Atomic Measures on Irreducible Truncated Moment Varieties

Definitions and mathematical statement

To describe the question, use the following conventional real multivariate notation. Let $d\geq 1$ be an ambient dimension and let $n\geq 0$ be a truncation order; neither is specified in the supplied record. For $\alpha\in{\mathbb{Z}}_{\geq 0}^d$, write $|\alpha|=\sum_i\alpha_i$ and $x^\alpha=\prod_i x_i^{\alpha_i}$. A real truncated moment sequence is an array $\beta=(\beta_\alpha)_{|\alpha|\leq 2n}$. Its moment matrix is \[ M^{(n)}(\beta)=(\beta_{\alpha+\gamma})_{|\alpha|,|\gamma|\leq n}. \] For $f\in{\mathbb{R}}[x_1,\ldots,x_d]$ of degree at most $n$, let $\widehat f$ denote its coefficient vector in the monomial indexing of this matrix. Under the usual convention, the matrix is recursively generated if \[ M^{(n)}(\beta)\widehat f=0 \quad\Longrightarrow\quad M^{(n)}(\beta)\widehat{fg}=0 \] whenever $f,g\in{\mathbb{R}}[x_1,\ldots,x_d]$ and $\deg f,\deg g,\deg(fg)\leq n$. Define its moment variety by \[ \mathcal V_n(\beta)=\{x\in{\mathbb{R}}^d:f(x)=0\text{ for every }f\in{\mathbb{R}}[x_1,\ldots,x_d]_{\leq n}\text{ with }M^{(n)}(\beta)\widehat f=0\}. \] A representing measure is a finite positive Borel measure $\mu$ on ${\mathbb{R}}^d$, with the required monomials integrable, such that \[ \beta_\alpha=\int_{{\mathbb{R}}^d}x^\alpha\,{\,\mathrm{d}}\mu(x)\qquad(|\alpha|\leq 2n). \] For $r=\operatorname{rank}M^{(n)}(\beta)$, an $r$-atomic representing measure has the form \[ \mu=\sum_{j=1}^{r}w_j\delta_{a_j},\qquad w_j>0,\quad a_j\in{\mathbb{R}}^d\text{ pairwise distinct}, \] with the empty sum interpreted as the zero measure when $r=0$.

The supplied question concerns an irreducible real algebraic set $V=Z_{{\mathbb{R}}}(p)$, where $Z_{{\mathbb{R}}}(p)=\{x\in{\mathbb{R}}^d:p(x)=0\}$. Here irreducible means nonempty and not the union of two proper real algebraic subsets. The record does not clarify whether $p$ is a single polynomial or notation for a family of defining equations.

Rank-atomic existence question, with an unresolved association hypothesis. Must every recursively generated $M^{(n)}(\beta)$ associated with such a $V$ have an $r$-atomic representing measure? If "associated with" means $\mathcal V_n(\beta)=V$, one precise conditional reading is \[ \begin{gathered} \text{for every }d,n,\beta\text{ as above, if }M^{(n)}(\beta)\text{ is recursively generated}\\ \text{and }\mathcal V_n(\beta)=Z_{{\mathbb{R}}}(p)\text{ is irreducible, then there exist distinct }a_1,\ldots,a_r\in{\mathbb{R}}^d\\ \text{and }w_1,\ldots,w_r>0\text{ such that } \beta_\alpha=\sum_{j=1}^{r}w_j a_j^\alpha\quad(|\alpha|\leq 2n), \qquad r=\operatorname{rank}M^{(n)}(\beta). \end{gathered} \] This reading is provisional: the source does not specify equality with the moment variety, positive semidefiniteness, prior existence of a representing measure, normalization, or whether support on $V$ is an explicit requirement. These hypotheses cannot be silently supplied. The background assessment also refers to an "$r$-to-the-$n$-atomic" measure, without defining $r$; that wording is not reconciled with the rank-atomic target in the statement fields.

Extension question. An extension is a real sequence $\widetilde\beta=(\widetilde\beta_\alpha)_{|\alpha|\leq 2n+2}$ satisfying $\widetilde\beta_\alpha=\beta_\alpha$ for $|\alpha|\leq 2n$, with moment matrix $M^{(n+1)}(\widetilde\beta)$. The source asks how the variety changes on passing to this extension. If the intended varieties are the moment varieties, the target is to characterize the possible $\mathcal V_{n+1}(\widetilde\beta)$ and their relation to $\mathcal V_n(\beta)$. The admissible extensions and the intended meaning of "change" are unspecified; in particular, positivity, recursive generation, flatness, and continued irreducibility are not stated as extension requirements.

Short English statement

For an irreducible real algebraic set, the supplied problem asks whether every associated recursively generated truncated moment matrix has a representing measure with exactly its rank many atoms, and how its variety can change under extension by one order. The association with the variety, positivity hypotheses, admissible extensions, and a conflicting atomic-count phrase remain unresolved. This is a statement-only draft requiring human mathematical review.

Sources

• Local UnsolvedMath registry, supplied complete record: permanent ID $30003441$, source problem code catalogue entry, title "Atomic Measures on Irreducible Truncated Moment Varieties," in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2017.
• Supplied report citation: Real Algebraic Geometry With a View Toward Moment Problems and Optimization (2018), Oberwolfach Reports, pp. 771--862, DOI 10.4171/owr/2017/14. https://doi.org/10.4171/owr/2017/14. The background attributes the collection to workshop reports and their open-problem discussions. No exact problem page or passage is supplied.
• Supplied literature source: "Curto--Fialkow--Moeller, extremal truncated moments." https://arxiv.org/abs/math/0610882.
• Supplied literature source: "Oberwolfach truncated-moment report." https://doi.org/10.4171/owr/2017/14.
• The literature assessment is supplied dated curation checked on 2026-08-21, not independently verified here. It labels the problem open and states that flat-extension and extremal cases give necessary and sufficient criteria in important regimes, while the irreducible-variety questions about an "$r$-to-the-$n$-atomic" measure and evolution under extension remain open in general. The supplied background calls this dated literature triage derived from OWR curation and requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified, explaining that notation, typography, and sentence structure were repaired while preserving mathematical content. This supplied editorial label does not establish the mathematical correctness or completeness of the formulation.