Border Comon's conjecture over the complex numbers
For a symmetric tensor $T \in \operatorname{Sym}^d(\mathbb{C}^n) \subset (\mathbb{C}^n)^{\otimes d}$ let $\underline{R}(T)$ be its border rank and $\underline{R}_{\mathrm{sym}}(T)$ its symmetric border rank, both over $\mathbb{C}$. Always $\underline{R}(T) \le \underline{R}_{\mathrm{sym}}(T)$. The analogous equality for rank, Comon's conjecture, is false (Shitov). Equality holds when $d = 3$ and $T$ is 1-generic of border rank $n$ (Landsberg), and when $T$ is concise of border rank $n$ and $n \le d + 1$ (Mańdziuk and Ventura).
Problem. Is $\underline{R}(T) = \underline{R}_{\mathrm{sym}}(T)$ for every $d \ge 3$, every $n \ge 2$ and every $T \in \operatorname{Sym}^d(\mathbb{C}^n)$?
Source: Asked in Oeding's report of the 2008 AIM workshop (2008), §2, open problem (1), and by Buczyński, Ginensky and Landsberg (2013), §1.3; stated as a conjecture for order three by Landsberg (2017), Conjecture 5.6.1.5, and in this form by Mańdziuk and Ventura (2026).
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- AG — Algebraic Geometry
- Secondary
- NA
- Keywords
- border ranks, OpenProblemsInNLA, secant varieties, symmetric tensors
- Literature checked
- Oct. 6, 2026 by clemens-admin
References
@misc{oeding2008,
author = {Oeding, Luke},
title = {Report on ``{G}eometry and representation theory of tensors for computer science, statistics and other areas''},
year = {2008},
eprint = {0810.3940},
archivePrefix = {arXiv},
}
@article{buczynski2013,
author = {Buczy{\'n}ski, Jaros{\l}aw and Ginensky, Adam and Landsberg, J. M.},
title = {Determinantal equations for secant varieties and the {E}isenbud--{K}oh--{S}tillman conjecture},
journal = {J. Lond. Math. Soc. (2)},
volume = {88},
number = {1},
year = {2013},
pages = {1--24},
doi = {10.1112/jlms/jds073},
eprint = {1007.0192},
archivePrefix = {arXiv},
}
@book{landsberg2017,
author = {Landsberg, J. M.},
title = {Geometry and Complexity Theory},
series = {Cambridge Stud. Adv. Math.},
volume = {169},
publisher = {Cambridge University Press},
address = {Cambridge},
year = {2017},
doi = {10.1017/9781108183192},
}
@article{mandziuk2026,
author = {Ma{\'n}dziuk, Tomasz and Ventura, Emanuele},
title = {Symmetrization maps and minimal border rank {C}omon's conjecture},
journal = {Selecta Math. (N.S.)},
volume = {32},
number = {4},
year = {2026},
pages = {86},
doi = {10.1007/s00029-026-01182-6},
eprint = {2411.05721},
archivePrefix = {arXiv},
}
@article{shitov2018,
author = {Shitov, Yaroslav},
title = {A counterexample to {C}omon's conjecture},
journal = {SIAM J. Appl. Algebra Geom.},
volume = {2},
number = {3},
year = {2018},
pages = {428--443},
doi = {10.1137/17M1131970},
eprint = {1705.08740},
archivePrefix = {arXiv},
}
@misc{openproblemsinnla-tr10,
author = {Townsend, Alex},
title = {{TR-10}: Border Comon's conjecture over the complex numbers},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/tensor-computations/TR-10/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-00127, "Border Comon's conjecture over the complex numbers", clopen.solutions.