OP-00211 Solution claimed

Comon's exact-rank conjecture for Hankel tensors

For integers $m \ge 3$ and $n \ge 2$ and $h = (h_0, \dots, h_{m(n-1)}) \in \mathbb{C}^{m(n-1)+1}$, let $H$ be the Hankel tensor of order $m$ and dimension $n$ with entries $H_{i_1 \cdots i_m} = h_{i_1 + \cdots + i_m - m}$ for $1 \le i_1, \dots, i_m \le n$; it is symmetric. Let $R(H)$ be its rank over $\mathbb{C}$, the least number of terms $u_1 \otimes \cdots \otimes u_m$ with $u_l \in \mathbb{C}^n$ summing to $H$, and $R_{\mathrm{sym}}(H)$ its symmetric rank, the least number of terms $c\, v^{\otimes m}$ with $c \in \mathbb{C}$ and $v \in \mathbb{C}^n$; both are $0$ for $H = 0$. Always $R(H) \le R_{\mathrm{sym}}(H)$. Comon's conjecture, that the two ranks agree for every complex symmetric tensor, is false (Shitov). Nie and Ye proved that they agree for generic Hankel tensors of every even order and of order three.

Problem. Is $R(H) = R_{\mathrm{sym}}(H)$ for every $m \ge 3$, every $n \ge 2$ and every $h \in \mathbb{C}^{m(n-1)+1}$?

Source: Nie and Ye (2019), Conjecture 6.3.

Imported from github.com/ajt60gaibb/OpenProblemsInNLA.

Classification

Primary: AG — Algebraic Geometry
Secondary: NA
Keywords: Hankel tensors, OpenProblemsInNLA, symmetric tensors, tensor ranks
Literature checked: Oct. 8, 2026 by clemens-admin

References

@article{nie2019,
  author        = {Nie, Jiawang and Ye, Ke},
  title         = {Hankel Tensor Decompositions and Ranks},
  journal       = {SIAM J. Matrix Anal. Appl.},
  volume        = {40},
  number        = {2},
  year          = {2019},
  pages         = {486--516},
  doi           = {10.1137/18M1168285},
  eprint        = {1706.03631},
  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-tr14,
  author       = {Townsend, Alex},
  title        = {{TR-14}: Comon's exact-rank conjecture for Hankel tensors},
  howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
  year         = {2026},
  url          = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/tensor-computations/TR-14/README.md},
  note         = {Read at commit eb7c9d6, 2026-10-07},
}

Resolution claims

Exact Rank of Complex Hankel Tensors v1 GitHub ↗

Sidney Holden

Let $H_{i_1 \cdots i_m} = h_{i_1 + \cdots + i_m}$ for $0 \le i_j \le n-1$, over $\mathbb{C}$. Write $D = m(n-1)$, and let $r$ be the rank of the middle Hankel catalecticant. For $H \ne 0$, choose a homogeneous binary apolar polynomial of smallest degree $r$, and let $s$ be its number of distinct projective zeros. We prove the …

Full resolution · not verified · fidelity not established · posted on GitHub 12 Sep 2026

Status history

When	Change	By	Reason
2026-10-08	Needs verification → Solution claimed	clemens-admin	Listed with the record of its solution, unreviewed.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00211, "Comon's exact-rank conjecture for Hankel tensors", clopen.solutions.