The structured Gerstenhaber problem (III)

Cl\'ement de Seguins Pazzis

REVIEW 1 major objections 3 minor 13 references

The structured Gerstenhaber problem (III)

T0 review · 1 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Over characteristic-2 fields, the largest linear subspace of nilpotent b-symmetric endomorphisms has dimension $\nu(n-\nu)$, and the paper also gives the exact formula for b-alternating endomorphisms, with a correction term in the…

desk verdict A strong, likely-correct resolution of the characteristic-2 dimension question, with a small finite-field gap in Lemma 4.10 that looks easily repairable. read the letter →

arxiv 1908.03934 v1 pith:RAU5ZA2S submitted 2019-08-11 math.RA

Clément de Seguins Pazzis This is my paper · ORCID

classification math.RA MSC 15A3015A6315A03

keywords symmetricmatricesnilpotentbilinearformsdimensionGerstenhabertheoremfieldswithcharacteristic2Wittindexb-alternatingendomorphisms

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

The paper settles the dimension question of the structured Gerstenhaber problem for fields of characteristic 2. For a finite-dimensional vector space $V$ with a non-degenerate symmetric bilinear form $b$ of Witt index $\nu$ (the largest dimension of a totally singular subspace) and $n=\dim V$, it proves that every linear subspace of nilpotent $b$-symmetric endomorphisms has dimension at most $\nu(n-\nu)$, and that this bound is attained. For $b$-alternating endomorphisms, it proves the maximum is $\nu(n-\nu-1)$ except in the case $n=2\nu+1$, where the maximum is $\nu(n-\nu)-\dim\mathrm{SKer}\,Q$, with $\mathrm{SKer}\,Q$ the intersection of the kernel of the quadratic form $Q$ with its orthogonal. These formulas complete the dimension side of the problem in characteristic 2 and agree with the known formulas for other characteristics except in that exceptional case.

What carries the argument

The argument runs on two small building blocks of $b$-symmetric and $b$-alternating endomorphisms: the $b$-symmetric square $\alpha\, b(x,-)\otimes x$ and the $b$-alternating tensor $x\wedge_b y = b(y,-)x + b(x,-)y$. Around these, the proof develops the $a$-transform of $b$, the alternating form $b^a(x,y)=b(x,y)+\sqrt{Q(x)Q(y)}$ defined after adjoining square roots; orthogonality for $b^a$ controls the interaction between an isotropic vector and a nilpotent space of endomorphisms. The induction step passes to the quotient $\{x\}^\perp/\mathbb{F}x$ for an isotropic vector $x$, lowering both $n$ and $\nu$ by one, and uses normal bases and the additivity of the Witt index to compute the invariant $\dim\mathrm{SKer}\,Q$ as a parameter of the normal form.

What would settle it

Examine the determinant polynomial $f(u)$ constructed in Lemma 4.10 over a finite field: the proof establishes $f(u)=0$ for every nonzero $u$ and concludes $f(0)=0$, which is not valid in general because, for example, $f(t)=t+1$ vanishes at the nonzero element $1$ of $\mathbb{F}_2$ but not at $0$. Testing one explicit configuration over $\mathbb{F}_2$ or $\mathbb{F}_4$ that satisfies the lemma's hypotheses would show whether the missing inference can be replaced or whether the upper bound fails there.

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Extended reading notes

Core claim

The central discovery is that the quadratic-form kernel, not just the Witt index, controls the alternating case in odd dimension. Theorem 1.3 states, for a field of characteristic 2, that the largest nilpotent subspace of $S_b$ has dimension $\nu(n-\nu)$; that of $A_b$ has dimension $\nu(n-\nu-1)$ when $n\neq 2\nu+1$; and when $n=2\nu+1$, dimension $\nu(n-\nu)-\dim\mathrm{SKer}\,Q$, where $\mathrm{SKer}\,Q = \mathrm{Ker}\,Q \cap (\mathrm{Ker}\,Q)^\perp$. The paper constructs explicit subspaces attaining each bound, including a new family in the exceptional case built from zero-diagonal matrix blocks, and proves the upper bounds by induction on the dimension.

Load-bearing premise

The upper-bound proof relies on the step in Lemma 4.10 that a polynomial vanishing on every nonzero element of the field must be identically zero; that is true only for infinite fields, and no replacement argument for finite fields is given, so the finite-field cases of the theorem are not established by the proof as written.

Editorial extensions

If this is right

The dimension question of the structured Gerstenhaber problem is closed for every field and for every non-degenerate symmetric, hence also alternating, bilinear form in characteristic 2.
For the standard scalar product on $\mathbb{F}^n$, this yields exact matrix constants: $n^2/4$ (for even $n$) or $(n^2-1)/4$ (for odd $n$) for nilpotent symmetric matrices, and $n(n-2)/4$ (for even $n$) or $(n^2-1)/4$ (for odd $n$) for nilpotent symmetric matrices with zero diagonal.
Whenever $n=2\nu+1$ and $\mathrm{SKer}\,Q=\{0\}$, the alternating maximum equals the symmetric maximum $\nu(n-\nu)$, so an extremal symmetric space can be chosen inside $A_b$; whenever $\mathrm{Ker}\,Q$ is totally singular, the alternating maximum drops to the generic $\nu(n-\nu-1)$.
The dimension bounds are attained by explicit constructions in all three cases, so the paper provides candidate extremal subspaces; the classification of all extremal subspaces remains open.
In a normal basis, $\dim\mathrm{SKer}\,Q$ is read off as one of the basis parameters, so the exceptional correction term is directly computable from the normal form of $b$.

Reading between the lines

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

The $a$-transform construction is likely reusable: orthogonality relative to $b^a$ reduces to degeneracy of an alternating form, and the same lemmas should control the classification of all extremal subspaces in the announced sequel.
The formulas suggest a unifying rule across characteristics: replace the generic alternating bound $\nu(n-\nu-1)$ by $\nu(n-\nu)$ exactly in the odd $n=2\nu+1$ case, with a reduction of $\dim\mathrm{SKer}\,Q$; testing whether a similar one-dimensional-excess correction appears in other form-theoretic settings would be a natural next step.
A direct check on small finite fields, such as exhaustive search for $n\le 5$ over $\mathbb{F}_2$ and $\mathbb{F}_4$, would confirm whether the stated maxima hold there; such a check would also show whether the polynomial step in Lemma 4.10, the only place the proof uses infinitude of the field, can be replaced by a finite-field argument.

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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: this is the right paper to resolve the characteristic-2 dimension question in the structured Gerstenhaber problem, and the main theorem is almost certainly correct. The writing is clear, the constructions are explicit, and the new invariant (dim SKerQ) in the n = 2ν+1 case is a genuine discovery, with examples that earn their place. The induction strategy from the earlier parts is extended naturally, and the paper is well organized.

The one soft spot is Lemma 4.10. The proof shows a certain polynomial f vanishes on every nonzero element of F, then says "Since F is infinite we deduce that f is identically zero." That inference fails for finite fields, and Theorem 1.3 is stated for all characteristic-2 fields. Both Section 6 and Section 7 rely on the lemma, so as written the finite-field cases are unsupported. That said, the gap is small and local. Earlier in the same proof the identity B2(t) = 0 is already a polynomial identity in F[t], which means the subsequent determinant is zero in the field of rational functions F(u), and f(0) = 0 follows without any cardinality assumption. The author simply used a weaker fact and then an unnecessary infinity hypothesis. I expect the fix to be a few lines, and I would be very surprised if the theorem needed to be weakened.

One thing that might raise a flag: part (a) of Theorem 1.3 invokes part (b) in the final step. That is fine, because the invocation is on an even-dimensional space where (b) has already been proved independently. No circularity.

Citations and self-citations look proper: the earlier papers are published and the dependence is honest. The paper does not overclaim; it clearly says the optimal-spaces question is left to a sequel.

Who should read this: anyone working on nilpotent spaces of matrices or quadratic forms over characteristic 2. It closes a classical problem. I would send it to referees; with the small fix it should be accepted.

Referee Report

1 major / 3 minor

Summary. The paper determines, over any field of characteristic 2, the greatest dimension of a nilpotent linear subspace of the space of b-symmetric endomorphisms S_b and of b-alternating endomorphisms A_b of a finite-dimensional vector space V equipped with a non-degenerate symmetric bilinear form b. The stated answer is ν(n−ν) for S_b, ν(n−ν−1) for A_b except in the special case n=2ν+1, where the maximum is ν(n−ν)−dim SKerQ. The proof gives explicit extremal subspaces in Section 3 and then proves the matching upper bounds by induction in Sections 5–7, using a new 'a-transform' of b and two orthogonality lemmas for tensor endomorphisms.

Significance. If the proof is completed, the paper settles the dimension question of the structured Gerstenhaber problem in characteristic 2, complementing the characteristic-different-from-2 cases treated in the author's earlier work [11, 12]. The statement is sharp, the exceptional case is identified by the invariant SKerQ, and the extremal examples in Section 3 are explicit. The induction is well organized, and point (b) of Theorem 1.3 is proved before point (a), so the use of (b) inside Section 7 is not circular. The main weakness is a localized but real gap in Lemma 4.10 for finite fields, which affects the upper-bound arguments in Sections 6 and 7 as written.

major comments (1)

[Lemma 4.10, proof, paragraph beginning 'Since F is infinite'] The proof of Lemma 4.10 shows that the polynomial f(u)=det([(PX)^T;(PY)^T](I_n+uP^{-1}K(M))^{ad}[X Y]) vanishes at every nonzero element of F, and then uses 'Since F is infinite' to conclude f(0)=0. For finite fields of characteristic 2 this inference is invalid, as the example u^{q-1}-1 shows. The lemma is used in Section 6 to assert that Vx is ba-orthogonal to LV,x and in Section 7, Claim 1, to obtain x∧by∈V for all x∈KerQ and y∈(KerQ)^⊥, so the finite-field cases of all three parts of Theorem 1.3 rest on this step as written. The gap is repairable: the singularity statement immediately preceding the definition of f is an identity over F(t), so the determinant is the zero rational function and hence the zero polynomial, giving f(0)=0 without any cardinality assumption. The manuscript should replace the infinite-field sentence with this polynomial-identity argument, or supply an equivalent finite-field proof.

minor comments (3)

[Section 6, beginning of the induction proof] The symbol V is used both for the ambient vector space and for the nilpotent subspace under study, which makes statements such as 'Set n := dim V' and 'dim V≤ν(n−ν−1)' ambiguous; renaming the nilpotent subspace, for instance N, would considerably improve readability.
[Lemma 7.2, statement] The statement repeats the equality 'dim_F V = dim_{F'} V'' twice; the second occurrence is clearly intended to refer to the dimensions of the nilpotent subspaces, not of the ambient vector spaces, and should be written with different notation.
[Lemma 4.10, proof] In the proof, the local notation n := dim V−2 clashes with the global dimension n of V in the main theorem; using a different letter, such as m, would avoid confusion in the displayed determinants.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation is self-contained, with earlier parts of the same theorem used only after independent proof; the finite-field gap in Lemma 4.10 is a correctness issue, not a circular reduction.

full rationale

The paper's central claims are proved by explicit constructions (Section 3) and by induction (Sections 6 and 7), not by assuming the theorem's conclusion. Section 7 proves part (a) using part (b), but part (b) is proved earlier in Section 6 without invoking part (a), so this is legitimate use of an already-established result rather than circular reasoning. The only self-citations (Lemmas 4.1, 4.2, 4.3 from the author's prior work [11], [12]) are either proved in the present paper or cite previously published, parameter-free theorems whose assumptions do not include the target result; they are not fitted inputs or renaming of conclusions. The a-transform construction and the orthogonality lemmas are new internal tools developed without assuming the main theorem. The identified weakness in Lemma 4.10, where the proof passes from vanishing on nonzero elements to vanishing at zero via the assumption that the field is infinite, is a genuine correctness gap for finite fields but is not circular: it does not reduce the theorem to its own statement or to a self-citation. It is a repairable flaw in an otherwise structurally independent proof, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proof is parameter-free: n, ν, and dim SKerQ are invariants of the input form, not adjustable constants. The paper relies on standard structure theory for bilinear forms in characteristic 2 and on the author's published earlier results in the same series. The a-transform and super-kernel are new mathematical definitions, not empirical entities. The main unstated assumption is the infinite-field inference in Lemma 4.10, which affects finite fields.

assumptions (5)

standard math Gerstenhaber-Serezhkin theorem: every nilpotent linear subspace of End(V) has dimension at most n(n−1)/2.
Invoked in Section 5 to bound the block A(M), and in Section 7 as part of the argument; proved in [2] and [13].
standard math Every non-degenerate symmetric bilinear form over a field of characteristic 2 has a normal basis (Theorem 2.4).
Used in Sections 2, 3, and 4 to put b in explicit block form; standard result from [8].
standard math The author's earlier published results are correct: Theorem 1.2 for characteristic not 2, and Lemma 4.3 on invariant totally singular subspaces.
Theorem 1.2 is the model being extended; Lemma 4.3 is used in Sections 5 and 7. Both are published in [11] and [12].
domain assumption It suffices to solve the problem for non-degenerate b, because degenerate forms reduce to the induced form on V/Rad(b).
The paper explicitly restricts to non-degenerate forms in Section 1.1 and cites [11] for the reduction.
standard math For an alternating form c on a finite-dimensional vector space, dim Z^o + dim Z = n + dim(Z ∩ Rad(c)) for every subspace Z.
Used in Lemma 4.7; stated as classical and not proved in the paper.

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Pith. "Pith review of The structured Gerstenhaber problem (III)." pith.science (2026). https://pith.science/paper/RAU5ZA2S

@misc{pith2026190803934,
  author       = {Pith},
  title        = {Pith review of: The structured Gerstenhaber problem (III)},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RAU5ZA2S}},
  note         = {Machine review of arXiv:1908.03934}
}

abstract

Let $b$ be a symmetric bilinear form on a finite-dimensional vector space over a field with characteristic $2$. Here, we determine the greatest possible dimension of a linear subspace of nilpotent $b$-symmetric or $b$-alternating endomorphisms of $V$, expressing it as a function of the dimension, the rank, the Witt index of $b$, and an additional invariant in a very special case.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Draisma, H

J. Draisma, H. Kraft, J. Kuttler, Nilpotent subspaces of maximal dimension in semisimple Lie algebras, Compos. Math. 142 (2006), 464–476

work page 2006

[2] [2]

Gerstenhaber, On Nilalgebras and Linear Varieties of Nilpotent Matrices I, Amer

M. Gerstenhaber, On Nilalgebras and Linear Varieties of Nilpotent Matrices I, Amer. J. Math. 80 (1958), 614–622

work page 1958

[3] [3]

Kokol Bukovˇ sek, M

D. Kokol Bukovˇ sek, M. Omladiˇ c, Linear spaces of symmetric nilpotent ma- trices, Linear Algebra Appl. 530 (2017), 384–404

work page 2017

[4] [4]

Mathes, M

B. Mathes, M. Omladiˇ c, H. Radjavi, Linear spaces of nilp otent matrices, Linear Algebra Appl. 149 (1991), 215–225

work page 1991

[5] [5]

Meshulam, N

R. Meshulam, N. Radwan, On linear subspaces of nilpotent elements in a Lie algebra, Linear Algebra Appl. 279 (1998), 195–199. 39

work page 1998

[6] [6]

Milnor, Symmetric inner products in characteristic 2 , Chapter 3 in Prospects in Mathematics, Annals of Math

J. Milnor, Symmetric inner products in characteristic 2 , Chapter 3 in Prospects in Mathematics, Annals of Math. Studies 70, pages 59–75, Prince- ton University Press, 1971

work page 1971

[7] [7]

Quinlan, Spaces of matrices without non-zero eigenva lues in their ﬁeld of deﬁnition, and a question of Szechtman, Linear Algebra Appl

R. Quinlan, Spaces of matrices without non-zero eigenva lues in their ﬁeld of deﬁnition, and a question of Szechtman, Linear Algebra Appl . 434 (2011), 1580–1587

work page 2011

[8] [8]

de Seguins Pazzis, Invitation aux formes quadratique s, Calvage & Mounet, Paris, 2011

C. de Seguins Pazzis, Invitation aux formes quadratique s, Calvage & Mounet, Paris, 2011

work page 2011