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Slice rank and partition rank of the determinant

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Pith's one-line read A logarithmic lower bound on the partition rank of the determinant yields the first unbounded separation between partition rank and analytic rank.

desk verdict First asymptotic separation of partition and analytic rank via the determinant, with a clean slice-rank characterization and a real but isolated gap in the random-tensor theorem. read the letter →

arxiv 2509.06294 v1 pith:O5R5BF56 submitted 2025-09-08 math.CO cs.CC

Amichai Lampert , Guy Moshkovitz This is my paper · ORCID
classification math.COcs.CC MSC 15A6915A15
keywords determinantslicerankpartitionanalyticLaplaceexpansiontensorsfinitefieldsstructureversusrandomness
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 asks how few product terms can express an n-by-n determinant under different restrictions on the factors. It proves that if every term must contain a linear factor, exactly n terms are necessary and every n-term expansion is essentially the Laplace expansion. If the factors only need to be multilinear, at least log2(n)+1 terms are necessary, and for 4-by-4 matrices three terms already suffice over any field. Because the determinant's analytic rank stays near 2 while its partition rank grows, the ratio between these two structure measures is unbounded as the matrix size grows—a new phenomenon for tensors. The paper also shows that random tensors do not produce such separations, making the determinant a rare explicit witness.

What carries the argument

Three mechanisms carry the argument. For slice rank, a subspace-of-matrices bound forces any short decomposition to vanish on a large space of low-rank matrices, and an alternating-syzygy lemma converts the extremal case into the Laplace expansion. For the logarithmic partition-rank lower bound, an induction fixes a minimal block of rows: a linear transformation zeroes one summand while leaving a determinant minor of size at least n/2, so each step costs a constant factor in size and yields a logarithm. For the 4x4 upper bound, the four-index alternating-symbol identity epsilon_{i,j,k,l}=epsilon_{i,j}epsilon_{k,l}-epsilon_{i,k}epsilon_{j,l}+epsilon_{i,l}epsilon_{j,k} gives a three-term quadr

What would settle it

For n=5 over F_2 or F_3, use a computer search to decide whether det_5 can be written as a sum of three products of multilinear forms; the lower bound says no. Separately, simulate the two candidate random models in Theorem 4.5 for small n and compare their expected bias; a mismatch would indicate the theorem's statement is not the model proved.

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Extended reading notes

Core claim

The central claim is that the determinant polynomial det_n has slice rank exactly n, with all minimum slice-rank decompositions equivalent to the Laplace expansion, while its partition rank is at least log2(n)+1 and equals 3 for n=4 over every field. Since the analytic rank of det_n is at most 2, the extremal ratio A(d) between partition rank and analytic rank is at least (log2 d +1)/2, so it is unbounded as d grows—the first asymptotic separation between the two ranks. Complementing this, a tensor built as a sum of r randomly chosen reducible forms has analytic rank r-o(1) with high probability, so random constructions cannot account for the separation.

Load-bearing premise

The random-tensor estimate is proved for a generative model in which the two factors of each summand are independently randomized, while the theorem is phrased as drawing a uniformly random tensor of partition rank r; if these two distributions differ, that part of the separation story is not established by the proof as written.

Editorial extensions

If this is right

  • The Laplace expansion is not just the standard expansion: for slice-rank decompositions it is essentially the only minimal one, up to invertible row and column changes and syzygies.
  • For every n, any multilinear product expansion of det_n needs at least log2(n)+1 summands, so the Laplace expansion is within a logarithmic factor of optimal.
  • The extremal ratio A(d) grows at least like (1/2)log d, so no converse inequality with a constant depending only on d can be valid.
  • Random tensors of partition rank r have analytic rank roughly r, so large separations between the two ranks cannot be found by random construction; explicit polynomials are needed.
  • For 4x4 matrices, partition rank is exactly 3 while slice rank is 4, showing the two ranks genuinely differ for a natural symmetric polynomial.

Reading between the lines

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

  • The determinant's behavior suggests that other polynomials with small analytic rank but strong symmetry under row operations might serve as additional explicit separators; testing the permanent or other SL-invariant forms would be a natural next step.
  • The four-index alternating-symbol identity offers a template: related Grassmann-Plucker or Pfaffian identities might yield low partition-rank expansions for other matrix functions, though the paper's logarithmic lower bound prevents such expansions from being too short while using the same inductive method.
  • A concrete testable extension is to run exact or SAT-based searches for a 3-term multilinear expansion of det_5 over a small field; the theorem predicts none exists, so success would force a revision of the logarithmic lower bound.
  • For algorithms that approximate tensor structure using analytic rank, the determinant is a useful stress test: analytic rank alone would classify it as nearly random even though its partition rank grows.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies the minimal number of summands needed to express the n×n determinant as a sum of products, under three notions of rank. It proves that the slice rank of det_n is exactly n and, moreover, that every minimum slice-rank expansion is equivalent to the Laplace expansion (Theorem 2.6). It proves a lower bound prk(det_n) ≥ log_2(n)+1 (Theorem 3.2) and exhibits a quadratic three-summand expansion of det_4 over any field (Theorem 3.5), so prk(det_4)=3. Combining the lower bound with the observation ⌈ark(det_n)⌉=2 yields the first asymptotic separation between partition rank and analytic rank: A(d) ≥ (log_2 d+1)/2, hence A(d) is unbounded (Corollary 4.3). The paper also claims a random-tensor result: if T is a random d-linear form of partition rank r, then ark(T) ≈ r with high probability (Theorem 4.5 and Corollary 4.6).

Significance. The determinant-based separation is the main contribution and, if correct, settles a natural open problem in the structure-versus-randomness program for tensors. The proof of Theorem 3.2 is a genuinely new induction that fixes many variables at once, and the explicit det_4 expansion is elegant and connects to known counterexamples to the Gowers inverse conjecture. The slice-rank uniqueness theorem is also strong and carefully formulated. The random-tensor theorem is secondary but is advertised in the abstract; as currently written it has a statement/proof mismatch. The determinant part is sound and well supported, so the core contribution is valuable, but the random-tensor claim needs repair before the paper can be accepted as a whole.

major comments (3)
  1. [Section 4.2, Theorem 4.5] The random model in the statement is not the model analyzed. The statement says T^(r) is obtained by summing r reducible forms chosen independently and uniformly at random from T_{n,d}(F), and the abstract says 'random tensor of partition rank r'. The proof analyzes a generative model in which each summand is R_i S_i with R_i forced to depend on x^(d), and the coefficients of the R_i and S_i are drawn independently and uniformly. No argument is given that this model coincides with a uniform distribution over reducible forms, nor with the conditional distribution given partition rank r. As written, Theorem 4.5 and Corollary 4.6 are not established for the stated distribution. Please define the distribution explicitly (including the choice of variable partition and factor degrees) and either prove the proof's model matches it or restate the theorem for the model actually analyzed.
  2. [Section 4.2, proof of Theorem 4.5, event F] The proof sets Pr[F]=q^{-(d-1)n}. If F is 'x is trivial' in the previously defined sense (at least one x^(i)=0), the probability is 1-(1-q^{-n})^{d-1}; if F means x=0, then Lemma 4.8 cannot be applied on F^c because some but not all x^(i) may be zero. The argument can likely be repaired, since the corrected probability is still o(q^{-r}) under r≤(1-ε)n/2, but as written the proof of (5) and (6) is not valid.
  3. [Section 3.2, Theorem 3.5 (comparison with Laplace expansion)] The proof that expansion (1) is genuinely different from the two-row Laplace expansion relies on the claim that any linear map T with det_4∘T=det_4 is an isomorphism. The argument given is sound, but the sentence 'Since the construction of C was used in the first inequality' is terse: it would help to spell out that C shares a row with A and hence det_4(C)=0 because A is in the kernel of T but C differs from A only in rows that leave the determinant unchanged. This is a clarity issue, not a correctness issue.
minor comments (5)
  1. [Section 3.1, proof of Theorem 3.2] The step r>n-k ≥ prk(det_{n-k}) uses the fact that prk(det_m) ≤ m, via the ordinary Laplace expansion. This inequality is not stated before the proof; it should be mentioned for completeness.
  2. [Section 4.1, Corollary 4.2] The sentence 'Trivially c_q≤q, so 1<ark(det_n)≤2' is too compressed. To get the strict lower bound one must note that the upper bound in Lemma 4.1 is strict, so bias(det_n)<q^{-1}. Please spell this out.
  3. [Section 4.2, proof of Theorem 4.5] The notation for the event F is inconsistent with the dimension of x. The proof uses 'nontrivial x∈V^d' when applying Lemma 4.8 to S_i and ∇R_i, but these are functions on V^{d-1}. The intended meaning is clear, but the indexing should be corrected.
  4. [Remark 4.7] There is a typo: 'As fFor an infinite field' should read 'As for an infinite field'.
  5. [Section 1.1 and Corollary 4.3] The notation A(d) is defined with a ceiling on ark, and Corollary 4.3 uses ⌈ark(det_d)⌉=2. This is consistent, but it may be helpful to note explicitly that the ceiling does not affect the unboundedness conclusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: determinant rank bounds and analytic-rank computation are derived in-paper or from external theorems; self-citations are contextual.

full rationale

The load-bearing derivations do not reduce to their own inputs by construction or through self-citation. Theorem 2.6's slice-rank characterization uses Meshulam's subspace-of-matrices theorem as an external input, and the uniqueness argument is carried out with the paper's own reduction rules. Theorem 3.2's logarithmic partition-rank lower bound is a self-contained induction: Lemma 3.1 zeroes the Q_i terms, the restricted determinant det_{n-k} is expressed as a partition-rank decomposition of length r-ell < r, and the strict inequality prk(det_n) > prk(det_{n-k}) is obtained without assuming the conclusion. The det_4 quadratic expansion (Theorem 3.5) is verified directly through the Levi-Civita 4-to-2 identity (2), which is proved case by case. Corollary 4.2's bound ceil(ark(det_n))=2 follows from the standard bias-gradient identity and the elementary rank-probability estimate Lemma 4.1. The self-citations present in the paper (e.g., [4], [5], [12], [20], [21]) are contextual or support auxiliary remarks, not load-bearing premises for the central determinant-separation theorem. Theorem 4.5 has a possible gap between the stated random model and the restricted generative model analyzed in the proof, but this is a correctness/interpretation issue rather than a circular reduction: the proof does not define its random model in terms of the conclusion, and no fitted quantity is later renamed a prediction. Therefore no circular step meets the evidentiary standard.

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

No free parameters are fitted. The paper introduces no new entities, particles, or forces. The main external inputs are Meshulam's theorem and the standard bias-gradient identity. All other lemmas (Lemma 3.1, Lemma 4.1, Lemma 4.8, the det_4 identity) are proven in the text.

assumptions (4)
  • standard math Meshulam's theorem: a subspace of n x n matrices with dim > s*n contains a matrix of rank > s; if dim = s*n and all ranks <= s, the subspace is E tensor F^n or its transpose, for an s-dimensional subspace E.
    Cited as [18], used in Corollary 2.8 to prove the lower bound r >= n and to classify the zero set M, which underpins Theorem 2.6's characterization of minimal slice-rank decompositions.
  • standard math For a d-linear form T over a finite field, the bias of T equals the probability that its gradient (with respect to the last vector variable) vanishes at a random point.
    Used without proof (cited to [17]) in Corollary 4.2 to compute ark(det_n)=2 and in Theorem 4.5 to express E[bias(T)] as Pr[A]. This is load-bearing for the analytic-rank separation.
  • domain assumption A partition-rank decomposition of a multilinear form represents it as a sum of products of two multilinear forms each of positive degree, so each summand splits the variable set into two nonempty parts.
    Implicit in the definition of partition rank (Section 3); used in Theorem 3.2 to assume every factor depends on at most n/2 variables and to rule out constant factors. If constant factors were allowed, the inductive argument would fail.
  • standard math The Laplace expansion gives prk(det_m) <= m for every m.
    Constructive observation used in the induction step of Theorem 3.2 to conclude prk(det_n) > prk(det_{n-k}) when k+r>n.

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Pith. "Pith review of Slice rank and partition rank of the determinant." pith.science (2026). https://pith.science/paper/O5R5BF56

@misc{pith2026250906294,
  author       = {Pith},
  title        = {Pith review of: Slice rank and partition rank of the determinant},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O5R5BF56}},
  note         = {Machine review of arXiv:2509.06294}
}
abstract

The Laplace expansion expresses the $n \times n$ determinant $\det_n$ as a sum of $n$ products. Do shorter expansions exist? In this paper we: - Fully determine the slice rank decompositions of $\det_n$ (where each product must contain a linear factor): In this case, we show that $n$ summands are necessary, and moreover, the only such expansions with $n$ summands are equivalent (in a precise sense) to the Laplace expansion. - Prove a logarithmic lower bound for the partition rank of $\det_n$ (where each product is of multilinear forms): In this case, we show that at least $\log_2(n)+1$ summands are needed. We also explain why existing techniques fail to yield any nontrivial lower bound, and why our new method cannot give a super-logarithmic lower bound. - Separate partition rank from slice rank for $\det_n$: we find a quadratic expansion for $\det_4$, over any field, with fewer summands than the Laplace expansion. This construction is related to a well-known example of Green-Tao and Lovett-Meshulam-Samorodnitsky disproving the naive version of the Gowers Inverse conjecture over small fields. An important motivation for these questions comes from the challenge of separating structure and randomness for tensors. On the one hand, we show that the random construction fails to separate: for a random tensor of partition rank $r$, the analytic rank is $r-o(1)$ with high probability. On the other hand, our results imply that the determinant yields the first asymptotic separation between partition rank and analytic rank of $d$-tensors, with their ratio tending to infinity with $d$.

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