Sum of Squares Symmetric Quartics in Higher Dimensions

Definitions and mathematical statement

For a real homogeneous quartic polynomial $F\in{\mathbb{R}}[x_1,\ldots,x_n]$, positive definiteness means $F(x)>0$ for every $x\in{\mathbb{R}}^n\setminus\{0\}$. Such a polynomial is a sum of squares if there exist an integer $r\geq1$ and real homogeneous quadratic polynomials $q_1,\ldots,q_r$ such that \[ F=\sum_{j=1}^{r}q_j^2 \] as a polynomial identity. These definitions describe the usual real-form interpretation; the supplied record does not explicitly specify the variable domain or the meanings and normalizations of $M_1,M_3,M_4$.

The recorded expression is \[ P_n(x)=8M_1(x)M_3(x)+nM_4(x). \] The source record describes it as a symmetric quartic form and asserts that it is positive definite and a sum of squares for $n=2,3$. These assertions are supplied background, not independently verified facts here. Under the interpretation that $n$ is the number of real variables, the extracted question asks whether, for every integer $n\geq4$, there exist $r_n\geq1$ and real homogeneous quadratic polynomials $q_{n,1},\ldots,q_{n,r_n}$ satisfying \[ 8M_1(x)M_3(x)+nM_4(x) =\sum_{j=1}^{r_n}q_{n,j}(x)^2. \] This is only a conditional reconstruction: the missing definitions of the $M_k$, their domains, and their dependence on $n$ prevent a standalone precise polynomial statement.

There is also an unresolved source-status conflict. The supplied registry labels the record open, but its statement-verification field says that the official report immediately answers the higher-dimensional question with an identity valid in every dimension, and marks the extraction unrecoverable. That identity is not supplied. Consequently, the displayed question is retained for provenance and is not asserted here to be an unresolved mathematical problem.

Short English statement

The extracted question asks whether the displayed symmetric quartic is a sum of squares in every dimension $n\geq4$, given the supplied claims for $n=2,3$. The notation $M_k$ is undefined, and the registry also reports that the original report already answers the question. A precise standalone open problem cannot be recovered from this record without further source clarification.

Sources

• Local UnsolvedMath registry, permanent ID $30003448$, source problem code catalogue entry, title Sum of Squares Symmetric Quartics in Higher Dimensions; collection: Oberwolfach Reports Open Problems. The supplied record gives proposed year 2017 and no proposer attribution.
• 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. Supplied source URL: https://doi.org/10.4171/owr/2017/14.
• Supplied literature source, labeled "Original Oberwolfach report": https://doi.org/10.4171/owr/2017/14.
• Literature assessment dated exactly 2026-08-21: supplied dated curation, not independently verified. It assigns open status under a triage rule because it describes the extraction as only "What about larger n?" and therefore lacking a standalone claim. The supplied statement field contains a longer quartic question. The background requests independent verification of time-sensitive status and cited claims.
• The supplied statement-verification field instead says that the report immediately gives an identity answering the question in every dimension, that no separate clean statement is promoted, and that the original record remains publicly visible for provenance. This source-verification assessment is supplied evidence, not an independently checked conclusion.