Duplicate Moment-Matrix Question

Definitions and mathematical statement

The supplied question is: What moment inequalities can be deduced from positive semidefiniteness of moment matrices? The record does not specify the underlying moment domain, matrix indexing, truncation orders, normalization, support conditions, or permitted forms of inequalities. Consequently, it does not determine a unique concrete mathematical problem.

An abstract formulation, conditional on supplying these missing data, is as follows. Let $D$ be a specified set of admissible moment data, let $\Lambda$ index the moment matrices under consideration, and let $M_\lambda(y)$ be a finite Hermitian matrix for each $y\in D$ and $\lambda\in\Lambda$. For a Hermitian matrix $A$, write \[ A\succeq0 \quad\Longleftrightarrow\quad v^*Av\geq0\text{ for every complex vector }v\text{ of the appropriate dimension}. \] This convention also accommodates real symmetric matrices. Given a specified class $\mathcal C$ of real-valued expressions on $D$, the characterization question is to determine \[ \left\{F\in\mathcal C:\ \forall y\in D,\quad \left(M_\lambda(y)\succeq0\text{ for every }\lambda\in\Lambda\right) \Longrightarrow F(y)\geq0\right\}. \] Here $D$, $\Lambda$, the matrix-entry rules, and $\mathcal C$ are unresolved parameters, not hypotheses recovered from the source. In particular, whether $D$ consists of formal moment sequences or moments of actual measures is unspecified.

The supplied verification labels this record unrecoverable as a separate statement: it duplicates the first question represented canonically by OWR-17475-006, where a localization-matrix extension is also represented. No separate clean statement is promoted for the present record. That extension is mentioned for provenance only; its mathematical formulation was not supplied.

Short English statement

The question asks which inequalities between moments are forced by positive semidefiniteness of the associated moment matrices. The source leaves the moment setting and the class of inequalities unspecified, so only a conditional characterization framework can be stated. This record is retained as a duplicate requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004441$, exact source problem code catalogue entry, title Duplicate Moment-Matrix Question, in the Oberwolfach Reports Open Problems collection. The recorded proposed year is 2020; no individual proposer is supplied.
• Supplied source citation: Real Algebraic Geometry with a View Toward Hyperbolic Programming and Free Probability (2021), Oberwolfach Reports, pp. 639--712, DOI 10.4171/owr/2020/12. https://doi.org/10.4171/owr/2020/12. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: Blekherman--Rincon--Sinn--Vinzant--Yu (2025). https://doi.org/10.1112/jlms.70311.
• Supplied literature reference: Oberwolfach Report 12/2020. https://doi.org/10.4171/owr/2020/12.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved and states that known tropical and moment-cone results answer substantial special cases and exhibit permanent SOS limitations, while the broad characterization remains incomplete. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; its status and cited claims require independent verification.
• Supplied statement-verification note: this record repeats the first question represented canonically, together with its localization-matrix extension, by OWR-17475-006. The original remains publicly visible for provenance, and no separate clean statement is promoted.