Moment Vanishing and Polynomial Abel Equations

Definitions and mathematical statement

Let $n\geq 1$ be an integer and let $\Omega\subset{\mathbb{R}}^n$ be a bounded domain. For polynomial functions $P$ and $q$ on ${\mathbb{R}}^n$, define \[ m_k(P,q;\Omega)=\int_{\Omega}P(x)^kq(x)\,{\,\mathrm{d}} x, \qquad k\in{\mathbb{Z}}_{\geq 0}, \] where ${\,\mathrm{d}} x$ denotes Lebesgue measure and $P^0$ denotes the constant polynomial $1$. These integrals are finite because $\Omega$ is bounded and polynomial functions are bounded on its closure. The source does not specify whether the polynomial coefficients are real or complex; the expression makes sense in either case, but the intended coefficient class remains unresolved.

The stated moment problem is to determine necessary and sufficient conditions on $(\Omega,P,q)$ for \[ \forall k\in{\mathbb{Z}}_{\geq 0},\qquad m_k(P,q;\Omega)=0. \] No additional regularity or algebraic hypotheses on $\Omega$ are supplied.

The record also requests corresponding closed-solution conditions for the polynomial Abel equation \[ \frac{{\,\mathrm{d}} y}{{\,\mathrm{d}} x}=p(x)y(x)^2+q(x)y(x)^3, \qquad P'(x)=p(x). \] This derivative and ordinary differential equation require a one-variable interpretation. The source does not specify an interval, endpoint conditions, the meaning of a closed solution, the class of solutions being quantified over, or how this one-variable setting is related to the preceding $n$-dimensional moment problem. Consequently, the Abel-equation request is an unresolved part of the formulation: no precise boundary condition or equivalence with moment vanishing can be asserted from the supplied record.

Short English statement

Characterize when every integral of $P^kq$ over a bounded domain vanishes. The record also asks for related closed-solution conditions for a polynomial Abel equation, but omits the interval, closure condition, solution quantifiers, and precise connection to the moments. This is a statement-only draft requiring human mathematical review.

Sources

• Local UnsolvedMath registry, supplied complete record: permanent ID $30002561$, title Moment Vanishing and Polynomial Abel Equations, exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. The proposed year is recorded as 2014; no individual proposer is supplied.
• Supplied report citation: Real Algebraic Geometry With A View Toward Systems Control and Free Positivity (2015), Oberwolfach Reports, pp. 977--1045, DOI 10.4171/owr/2014/17. Supplied source URL: https://doi.org/10.4171/owr/2014/17. The background describes these reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature source: Oberwolfach Report 2014/17, https://doi.org/10.4171/owr/2014/17. Literature checked at: 2026-08-21. This assessment is supplied dated curation, not independently verified. It labels the record open because the extracted passage is an incomplete setup for a moment-vanishing problem and contains no verifiable final assertion; it classifies the unresolved or unclear content as open and requests independent verification of time-sensitive status and cited claims.
• The registry separately labels the statement corrected_verified, reporting manual semantic, typographic, and LaTeX review and describing the cleanup as self-contained and meaning-preserving. This supplied review label does not resolve the formulation gaps identified above or establish mathematical correctness. No report passage or pinpoint location within the supplied page range is provided.