Single Polynomial Description of Binary Tensor Gram Loci

Definitions and mathematical statement

For a positive integer $n$, a real binary tensor with $n$ factors is an element of \[ T_n=({\mathbb{R}}^2)^{\otimes n}. \] The supplied record does not define the associated Gram locus, its coordinates, or any normalization imposed on tensors. Consequently, its intended mathematical statement cannot be specified uniquely from this record alone.

To express the proposed property without silently resolving those omissions, let $G_n\subseteq A_n\subseteq{\mathbb{R}}^{m_n}$ denote the intended Gram locus and its intended ambient domain, where the coordinate dimension $m_n$, the set $A_n$, and the construction of $G_n$ remain to be supplied. For each fixed format covered by the conjecture, the proposed single-polynomial characterization has the form \[ \exists\,p_n\in{\mathbb{R}}[x_1,\ldots,x_{m_n}]\quad \forall\,x\in A_n,\qquad x\in G_n\ \Longleftrightarrow\ p_n(x)\geq 0. \] Here real coefficients are the unrestricted interpretation of a real polynomial inequality; the record supplies no coefficient restrictions. This display is a conditional formulation of the requested characterization, not a complete definition of the conjectured locus. In particular, equality with $\{x\in{\mathbb{R}}^{m_n}:p_n(x)\geq0\}$ is asserted only if the intended ambient domain is all of ${\mathbb{R}}^{m_n}$. If additional ambient constraints are intended, they must be stated explicitly.

The supplied dated assessment identifies $n\geq4$ as the remaining conjectural range, reports a characterization for three binary factors, and reports that the convex hull is known in general. These are supplied literature-status claims, not independently verified mathematical conclusions. The record does not specify whether a general characterization means existence of a separate polynomial for each fixed $n$, or a uniform construction of such polynomials.

Short English statement

For each fixed binary tensor format, the question asks whether membership in its Gram locus can be characterized by one polynomial being nonnegative. The supplied record leaves the Gram locus, normalization, coordinates, and ambient constraints undefined, so the displayed formulation is conditional. Its dated curation labels the problem partially solved and identifies $n\geq4$ as conjectural; human mathematical review is required.

Sources

• Supplied local registry record, permanent UnsolvedMath ID $30003480$, source problem code catalogue entry, title "Single Polynomial Description of Binary Tensor Gram Loci," in the collection "Oberwolfach Reports Open Problems." No individual proposer is supplied; the proposed year is recorded as 2017.
• Supplied report citation: Algebraic Statistics (2018), Oberwolfach Reports, pp. 1207--1279, DOI 10.4171/owr/2017/20. Supplied source URL and literature-source link labeled "original OWR report": https://doi.org/10.4171/owr/2017/20. The background attributes the record to workshop open-problem discussions documented in Oberwolfach Reports.
• Supplied literature-source link labeled "Seigal on Gram determinants": https://pmc.ncbi.nlm.nih.gov/articles/PMC6051437/. The record supplies no fuller bibliographic citation for this link.
• Literature checked at: 2026-08-21. This is supplied dated curation, not an independently verified assessment. It labels the problem "partially solved" and states that the Gram locus is characterized for three binary factors, its convex hull is known in general, and the proposed single-polynomial description for $n\geq4$ remains conjectural. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says that mathematical content was preserved while notation, typography, and sentence structure were repaired for publication. This supplied editorial label does not establish independent mathematical verification of this draft.