Strassen's asymptotic rank conjecture for tight tensors
A tensor $T \in (\mathbb{C}^d)^{\otimes 3}$ is concise if its three flattenings all have rank $d$, and tight if in some bases there are injective maps $a, b, c : \{1, \dots, d\} \to \mathbb{Z}$ with $a(i) + b(j) + c(k) = 0$ whenever $T_{ijk} \ne 0$. Let $R$ be tensor rank over $\mathbb{C}$, and $T^{\otimes m} \in (\mathbb{C}^{d^m})^{\otimes 3}$ the $m$-th Kronecker power. The conjecture below would imply that the exponent of matrix multiplication is $2$, since the $2 \times 2$ matrix multiplication tensor is concise and tight. Björklund and Kaski showed that it and the Set Cover Conjecture are not both true.
Problem. Is $\lim_{m \to \infty} R(T^{\otimes m})^{1/m} = d$ for every $d \ge 1$ and every concise tight tensor $T \in (\mathbb{C}^d)^{\otimes 3}$?
Source: Strassen (1994), Conjecture 5.3, stated there through support functionals; in this form, Conner, Gesmundo, Landsberg, Ventura and Wang (2021), Conjecture 1.4.
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- AG — Algebraic Geometry
- Secondary
- NA
- Keywords
- asymptotic rank, matrix multiplication, OpenProblemsInNLA, tensor rank
- Literature checked
- Oct. 2, 2026 by clemens-admin
References
@incollection{strassen1994,
author = {Strassen, Volker},
title = {Algebra and complexity},
booktitle = {First European Congress of Mathematics, Vol. II (Paris, 1992)},
series = {Progr. Math.},
volume = {120},
publisher = {Birkh{\"a}user},
address = {Basel},
year = {1994},
pages = {429--446},
doi = {10.1007/978-3-0348-9112-7_18},
}
@article{conner2021,
author = {Conner, Austin and Gesmundo, Fulvio and Landsberg, Joseph M. and Ventura, Emanuele and Wang, Yao},
title = {Towards a geometric approach to {S}trassen's asymptotic rank conjecture},
journal = {Collect. Math.},
volume = {72},
number = {1},
year = {2021},
pages = {63--86},
doi = {10.1007/s13348-020-00280-8},
}
@inproceedings{bjorklund2024,
author = {Bj{\"o}rklund, Andreas and Kaski, Petteri},
title = {The Asymptotic Rank Conjecture and the Set Cover Conjecture Are Not Both True},
booktitle = {Proceedings of the 56th Annual ACM Symposium on Theory of Computing (STOC 2024)},
publisher = {ACM},
year = {2024},
pages = {859--870},
doi = {10.1145/3618260.3649656},
}
@misc{lee2026,
author = {Lee, Kisun},
title = {Asymptotic rank bounds: a numerical census},
year = {2026},
eprint = {2601.08119},
archivePrefix = {arXiv},
}
@misc{openproblemsinnla-ac05,
author = {Townsend, Alex},
title = {{AC-05}: Strassen's asymptotic rank conjecture for tight tensors},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/arithmetic-and-complexity/AC-05/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-00068, "Strassen's asymptotic rank conjecture for tight tensors", clopen.solutions.