Mixed Odd Powers of Sum-of-Squares Polynomials

Definitions and mathematical statement

The coefficient domain and exponent ranges are not explicit in the supplied record. The following is a provisional real-polynomial formulation, requiring confirmation against the source: let $n\geq 1$, let $p,q\in{\mathbb{R}}[x_1,\ldots,x_n]$, and let $i,j,k\in{\mathbb{Z}}_{\geq 0}$. A polynomial $f\in{\mathbb{R}}[x_1,\ldots,x_n]$ is a sum of squares if there exist an integer $r\geq 0$ and polynomials $h_1,\ldots,h_r$ in the same ring such that \[ f=\sum_{\ell=1}^{r}h_\ell^2. \] The first question is whether, for every such $n,p,q,i,j$, \[ \left(p^{2i+1}\text{ is a sum of squares}\right) \ \text{and}\ \left(q^{2j+1}\text{ is a sum of squares}\right) \quad\Longrightarrow\quad (p+q)^{2i+2j+1}\text{ is a sum of squares}. \] For the second question, under the same provisional interpretation of sums of squares over ${\mathbb{R}}$, put \[ M_c(x,y)=x^4y^2+x^2y^4+1-cx^2y^2, \qquad c\in{\mathbb{R}}. \] The supplied statement posits, for each admissible index $k$, a number $c_k$ satisfying $0\leq c_k<3$ and \[ \left\{c\in{\mathbb{R}}:M_c^{\,2k+1}\text{ is a sum of squares in }{\mathbb{R}}[x,y]\right\} = \left\{c\in{\mathbb{R}}:c\leq c_k\right\}. \] Determine \[ \lim_{k\to\infty}c_k. \] The displayed threshold equality, including attainment of the endpoint, is part of the supplied formulation; its justification is not supplied. The starting index for $k$ is also unspecified.

As background, the original extraction labels the following assertion a theorem: if $p^{2k+1}$ and $q$ are sums of squares, then $(p+q)^{2k+1}$ is a sum of squares. Separately, the supplied literature assessment describes the proved result as a common-exponent special case. These descriptions are retained as source reports, without asserting their precise relationship or independently verifying either. The extracted proof hint is corrupted, and an added remark is truncated; neither supplies a reliable additional mathematical statement.

Short English statement

Ask whether odd powers of two polynomials being sums of squares forces the indicated odd power of their sum to be a sum of squares. Also determine the limiting sum-of-squares threshold for the displayed bivariate family. The real coefficient domain and nonnegative integer exponent ranges are provisional interpretations requiring review.

Sources

• Local UnsolvedMath registry: supplied record with permanent ID $30005461$, title Mixed Odd Powers of Sum-of-Squares Polynomials, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is identified; the supplied proposed year is 2023.
• Supplied report citation: Real Algebraic Geometry with a View toward Koopman Operator Methods (2023), Oberwolfach Reports, pp. 741--811, DOI 10.4171/owr/2023/14. Source URL: https://doi.org/10.4171/owr/2023/14. The supplied literature-source entry, labelled "OWR report and stated conjecture," uses the same link.
• Supplied literature check date: 2026-08-22. This is supplied dated curation, not an independently verified assessment. It labels the problem open, describes only the common-exponent special case as proved, and reports that no later primary result located in that curation settles either the mixed-exponent conjecture or the limit question. Its background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry supplies the statement-status label corrected_verified and reports that the official report was checked directly to restore definitions and remove extraction artifacts. These are supplied provenance claims; no source visit or independent verification is claimed in this draft.