Irreducibility of asymptotic tensor-rank sublevel varieties
Let $\mathbb{F}$ be an algebraically closed field, $k \ge 3$, and $V = \mathbb{F}^{d_1} \otimes \cdots \otimes \mathbb{F}^{d_k}$. For $T \in V$ let $$\widetilde R(T) = \lim_{m \to \infty} R(T^{\boxtimes m})^{1/m}$$ be its asymptotic rank, where $R$ is tensor rank over $\mathbb{F}$, with $R(0) = 0$, and $T^{\boxtimes m}$ is the $m$-th Kronecker power. For real $a \ge 0$ let $$V_{\le a} = \{T \in V : \widetilde R(T) \le a\},$$ which contains $0$. Christandl, Hoeberechts, Nieuwboer, Vrana and Zuiddam proved that these sets are Zariski closed, over any field. For matrices ($k = 2$) asymptotic rank is matrix rank, and these sets are irreducible determinantal varieties. They observe that if every $V_{\le a}$ is irreducible, then at most $d_1 \cdots d_k + 1$ values of $\widetilde R$ occur on $V$.
Problem. Is $V_{\le a}$ irreducible for every algebraically closed field $\mathbb{F}$, every $k \ge 3$, all positive integers $d_1, \dots, d_k$ and every real $a \ge 0$?
Source: Christandl, Hoeberechts, Nieuwboer, Vrana and Zuiddam (2025), Section 5.
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- AG — Algebraic Geometry
- Secondary
- NA
- Keywords
- asymptotic ranks, irreducible varieties, OpenProblemsInNLA
- Literature checked
- Oct. 6, 2026 by clemens-admin
References
@inproceedings{christandl2025,
author = {Christandl, Matthias and Hoeberechts, Koen and Nieuwboer, Harold and Vrana, P{\'e}ter and Zuiddam, Jeroen},
title = {Asymptotic Tensor Rank Is Characterized by Polynomials},
booktitle = {Proceedings of the 57th Annual ACM Symposium on Theory of Computing (STOC 2025)},
publisher = {ACM},
year = {2025},
pages = {750--755},
doi = {10.1145/3717823.3718122},
eprint = {2411.15789},
archivePrefix = {arXiv},
}
@misc{openproblemsinnla-tr23,
author = {Townsend, Alex},
title = {{TR-23}: Irreducibility of asymptotic tensor-rank sublevel varieties},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/tensor-computations/TR-23/README.md},
note = {Read at commit e375a6f, 2026-09-30},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00133, "Irreducibility of asymptotic tensor-rank sublevel varieties", clopen.solutions.