Real Subspaces in Zeros of Elementary Symmetric Polynomials

Definitions and mathematical statement

For a positive integer $m$ and an integer $r$ with $1\le r\le m$, define the elementary symmetric polynomial by \[ e_r(y_1,\ldots,y_m)=\sum_{1\le i_1<\cdots<i_r\le m}y_{i_1}\cdots y_{i_r}, \qquad (y_1,\ldots,y_m)\in{\mathbb{R}}^m, \] and let \[ Z_{m,r}=e_r^{-1}(0)=\{y\in{\mathbb{R}}^m:e_r(y)=0\}. \] A real vector subspace here means a linear subspace of ${\mathbb{R}}^m$, with dimension over ${\mathbb{R}}$.

The supplied conjecture, expressed in the conventional positive-degree range, asks whether for every positive integer $m$ and every even integer $r$ satisfying $2\le r\le m$, \[ \nexists\,L\subseteq{\mathbb{R}}^m\quad\text{such that }L\text{ is a real linear subspace},\quad \dim_{{\mathbb{R}}}L=r,\quad L\subseteq Z_{m,r}. \] Equivalently, every real linear subspace $L\subseteq{\mathbb{R}}^m$ of dimension $r$ must contain a vector $y$ with $e_r(y)\ne0$. The extracted conjecture does not explicitly state the ranges of $m$ and $r$; the displayed ranges make explicit the conventional nontrivial interpretation and require confirmation against the cited source.

The report attributes the conjecture to the author of its reference [5]. It also reports a proved special case, labeled Theorem 7: for $m=2k+1$, with $k\in{\mathbb{N}}$, the set $Z_{m,m-1}$ contains no real vector subspace of dimension $m-1$. The report says that its proof of Theorem 4 includes this special case, which reference [5] had described as particularly difficult. These are supplied source claims, not independently verified conclusions of this draft. The source does not specify the convention for ${\mathbb{N}}$.

The preceding source passage separately says that extending its Theorem 6(b) to $m=2k+1>3$ is technically more complicated and had not been proved. The statement of that theorem is not supplied, so its precise relationship to the conjecture cannot be reconstructed here.

Short English statement

The conjecture asks whether, for every even degree $2\le r\le m$, each $r$-dimensional real linear subspace of ${\mathbb{R}}^m$ contains a point where the elementary symmetric polynomial $e_r$ is nonzero. The report claims the special case $r=m-1$ for odd $m$. The general open status is supplied dated curation, and the intended parameter ranges and source claims require human review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005519$, source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. The project identifies data/unsolvedmath/problems.json and catalog/problems.json as its local registries; their contents were not independently inspected for this draft.
• Tomographic Inverse Problems: Mathematical Challenges and Novel Applications (2023), Oberwolfach Reports, pp. 1105--1194, DOI 10.4171/owr/2023/21. Supplied source report and literature-source link: https://doi.org/10.4171/owr/2023/21.
• Supplied literature source labeled "real symmetric polynomial paper": https://www.math.nthu.edu.tw/~amen/2006/051014-3.pdf. The excerpt attributes the conjecture to the author of reference [5], but supplies neither that author's name nor a complete bibliographic entry establishing the identification of [5] with this link.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the general assertion open and conjectural despite special-case results. This is supplied dated curation, not independently verified literature assessment. The background expressly requests independent verification of time-sensitive status and cited claims.
• The supplied registry labels the statement corrected_verified and records that the official report was checked directly during curation. This draft preserves that provenance claim without claiming an independent source visit or verification. No proposer is named; the supplied proposed year is 2023.