Kronecker Products of Symmetric Persistent Tensors

Masoud Gharahi

REVIEW 3 major objections 3 minor 10 references

Kronecker Products of Symmetric Persistent Tensors

T0 review · 3 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper proves that the Kronecker product of any two symmetric persistent tensors is again persistent, closing the symmetric case of a conjecture that fails in general.

desk verdict A nice result is likely true, but the proof of the key step has a real gap: equality on decomposable tensors does not extend to all tensors without an argument about the Segre ideal. read the letter →

arxiv 2608.11182 v1 pith:UGN56N6S submitted 2026-08-11 math.AG math-phmath.ACmath.MP

Masoud Gharahi This is my paper · ORCID

classification math.AGmath-phmath.ACmath.MP MSC 15A6915A0315A7281P40

keywords symmetricpersistenttensorsKroneckerproductsHessiandeterminantstensorranklowerboundssubstitutionmethodperfect-poweridentity

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Persistent tensors are a recursively defined class that yields certified lower bounds on tensor rank through repeated substitution. Earlier work conjectured that the class is closed under Kronecker products, but a nonsymmetric counterexample showed the unrestricted conjecture is false. This paper establishes the symmetric case: if $f$ and $g$ are symmetric persistent tensors of the same order $n$, then their Kronecker product $f\boxtimes g$ is persistent. The proof first shows that the Hessian matrix of a Kronecker product is the Kronecker product of the factor Hessians up to a normalization, then demonstrates that for persistent factors the polarized Hessian determinant is a perfect power. It follows that symmetric persistence is closed under iterated Kronecker products and Kronecker powers, generating new infinite families of tensors with certified tensor-rank lower bounds.

What carries the argument

The load-bearing machinery is the Hessian characterization of symmetric persistence, combined with two identities for Kronecker products of homogeneous polynomials. The Hessian characterization (Theorem 3, from [GO]) says $f\in\operatorname{Sym}^n\mathbb{C}^d$ is persistent iff there is a nonzero multihomogeneous polynomial $P_f$ of multidegree $(1,\ldots,1)$ such that $\operatorname{Hess}\big(f_{v^{(1)},\ldots,v^{(n-2)}}(x)\big) = P_f(v^{(1)},\ldots,v^{(n-2)})^d$ for every $(n-2)$-tuple of vectors. Proposition 7 and Corollary 8 show that partial differentiation and Hessians are compatible with Kronecker products: $H_{f\boxtimes g} = \frac{1}{n(n-1)} H_f \boxtimes H_g$. Proposition 10 then combines these: for persistent factors, the determinant of the polarized Hessian of $f\boxtimes g$ reduces, on decomposable tuples $v^{(i)}\otimes w^{(i)}$, to the $(d_1d_2)$-th power of the product $\frac{1}{n!}P_fP_g$; multilinearity in each slot lets the author define $P_{f\boxtimes g}$ on the full space $(\mathbb{C}^{d_1}\otimes\mathbb{C}^{d_2})^{\times(n-2)}$, which is exactly the data the Hessian criterion requires.

What would settle it

Compute the $(n-2)$-fold polarized Hessian determinant of $f\boxtimes g$ at a tuple $U^{(1)},\dots,U^{(n-2)}\in\mathbb{C}^{d_1d_2}$ in which at least one $U^{(i)}$ is not of the form $v\otimes w$ — for the quartics of Example 13, take $U^{(1)}=e_0\otimes e_0+e_1\otimes e_1$. If the determinant is not the $(d_1d_2)$-th power of a multihomogeneous expression in the $U^{(i)}$, Proposition 10 and Theorem 11 are false.

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Extended reading notes

Core claim

The paper's central claim is Theorem 11: for $f\in \operatorname{Sym}^n\mathbb{C}^{d_1}$ and $g\in\operatorname{Sym}^n\mathbb{C}^{d_2}$ that are persistent, $f\boxtimes g\in\operatorname{Sym}^n(\mathbb{C}^{d_1}\otimes\mathbb{C}^{d_2})$ is persistent. The proof rests on the Hessian characterization of symmetric persistence from [GO]: a symmetric tensor is persistent exactly when the Hessian determinant of every $(n-2)$-fold partial polarization is a perfect power. Using the differentiation identity $\partial(f\boxtimes g)/\partial z_{ij} = \frac{1}{n}(\partial f/\partial x_i)\boxtimes(\partial g/\partial y_j)$, and hence $H_{f\boxtimes g} = \frac{1}{n(n-1)}H_f\boxtimes H_g$, the author derives a polarized perfect-power identity: the Hessian determinant of any partial polarization of $f\boxtimes g$ is the $(d_1d_2)$-th power of a single multihomogeneous polynomial built from the corresponding polynomials of $f$ and $g$. This exactly matches the Hessian criterion, so persistence is inherited.

Load-bearing premise

The proof depends on a step in which an identity verified only for tuples of the form one vector in the first space times one vector in the second is asserted to hold for every vector in the larger tensor space, even though Remark 9 notes that a similar determinant identity can fail for vectors not of that product form.

Editorial extensions

If this is right

If $f_1,\dots,f_k$ are persistent symmetric tensors of the same order $n$, the iterated Kronecker product $f_1\boxtimes\cdots\boxtimes f_k$ is persistent.
Every Kronecker power $f^{\boxtimes k}$ of a persistent symmetric tensor is persistent, so repeated self-products create persistent tensors in exponentially growing dimensions.
The binary W-state tensor $W_n = x_0^{n-1}x_1$ yields persistent Kronecker powers in $\operatorname{Sym}^n(\mathbb{C}^{2^k})$, giving explicit infinite families of tensors with certified lower bounds on tensor rank.
The Hessian-matrix identity $H_{f\boxtimes g} = \frac{1}{n(n-1)}H_f\boxtimes H_g$ holds for arbitrary symmetric tensors and gives a new tool for studying differential invariants of Kronecker products.
The closure converts the recursive persistence test into a practical certificate: to certify a large Kronecker product, it suffices to certify its smaller factors.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: if the extension from decomposable to arbitrary tuples is supplied, the same Hessian-perfect-power route could prove persistence for other symmetry classes closed under Kronecker products, since the obstruction is purely about the embedding of decomposable tensors being nondegenerate.
Editorial inference: Corollary 18's one-sided triangularizability criterion suggests that for cubic forms the product may be persistent under conditions weaker than persistence of both factors, such as triangularizability of one normalized Hessian space; this is not explored in the paper.
Editorial inference: the Kronecker powers of the W-state give explicit persistent tensors in dimensions $2^k$, which may be useful as test cases for numerical tensor-rank algorithms because persistence provides a lower bound that can be compared against constructive upper bounds.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the paper is aiming at a good theorem, and the first half is solid, but the proof of the main closure result has a real gap. I'd send it to a referee, but the referee should be told to focus on Proposition 10.

What's actually new: the Hessian-matrix identity (Corollary 8) is a clean and correct global statement, and the differentiation rules (Proposition 7) are useful. The strategy—use the [GO] Hessian criterion and prove a polarized perfect-power identity for the Kronecker product—is sensible. For decomposable direction tensors U^(i)=v^(i)⊗w^(i), the computation goes through: the determinant reduces to (1/n! P_f P_g)^{d1d2}, which is exactly what you want. The binary quartic example is worked carefully and checks out, though it falls in the rank-one case.

The soft spot is the last step of Proposition 10. The paper proves (22) on the product of Segre cones, then defines P_{f⊠g} on all of (V⊗W)^r via the universal property, and then says 'performing the Kronecker substitution therefore gives (22).' That 'therefore' doesn't follow. The two sides of (22) are polynomials on (V⊗W)^r; agreeing on the Segre cone only tells you their difference lies in the Segre ideal. The Segre cone is not Zariski dense when both factors have dimension >1, so equality there doesn't force global equality. Remark 9 explicitly acknowledges this issue for the ordinary Hessian determinant, and the same obstruction applies here. The proof supplies no argument that the difference vanishes modulo the Segre ideal. Theorem 11 is a direct application and inherits the gap. The main result may be true—the examples are consistent, and the structure is plausible—but the submitted argument doesn't prove it.

One more caveat: the theorem depends essentially on the Hessian characterization from [GO], which is a preprint from the same group. That is not circular, but it does mean the paper is conditional on an unverified result. That's a normal state of affairs but worth noting.

Bottom line: a serious referee should see this. The result is significant if true, and the gap is possibly repairable. If I were the editor, I'd send it out with a clear instruction to scrutinize Proposition 10.

Referee Report

3 major / 3 minor

Summary. The paper defines Kronecker products of symmetric tensors and claims that if f∈Sym^n C^{d1} and g∈Sym^n C^{d2} are persistent, then f⊠g is persistent (Theorem 11). The proof route is: Proposition 7/Corollary 8 establish differentiation identities for Hessian matrices; Proposition 10 claims a polarized perfect-power identity for Hessian determinants; Theorem 11 applies the Hessian characterization of persistence from [GO]. An appendix discusses normalized Hessian spaces for cubic forms.

Significance. The claimed closure result would be significant: it would yield iterated Kronecker products and powers of persistent symmetric tensors, hence new families of tensors with certified lower bounds on tensor rank. The differentiation identities (Prop. 7, Cor. 8) are clean, and the determinant computation on decomposable tuples in Prop. 10 is correct as far as it goes. However, the key extension from decomposable to all direction vectors is not justified, so the central claim is not established by the submitted argument.

major comments (3)

[§3, Proposition 10, Eq. (22)] The proof verifies Eq. (22) only for tuples U^{(i)}=v^{(i)}⊗w^{(i)}. The sentence "Performing the Kronecker substitution therefore gives (22)" assumes that agreement on the product of Segre cones implies equality as polynomials on (V⊗W)^{×r}. This is false: the affine Segre cone is a proper closed subvariety whenever d1,d2≥2, and its ideal is generated by the 2×2 minors, as Remark 9 explicitly notes. Both sides of (22) are multihomogeneous of degree d1d2 in each slot, so the difference lies in the Segre ideal but need not vanish. No argument shows that the difference vanishes modulo that ideal, nor that persistence of f and g forces it to vanish. Thus Proposition 10 is unproved.
[§3, Theorem 11 and Corollaries 14–15] The proofs of Theorem 11 and its corollaries are direct applications of Proposition 10 and therefore inherit the gap. The worked Example 13 does not test the gap: it verifies the identity only for decomposable U^{(1)},U^{(2)}, and in that example P_f and P_g are rank-one, so the special case of Remark 12 applies. The general case with non-rank-one P_f or P_g is precisely where the missing extension is needed.
[§3, Remark 12, Eq. (23)] The diagonal identity (23) is presented as a consequence of Proposition 10 and is then used to derive (29). Since (22) is not established for non-decomposable U, Eq. (23) is conditional, and so is the derivation of (29) even in the rank-one case. The direct computation in Example 13 verifies a single instance but does not replace a proof of the general statement.

minor comments (3)

[§3, Proposition 10, proof] There is a typographical error in the sentence defining the polynomials: "P_f (v(1), . . . , v(r) and P_g(w(1), . . . , w(r))" is missing a closing parenthesis after v^{(r)}.
[§1 and References] The paper relies on the Hessian characterization of [GO], an arXiv preprint by the same research group. Since this is a load-bearing external result, the authors should state explicitly that it is a preprint and provide the latest version or a proof sketch.
[§3, Corollary 8 and Remark 12] The symbol ⊠ is used for both the polynomial Kronecker product and, after evaluation, the ordinary matrix Kronecker product. The distinction is explained, but the multiple uses make Remark 12 harder to follow; a dedicated notation for the entrywise polynomial Kronecker product would improve clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: Theorem 11 is not presupposed by its inputs; the only flagged weakness is a proof gap in Proposition 10, which is a soundness issue rather than a circularity issue.

full rationale

The claimed derivation chain is not circular. Theorem 11 concludes that f⊔g is persistent by invoking the Hessian characterization of symmetric persistence (Theorem 3, from [GO]) and then constructing a polynomial P_{f⊔g} that satisfies the required perfect-power identity. The inputs are the persistence of f and g, and the construction uses the corresponding P_f and P_g, together with Proposition 7 and Corollary 8 for differentiation and Hessian-matrix identities. None of these steps defines persistence of f⊔g in terms of itself, nor does the paper fit a parameter and then rename it as a prediction. The proof of Proposition 10 contains a genuine gap: it proves identity (22) only for decomposable tuples U^(i)=v^(i)⊗w^(i), and the sentence 'Performing the Kronecker substitution therefore gives (22)' does not extend the polynomial equality to all tuples. Remark 9 explicitly concedes that equality on the Segre variety does not imply equality as polynomials on V⊗W, since the difference may lie in the Segre ideal. This is an invalid inference in the proof as written, but it is not circularity: it does not assume the conclusion or reduce the target result to an equivalent input. The cited Hessian characterization [GO] is by the same research group and is load-bearing, but it is a parameter-free theorem whose assumptions do not include Kronecker-product closure; under the review rules, such a citation is treated as independent evidence rather than as a circularity-raising self-citation. Therefore the central claim has independent mathematical content and no step is equivalent to its own input by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted parameters and no invented entities. The proof depends on the Hessian characterization of symmetric persistence from [GO], a preprint by the same research group, and on standard facts about determinants of Kronecker products and spanning properties of the Segre-Veronese variety. The chief unproved step is the extension of the polarized identity from decomposable directions to all directions.

assumptions (4)

domain assumption Hessian characterization of symmetric persistence: a symmetric tensor f is persistent iff the Hessian determinant of every (n-2)-fold partial polarization is the d-th power of a multihomogeneous polynomial of multidegree (1,...,1).
Imported from [GO, Theorem 2(c)], a preprint by the same first author. The main theorem is proved only modulo this criterion, and no proof or formalization is included here.
standard math Kronecker determinant identity: det(A⊠B) = det(A)^{d2} det(B)^{d1}.
Used in Proposition 10 to pass from Hessian matrices to Hessian determinants.
standard math Pure powers span symmetric powers and the Segre-Veronese variety is linearly nondegenerate.
Used in Lemma 6 to prove the differentiation identities for Kronecker products.
standard math Restitution convention and the partial polarization formula (4).
Defines the relationship between a symmetric tensor, its associated polynomial, and its partial polarizations used throughout the Hessian criteria.

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Cite this review

Pith. "Pith review of Kronecker Products of Symmetric Persistent Tensors." pith.science (2026). https://pith.science/paper/UGN56N6S

@misc{pith2026260811182,
  author       = {Pith},
  title        = {Pith review of: Kronecker Products of Symmetric Persistent Tensors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UGN56N6S}},
  note         = {Machine review of arXiv:2608.11182}
}

abstract

Persistent tensors form a recursively defined class adapted to the substitution method and yield nontrivial lower bounds on tensor rank. Although persistence is not preserved under Kronecker products in general, we prove that it is preserved in the symmetric setting: the Kronecker product of any two symmetric persistent tensors is again persistent. We first establish a global differentiation identity showing that the Hessian matrix of the Kronecker product of arbitrary homogeneous polynomials is the Kronecker product of the Hessian matrices of the factors, up to the normalization dictated by the restitution convention. For persistent factors, we then prove a polarized perfect-power identity for the Hessian determinants of all $(n-2)$-fold partial polarizations. Combined with the Hessian characterization of symmetric persistence, this yields closure under Kronecker products. As consequences, symmetric persistence is closed under iterated Kronecker products and Kronecker powers.

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Works this paper leans on

10 extracted references · 5 canonical work pages

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Dolgachev I., Classical Algebraic Geometry: A Modern View, Cambridge University Press (2012) https://doi.org/10.1017/CBO9781139084437

doi:10.1017/cbo9781139084437 2012

[2] [2]

I., Three qubits can be entangled in two inequivalent ways, Phys

D\"ur W., Vidal G., and Cirac J. I., Three qubits can be entangled in two inequivalent ways, Phys. Rev. A 62, 062314 (2000) https://doi.org/10.1103/PhysRevA.62.062314

doi:10.1103/physreva.62.062314 2000

[3] [3]

Gharahi M., Classifying entanglement by algebraic geometry, Int. J. Quant. Inf. 22 (2024), 2350047 https://doi.org/10.1142/S0219749923500478

doi:10.1142/s0219749923500478 2024

[4] [4]

and Lysikov V., Persistent Tensors and Multiqudit Entanglement Transformation, Quantum 8 (2024), 1238 https://doi.org/10.22331/q-2024-01-31-1238

Gharahi M. and Lysikov V., Persistent Tensors and Multiqudit Entanglement Transformation, Quantum 8 (2024), 1238 https://doi.org/10.22331/q-2024-01-31-1238

work page doi:10.22331/q-2024-01-31-1238 2024

[5] [5]

and Ottaviani G., Symmetric Persistent Tensors and their Hessian, arXiv:2510.07404 (2025) https://doi.org/10.48550/arXiv.2510.07404

Gharahi M. and Ottaviani G., Symmetric Persistent Tensors and their Hessian, arXiv:2510.07404 (2025) https://doi.org/10.48550/arXiv.2510.07404

work page doi:10.48550/arxiv.2510.07404 2025

[6] [6]

Iarrobino and V

A. Iarrobino and V. Kanev, Power Sums, Gorenstein Algebras, and Determinantal Loci, Lecture Notes in Mathematics, Vol. 1721, Springer-Verlag, Berlin (1999) https://doi.org/10.1007/BFb0093426

doi:10.1007/bfb0093426 1999

[7] [7]

J. M. Landsberg, Tensors: Geometry and Applications, Graduate Studies in Mathematics, Vol. 128, American Mathematical Society (2012) https://doi.org/10.1090/gsm/128

doi:10.1090/gsm/128 2012

[8] [8]

Mathes, M

B. Mathes, M. Omladi c , and H. Radjavi, Linear spaces of nilpotent matrices, Linear Algebra Appl. 149 (1991), 215 https://doi.org/10.1016/0024-3795(91)90335-T

work page doi:10.1016/0024-3795(91)90335-t 1991