Border-rank deficiency forcing strict submultiplicativity at the tensor square
Let $V$ be a finite-dimensional complex vector space and $X \subset \mathbb{P}(V)$ an irreducible, reduced projective variety that is nondegenerate, that is, not contained in a hyperplane. For $p \in \mathbb{P}(V)$ the $X$-rank $R_X(p)$ is the least number of points of $X$ whose linear span contains $p$, and the border $X$-rank $\underline{R}_X(p)$ is the least $s$ such that $p$ lies in the Zariski closure of $\{q \in \mathbb{P}(V) : R_X(q) \le s\}$. Let $X \times X \subset \mathbb{P}(V \otimes V)$ be the image of $X \times X$ under the Segre embedding $([x], [y]) \mapsto [x \otimes y]$, and let $p \otimes p = [v \otimes v]$ for any representative $v$ of $p$; when $X$ is a variety of tensors, both sets of factors are kept, so this is not a Kronecker product. Always $R_{X \times X}(p \otimes p) \le R_X(p)^2$. Ballico, Bernardi, Gesmundo, Oneto and Ventura proved the implication below when $X$ is a rational normal curve or the image of the third Veronese embedding of $\mathbb{P}^2$, and observed that if $\underline{R}_X(p) < R_X(p)$, then $R_{X^{\times k}}(p^{\otimes k}) < R_X(p)^k$ for some $k$, where $X^{\times k} \subset \mathbb{P}(V^{\otimes k})$ is the $k$-fold Segre product.
Problem. Does $\underline{R}_X(p) < R_X(p)$ imply $R_{X \times X}(p \otimes p) < R_X(p)^2$ for every such $V$ and $X$ and every $p \in \mathbb{P}(V)$?
Source: Ballico, Bernardi, Gesmundo, Oneto and Ventura (2021), Conjecture 1.1; restated by Oneto and Ventura (2025), Conjecture 4.9.
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- AG — Algebraic Geometry
- Secondary
- NA
- Keywords
- border ranks, OpenProblemsInNLA, secant varieties, tensor products
- Literature checked
- Oct. 8, 2026 by clemens-admin
References
@article{ballico2021,
author = {Ballico, Edoardo and Bernardi, Alessandra and Gesmundo, Fulvio and Oneto, Alessandro and Ventura, Emanuele},
title = {Geometric conditions for strict submultiplicativity of rank and border rank},
journal = {Ann. Mat. Pura Appl. (4)},
volume = {200},
number = {1},
year = {2021},
pages = {187--210},
doi = {10.1007/s10231-020-00991-6},
eprint = {1909.03811},
archivePrefix = {arXiv},
}
@article{oneto2025,
author = {Oneto, Alessandro and Ventura, Emanuele},
title = {Ranks of tensors: geometry and applications},
journal = {Boll. Unione Mat. Ital.},
volume = {18},
number = {3},
year = {2025},
pages = {751--778},
doi = {10.1007/s40574-025-00472-9},
}
@misc{openproblemsinnla-tr27,
author = {Townsend, Alex},
title = {{TR-27}: Border-rank deficiency forcing strict submultiplicativity at the tensor square},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/tensor-computations/TR-27/README.md},
note = {Read at commit eb7c9d6, 2026-10-07},
}
Resolution claims
An explicit smooth, irreducible, nondegenerate curve $X \subset \mathbb{P}^{11}$ and a point $p$ satisfy $\underline{R}_X(p) = 2$, $R_X(p) = 3$, and $R_{X \times X}(p \otimes p) = 9$. Thus the implication conjectured by Ballico, Bernardi, Gesmundo, Oneto and Ventura is false for its stated class of projective varieties. More generally, for every $r \ge 3$ and every $m \ge 2$, …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-08 | Needs verification → Solution claimed | clemens-admin | Listed with the record of its solution, unreviewed. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00216, "Border-rank deficiency forcing strict submultiplicativity at the tensor square", clopen.solutions.