Discreteness from below of asymptotic tensor rank
For a complex tensor $T \in \mathbb{C}^{d_1} \otimes \cdots \otimes \mathbb{C}^{d_k}$ let $$\widetilde R(T) = \lim_{m \to \infty} R(T^{\boxtimes m})^{1/m}$$ be its asymptotic rank, where $R$ is tensor rank over $\mathbb{C}$, with $R(0) = 0$, and $T^{\boxtimes m} \in \mathbb{C}^{d_1^m} \otimes \cdots \otimes \mathbb{C}^{d_k^m}$ is the $m$-th Kronecker power. Christandl, Hoeberechts, Nieuwboer, Vrana and Zuiddam proved that every non-increasing sequence of asymptotic ranks of order-$k$ tensors is eventually constant (discreteness from above), and that the limit of every convergent sequence of them is again such an asymptotic rank. They ask whether asymptotic rank is also discrete from below; the problem below asks this in each fixed format. A positive answer would follow if all the sets $\{T : \widetilde R(T) \le r\}$, which they proved Zariski closed, were irreducible.
Problem. Is $$\{\widetilde R(T) : T \in \mathbb{C}^{d_1} \otimes \cdots \otimes \mathbb{C}^{d_k}\}$$ a finite set for every $k \ge 3$ and all positive integers $d_1, \dots, d_k$? Equivalently, is every convergent sequence of asymptotic ranks of tensors in one format eventually constant?
Source: Christandl, Hoeberechts, Nieuwboer, Vrana and Zuiddam (2025), Section 5, who ask whether asymptotic rank is discrete from below; stated as finiteness in each format by OpenProblemsInNLA, after their Corollary 4.8 (arXiv:2411.15789v2).
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- AG — Algebraic Geometry
- Secondary
- NA
- Keywords
- asymptotic ranks, OpenProblemsInNLA, tensor ranks
- 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-tr22,
author = {Townsend, Alex},
title = {{TR-22}: Discreteness from below of asymptotic tensor rank},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/tensor-computations/TR-22/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-00132, "Discreteness from below of asymptotic tensor rank", clopen.solutions.