alpha\in{\mathbb{R}}. \] Its Riesz functional is the linear map \[ L_\beta:{\mathbb{R}}[x]_{\leq m}\longrightarrow{\mathbb{R}},\qquad L_\beta\!\left(\sum_{|\alpha|\leq m}p_\alpha x^\alpha\right) =\sum_{|\alpha|\leq m}p_\alpha\beta_\alpha. \] Under the full-space interpretation, positivity means \[ L_\beta(p)\geq 0 \quad\text{for every }p\in{\mathbb{R}}[x]_{\leq m} \text{ such that }p(x)\geq 0\text{ for every }x\in{\mathbb{R}}^n. \] This condition must be distinguished from requiring only $L_\beta(q^2)\geq 0$ for every $q$ with $2\deg q\leq m$. The record does not explicitly specify which meaning is intended.

For an integer $d\geq 1$ and a sequence $\beta^{(2d)}$, define the moment matrix \[ M_d(\beta)=\left(\beta_{\alpha+\gamma}\right)_{|\alpha|,|\gamma|\leq d}. \] Here positive means positive semidefinite, denoted $M_d(\beta)\succeq 0$. Under the usual adjacent-order rank convention, this matrix is flat when \[ \operatorname{rank}M_d(\beta)=\operatorname{rank}M_{d-1}(\beta), \] where $M_{d-1}(\beta)$ is its principal submatrix indexed by monomials of degree at most $d-1$. With this convention, put \[ \mathcal S_{n,d}=\left\{M_d(\beta): \beta\in{\mathbb{R}}^{\binom{n+2d}{n}},\ M_d(\beta)\succeq 0, \ \operatorname{rank}M_d(\beta)=\operatorname{rank}M_{d-1}(\beta)\right\}, \qquad F_{n,d}=\overline{\mathcal S_{n,d}}, \] where the closure is taken in the finite-dimensional real vector space of moment matrices, with its Euclidean topology. Equivalently, $A\in F_{n,d}$ means that there exists a sequence $A_k\in\mathcal S_{n,d}$ converging entrywise to $A$.

Problem, subject to the stated conventions. Give concrete conditions on $\beta^{(m)}$ sufficient to ensure positivity of $L_\beta$, and give a concrete necessary-and-sufficient characterization of membership in $F_{n,d}$ for general $n$ and $d$.

The source calls the flat positive matrices a cone. The definition above uses the set of flat positive moment matrices itself; whether the source instead intends a conic hull must be checked. The source also does not specify normalization, a support set, the relation of $m$ to $2d$ beyond the matrix construction, or boundary conventions for $d=0$. No such additional hypotheses are imposed here.

Short English statement

Find concrete sufficient conditions ensuring that a truncated moment sequence has a positive Riesz functional, and characterize limits of flat positive moment matrices. The displayed formulation uses explicit conventional definitions that need source confirmation. The supplied status is partially solved; this is a statement-only draft requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003447$, exact source problem code catalogue entry, title: Positivity and Closure of Flat Moment Matrices. Collection: Oberwolfach Reports Open Problems. The record gives proposed year 2017 and names no individual proposer.
• Supplied source citation: Real Algebraic Geometry With a View Toward Moment Problems and Optimization (2018), Oberwolfach Reports, pp. 771--862, DOI 10.4171/owr/2017/14. Original Oberwolfach report: https://doi.org/10.4171/owr/2017/14. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: Positive Gorenstein ideals, https://doi.org/10.1090/S0002-9939-2014-12253-2. Authors and a precise supporting passage are not supplied in this record.
• Literature assessment date: 2026-08-21. This is supplied dated curation, not independently verified. The supplied assessment labels the problem partially solved and states that concrete characterizations are known in several low-dimensional cases and rank-based positivity criteria are known, while a general concrete characterization of $F$ remains open. The background describes this assessment as dated literature triage derived from the supplied OWR curation and calls for independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified, explaining that mathematical content was preserved while notation, typography, and sentence structure were repaired for publication. This is a supplied editorial label, not independent mathematical verification of this draft or its conventions.