Rational Certificates for Truncated Moment Nonrepresentability

Definitions and mathematical statement

Write ${\mathbb{R}}[x]={\mathbb{R}}[x_1,\ldots,x_n]$, and let $\Sigma_{{\mathbb{R}}}[x]$ be the set of finite sums of squares of polynomials in ${\mathbb{R}}[x]$. For real polynomials $g_1,\ldots,g_r$, define \[ K=\{x\in{\mathbb{R}}^n:g_i(x)\geq0\text{ for }1\leq i\leq r\},\qquad Q_{{\mathbb{R}}}(g)=\left\{\sigma_0+\sum_{i=1}^r\sigma_i g_i:\sigma_i\in\Sigma_{{\mathbb{R}}}[x]\right\}. \] Thus $p\in1+Q_{{\mathbb{R}}}(g)$ means that \[ p-1=\sum_j h_{0j}^2+\sum_{i=1}^r g_i\sum_j h_{ij}^2 \] for finitely many real polynomials $h_{ij}$. Such an identity is called a rational sum-of-squares certificate here if every $h_{ij}$ belongs to ${\mathbb{Q}}[x]$. The assertion that every certificate requires irrational coefficients means that a real certificate exists but no certificate of this rational form exists; it does not merely mean that some coefficients of the expanded multipliers are irrational.

For an integer $d\geq0$, let $A_d=\{\alpha\in{\mathbb{Z}}_{\geq0}^n:|\alpha|\leq d\}$. A rational truncated moment vector is $y=(y_\alpha)_{\alpha\in A_d}\in{\mathbb{Q}}^{A_d}$, with associated functional \[ L_y\!\left(\sum_{\alpha\in A_d}p_\alpha x^\alpha\right)=\sum_{\alpha\in A_d}p_\alpha y_\alpha. \] A representing measure on $K$ is a finite positive Borel measure supported on $K$, with the indicated monomials integrable, such that $y_\alpha=\int_K x^\alpha\,{\,\mathrm{d}}\mu(x)$ for every $\alpha\in A_d$. No normalization $y_0=1$ is supplied.

Under the usual strict interpretation of a separating polynomial, the extracted question asks whether there exist data as above and $p\in{\mathbb{Q}}[x]$ of degree at most $d$ such that $y$ has no representing measure on $K$, \[ L_y(p)<0,\qquad p\in1+Q_{{\mathbb{R}}}(g), \] but there are no finite families $q_{ij}\in{\mathbb{Q}}[x]$ satisfying \[ p-1=\sum_j q_{0j}^2+\sum_{i=1}^r g_i\sum_j q_{ij}^2. \] Here membership in $1+Q_{{\mathbb{R}}}(g)$ gives $p\geq1$ on $K$, so the displayed negative evaluation expresses separation from moment vectors of positive measures.

Unresolved qualifications. This is a conditional formalization of the supplied target: the record does not specify the coefficient field of the generators, the precise meaning of separation, the truncation index set, or any degree bound on certificate summands. In particular, rational generators must not be assumed without clarification, and the strict inequality above may not capture a source convention allowing a nonpositive evaluation. The total-degree indexing is an explicit interpretation of the unspecified truncation. The displayed certificate identities impose no degree bound on their summands. The original extracted sentence refers to an unspecified preceding phenomenon, so these interpretations require human review against the report context.

Short English statement

Can a rational truncated moment vector with no representing positive measure on $K$ have a rational separating polynomial $p\in1+Q(g)$ whose sum-of-squares membership certificates all require irrational coefficients? The displayed formulation uses strict separation and total-degree truncation explicitly; the source leaves these conventions, the coefficient field of the generators, and certificate degree restrictions unresolved. The supplied record labels the problem open; this status is not independently verified here.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005468$, title Rational Certificates for Truncated Moment Nonrepresentability, source code catalogue entry. The record gives proposed year 2023 and names no proposer. Its supplied statement status is corrected_verified; its verification note reports a direct check of the official report, which is not independently repeated here.
• Real Algebraic Geometry with a View toward Koopman Operator Methods (2023), Oberwolfach Reports, pp. 741--811, DOI 10.4171/owr/2023/14. Supplied report and literature-source link: https://doi.org/10.4171/owr/2023/14. The record describes Oberwolfach Reports as documenting workshops and their open-problem discussions; no exact problem page or passage location is supplied.
• Supplied literature reference: Scheiderer, JEMS, https://doi.org/10.4171/JEMS/620. No article title, theorem number, or detailed hypotheses are supplied.
• Literature assessment dated exactly 2026-08-22: supplied dated curation, not independently verified. It labels the problem open and attributes to Scheiderer a result that rational positive certificates can fail to have rational SOS certificates, while stating that occurrence for the specified truncated-moment certificates remains open. The background explicitly requires independent verification of time-sensitive status and cited claims.