Equality of ranks for generic odd-order Hankel tensors
Fix an odd integer $m \ge 5$ and an integer $n \ge 2$. For $h = (h_0, \dots, h_{m(n-1)}) \in \mathbb{C}^{m(n-1)+1}$ 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 $1 \le i_1, \dots, i_m \le n$; these tensors form a space identified with $\mathbb{C}^{m(n-1)+1}$. Over $\mathbb{C}$, let $R$ and $R_{\mathrm{sym}}$ be rank and symmetric rank, and $\underline{R}$ and $\underline{R}_{\mathrm{sym}}$ border rank and symmetric border rank, where the approximating tensors for $\underline{R}$ range over all of $(\mathbb{C}^n)^{\otimes m}$ and those for $\underline{R}_{\mathrm{sym}}$ over symmetric tensors. The Vandermonde rank $R_V(H)$ is the least $r$ with $H = \sum_{j=1}^{r} \lambda_j v(a_j, b_j)^{\otimes m}$, where $\lambda_j \in \mathbb{C}$, $(a_j, b_j) \in \mathbb{C}^2 \setminus \{0\}$ and $v(a, b) = (a^{n-1}, a^{n-2} b, \dots, b^{n-1})$. Always $\underline{R} \le R \le R_{\mathrm{sym}} \le R_V$ and $\underline{R} \le \underline{R}_{\mathrm{sym}} \le R_{\mathrm{sym}}$, and $R_V(H) = \lceil (m(n-1)+1)/2 \rceil$ for generic $H$ (Nie and Ye). Nie and Ye proved that all five ranks agree for generic Hankel tensors of every even order and of order three; their examples show that they need not agree for every Hankel tensor.
Problem. Is there, for every odd $m \ge 5$ and every $n \ge 2$, a nonempty Zariski-open subset of the space of Hankel tensors of order $m$ and dimension $n$ on which $$R(H) = R_{\mathrm{sym}}(H) = \underline{R}(H) = \underline{R}_{\mathrm{sym}}(H) = R_V(H)?$$
Source: Nie and Ye (2019), Question 6.1 and Conjecture 6.2.
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- AG — Algebraic Geometry
- Secondary
- NA
- Keywords
- border ranks, Hankel tensors, OpenProblemsInNLA, 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},
}
@misc{openproblemsinnla-tr13,
author = {Townsend, Alex},
title = {{TR-13}: Equality of ranks for generic odd-order Hankel tensors},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/tensor-computations/TR-13/README.md},
note = {Read at commit eb7c9d6, 2026-10-07},
}
Canonical Lean statement
A submission resolves this problem by proving the constant Problem_OP_00210.
namespace RegistryProblems.OP_00210
noncomputable section
open scoped BigOperators
open Filter
abbrev Tensor (m n : ℕ) := (Fin m → Fin n) → ℂ
abbrev Moments (m n : ℕ) := Fin (m * (n - 1) + 1) → ℂ
def pureTensor {m n : ℕ} (v : Fin m → Fin n → ℂ) : Tensor m n :=
fun i => ∏ j, v j (i j)
def symmetricTensor {m n : ℕ} (v : Fin n → ℂ) : Tensor m n :=
pureTensor (fun _ => v)
/-- Homogeneous rational-normal-curve coordinates, including the node at infinity. -/
def vandermondeVector (n : ℕ) (a b : ℂ) : Fin n → ℂ :=
fun i => a ^ (n - 1 - i.val) * b ^ i.val
def OrdinaryRankAtMost {m n : ℕ} (q : ℕ) (T : Tensor m n) : Prop :=
∃ v : Fin q → Fin m → Fin n → ℂ,
T = ∑ j, pureTensor (v j)
def SymmetricRankAtMost {m n : ℕ} (q : ℕ) (T : Tensor m n) : Prop :=
∃ (c : Fin q → ℂ) (v : Fin q → Fin n → ℂ),
T = ∑ j, c j • symmetricTensor (v j)
def VandermondeRankAtMost {m n : ℕ} (q : ℕ) (T : Tensor m n) : Prop :=
∃ (c a b : Fin q → ℂ), (∀ j, (a j, b j) ≠ (0, 0)) ∧
T = ∑ j, c j • symmetricTensor (vandermondeVector n (a j) (b j))
def OrdinaryBorderRankAtMost {m n : ℕ} (q : ℕ) (T : Tensor m n) : Prop :=
∃ U : ℕ → Tensor m n,
(∀ k, OrdinaryRankAtMost q (U k)) ∧ Tendsto U atTop (nhds T)
def SymmetricBorderRankAtMost {m n : ℕ} (q : ℕ) (T : Tensor m n) : Prop :=
∃ U : ℕ → Tensor m n,
(∀ k, SymmetricRankAtMost q (U k)) ∧ Tendsto U atTop (nhds T)
def ordinaryRank {m n : ℕ} (T : Tensor m n) : ℕ :=
sInf {q | OrdinaryRankAtMost q T}
def symmetricRank {m n : ℕ} (T : Tensor m n) : ℕ :=
sInf {q | SymmetricRankAtMost q T}
def vandermondeRank {m n : ℕ} (T : Tensor m n) : ℕ :=
sInf {q | VandermondeRankAtMost q T}
def ordinaryBorderRank {m n : ℕ} (T : Tensor m n) : ℕ :=
sInf {q | OrdinaryBorderRankAtMost q T}
def symmetricBorderRank {m n : ℕ} (T : Tensor m n) : ℕ :=
sInf {q | SymmetricBorderRankAtMost q T}
def hankel {m n : ℕ} (h : Moments m n) : Tensor m n :=
fun i => h ⟨∑ j, (i j).val, by
have hb : (∑ j : Fin m, (i j).val) ≤ m * (n - 1) := by
calc
(∑ j : Fin m, (i j).val) ≤ ∑ _j : Fin m, (n - 1) :=
Finset.sum_le_sum (fun j _ => Nat.le_pred_of_lt (i j).isLt)
_ = m * (n - 1) := by simp
omega⟩
/-- Ceiling of `(m * (n - 1) + 1) / 2`. -/
def expectedRank (m n : ℕ) : ℕ := (m * (n - 1) + 2) / 2
def AllRanksEqual {m n : ℕ} (T : Tensor m n) (r : ℕ) : Prop :=
ordinaryRank T = r ∧ symmetricRank T = r ∧
ordinaryBorderRank T = r ∧ symmetricBorderRank T = r ∧
vandermondeRank T = r
/-- An explicit principal Zariski-open subset of the finite affine moment space. -/
def principalOpen {m n : ℕ} (p : MvPolynomial (Fin (m * (n - 1) + 1)) ℂ) :
Set (Moments m n) :=
{h | MvPolynomial.eval h p ≠ 0}
/-- OP-00210: Equality of ranks for generic odd-order Hankel tensors -/
def _root_.RegistryProblems.Problem_OP_00210 : Prop :=
∀ m n : ℕ, 5 ≤ m → Odd m → 2 ≤ n →
∃ p : MvPolynomial (Fin (m * (n - 1) + 1)) ℂ,
p ≠ 0 ∧ (principalOpen p).Nonempty ∧
∀ h ∈ principalOpen p, ∃ r : ℕ, AllRanksEqual (hankel h) r
end
end RegistryProblems.OP_00210
Formalised by OpenAI Codex agents (Apache-2.0); reviewed by clemens-admin.
Available to match against in: Lean v4.33.1 with Mathlib v4.33.1.
Resolving this problem in Lean
Add the problems package as a dependency, import this problem's module, and prove the constant. Any published revision works.
In lakefile.toml:
[[require]] name = "RegistryProblems" git = "https://github.com/cthalhammer/registry-problems" rev = "753bebc58e3fee845f4fffec7d24e7c8cc2a689a"
In your proof:
import RegistryProblems.OP_00210 theorem resolution : RegistryProblems.Problem_OP_00210 := by sorry
In preprint.toml, mapping the claim to the declaration that establishes it:
[[statements]] label = "Theorem 1" lean = "resolution" latex = "..." problem = "OP-00210" claim = "full"
To disprove it instead, prove the negation and claim "disproof", or "counterexample" if the proof exhibits one:
theorem disproof : ¬ RegistryProblems.Problem_OP_00210 := by sorry
You can also write the statement out in full; the verifier checks it matches up to definitional equality. Partial results and reductions are not matched against this statement.
Resolution claims
Proof of the full original target, using Prony reconstruction and a Koszul determinant certificate; draft pending authoritative Linux verification and final independent review.
For every odd order $m \ge 3$ and dimension $n \ge 2$, the generic ordinary rank, symmetric rank, ordinary border rank, symmetric border rank and Vandermonde rank of a complex Hankel tensor all equal $\lceil (m(n-1)+1)/2 \rceil$. The lower bound is certified by an explicit three-slice Koszul flattening after grouping the tensor factors. Its rank is established using a two-spike …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-20 | Solution claimed → Lean verified | The registry's curators | TR-13: complete generic odd-order Hankel rank equality v1 recorded from OpenProblemsInNLA @ 5487945, TR-13/lean/. |
| 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-00210, "Equality of ranks for generic odd-order Hankel tensors", clopen.solutions.