Degree-five, six, and nine generation of the Salmon tensor ideal
Let $X \subset \mathbb{C}^4 \otimes \mathbb{C}^4 \otimes \mathbb{C}^4$ be the variety of tensors of border rank at most $4$, the affine cone over the fourth secant variety of the Segre variety $\mathbb{P}^3 \times \mathbb{P}^3 \times \mathbb{P}^3$. Let $I \subset \mathbb{C}[t_{ijk}]$, a polynomial ring in the $64$ entries of the tensor, be the ideal of polynomials vanishing on $X$, and $I_a$ the space of its homogeneous elements of degree $a$. Determining $I$ is Allman's salmon problem. Friedland and Gross proved that $X$ is the zero set of polynomials of degrees $5$, $6$ and $9$ in $I$, and $I_6$ is not contained in the ideal generated by $I_5$ (Bates and Oeding).
Problem. Is $I = \langle I_5 \cup I_6 \cup I_9 \rangle$, the ideal of $\mathbb{C}[t_{ijk}]$ generated by $I_5$, $I_6$ and $I_9$?
Source: Pachter and Sturmfels (2005), Conjecture 3.24, with degrees 5 and 9; in this revised form, Friedland and Gross (2012), §1, after Sturmfels (2009), §2, and Bates and Oeding (2011).
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- AG — Algebraic Geometry
- Secondary
- AC, NA
- Keywords
- border ranks, OpenProblemsInNLA, phylogenetic invariants, secant varieties
- Literature checked
- Oct. 6, 2026 by clemens-admin
References
@incollection{pachter2005,
author = {Pachter, Lior and Sturmfels, Bernd},
title = {Algebra},
booktitle = {Algebraic Statistics for Computational Biology},
editor = {Pachter, Lior and Sturmfels, Bernd},
publisher = {Cambridge University Press},
address = {Cambridge},
year = {2005},
pages = {85--124},
doi = {10.1017/CBO9780511610684.006},
}
@article{friedland2012,
author = {Friedland, Shmuel and Gross, Elizabeth},
title = {A proof of the set-theoretic version of the salmon conjecture},
journal = {J. Algebra},
volume = {356},
number = {1},
year = {2012},
pages = {374--379},
doi = {10.1016/j.jalgebra.2012.01.017},
eprint = {1104.1776},
archivePrefix = {arXiv},
}
@incollection{sturmfels2009,
author = {Sturmfels, Bernd},
title = {Open problems in algebraic statistics},
booktitle = {Emerging Applications of Algebraic Geometry},
editor = {Putinar, Mihai and Sullivant, Seth},
series = {IMA Vol. Math. Appl.},
volume = {149},
publisher = {Springer},
address = {New York},
year = {2009},
pages = {351--363},
doi = {10.1007/978-0-387-09686-5_10},
eprint = {0707.4558},
archivePrefix = {arXiv},
}
@article{bates2011,
author = {Bates, Daniel J. and Oeding, Luke},
title = {Toward a salmon conjecture},
journal = {Experiment. Math.},
volume = {20},
number = {3},
year = {2011},
pages = {358--370},
doi = {10.1080/10586458.2011.576539},
eprint = {1009.6181},
archivePrefix = {arXiv},
}
@misc{jagiella2026,
author = {Jagie{\l}{\l}a, Jakub and Jelisiejew, Joachim},
title = {Unrestrictions and concise secant varieties},
year = {2026},
eprint = {2604.24879},
archivePrefix = {arXiv},
}
@misc{openproblemsinnla-tr24,
author = {Townsend, Alex},
title = {{TR-24}: Degree-five, six, and nine generation of the Salmon tensor ideal},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/tensor-computations/TR-24/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-00134, "Degree-five, six, and nine generation of the Salmon tensor ideal", clopen.solutions.