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Lower bounds on the strength of the determinant

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Pith's one-line read The paper proves that for every prime p the determinant has strength exactly p, and that for all large n its strength and partition rank are nearly maximal.

desk verdict Settles the asymptotic partition rank of the determinant, gives exact strength for prime sizes, and the new intersection-theoretic method is sound. read the letter →

arxiv 2607.21015 v1 pith:E6WFQOOA submitted 2026-07-23 math.AC math.AGmath.CO

Qiyuan Chen , Yuhao Zhao This is my paper · ORCID
classification math.ACmath.AGmath.CO MSC 14C1514L3514M1215A6911N05
keywords strengthofpolynomialpartitionrankdeterminantChowringChernclassobstructionPGL_nprimegapssingularlocuscodimension
topics The Twin Prime Conjecture
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 proves that the determinant of a p×p matrix, viewed as a homogeneous degree-p polynomial, has strength exactly p for every prime p, and that for all sufficiently large n the strength of the determinant grows at least like $(1-o(1))n^{0.475}$. It further proves that the partition rank of the determinant, viewed as an n-linear form in its columns, is $n-o(n)$; the determinant cannot be written as a sum of products of lower-degree pieces except for a negligible number of terms. Because the singular locus of the determinant has codimension 4, this is the first explicit family of forms showing that any bound relating strength to that codimension and degree must genuinely depend on the degree. The results answer an earlier conjecture that determinant strength tends to infinity and an open problem about the asymptotics of determinant partition rank. The proof works through an intersection-theoretic contradiction: a short decomposition would create a nowhere-vanishing section of a split vector bundle, while a nonzero top Chern class in a Chow ring forbids such a section.

What carries the argument

The load-bearing object is the Chow ring of $\mathrm{PGL}_p$, the ring of algebraic cycle classes on the quotient of invertible p×p matrices by scalars, which is $\mathbb{Z}[x]/(px,x^p)$ when p is prime, with x the first Chern class of the tautological line bundle. The mechanism is a contrapositive supplied by Lemma 4.1: a strength decomposition with r summands produces a nowhere-vanishing global section of the split bundle $\bigoplus_{i=1}^r (\mathcal{O}(1)|_{\mathrm{PGL}_p})^{\otimes d_i}$, so if every such bundle has nonzero top Chern class, then $\operatorname{str}(\det_p)>p-1$. The top Chern class is $(\prod_{i=1}^{p-1} d_i)x^{p-1}$, which is nonzero in the Chow ring for $1\le d_i\le p-1$, contradicting the fact that a bundle with a nowhere-vanishing section has vanishing top Chern class. Two auxiliary mechanisms carry the results to all n: k-step weak monotonicity via higher-order directional derivatives transfers the exact prime bound to composite sizes, and monotonicity for partition rank under column specialization gives the near-maximal bound; the prime-gap estimate converts the largest prime $p(n)\le n$ into the quantitative exponent 0.475.

What would settle it

For a single prime p, find an explicit decomposition of $\det_p$ as a sum of at most p−1 products of homogeneous forms of degrees strictly between 0 and p; alternatively, compute $\mathrm{CH}^*(\mathrm{PGL}_p)$ by an independent method and check whether any element $(\prod_{i=1}^{p-1} d_i)x^{p-1}$ with $1\le d_i\le p-1$ vanishes. Either would break the obstruction.

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Extended reading notes

Core claim

The paper's central claim is the exact identity $\operatorname{str}(\det_p)=p$ for every prime p, together with the asymptotic bounds $\operatorname{str}(\det_n)\ge (1-o(1))n^{0.475}$ and $\operatorname{prk}(\det_n)=n-o(n)$. For prime p, the complement of the determinantal hypersurface in the projective space of p×p matrices is $\mathrm{PGL}_p$, and the argument uses the Chow ring computation $\mathrm{CH}^*(\mathrm{PGL}_p)\cong \mathbb{Z}[x]/(px,x^p)$. If $\det_p$ had a strength decomposition with at most p−1 summands, the summands would define a nowhere-vanishing section of a split vector bundle of rank p−1 whose top Chern class is $(\prod_{i=1}^{p-1} d_i)x^{p-1}$ with each $1\le d_i\le p-1$; this class is nonzero in that ring, whereas a bundle carrying a nowhere-vanishing section must have zero top Chern class. The extension to all n uses a weak monotonicity lemma proved with higher-order directional derivatives, together with the prime-gap bound stating that every large interval of length $x^{0.525}$ around x contains a prime; for partition rank, a stronger monotonicity under column specialization gives $\operatorname{prk}(\det_n)\ge p(n)$, the largest prime not exceeding n, and hence $n-n^{0.525}\le \operatorname{prk}(\det_n)\le n$.

Load-bearing premise

The proof depends on a previously computed algebraic invariant of the matrix space modulo scalars being exactly as claimed; if that invariant carried extra zero relations, the contradiction at the heart of the argument would fail.

Editorial extensions

If this is right

  • For every prime p, $\operatorname{str}(\det_p)=p$, so the determinant of prime order cannot be expressed as a sum of fewer than p products of lower-degree forms.
  • For all sufficiently large n, $\operatorname{str}(\det_n)\ge (1-o(1))n^{0.475}$, so determinant strength tends to infinity at least polynomially.
  • Since the singular locus of the determinant has codimension 4, any inequality of the form $\operatorname{str}(f)\le C(d)\operatorname{Brk}(f)$ must have $C(d)\ge d/4$ infinitely often, so a linear degree dependence in such bounds is unavoidable up to an absolute constant.
  • The partition rank of the determinant satisfies $n-n^{0.525}\le \operatorname{prk}(\det_n)\le n$, so $\operatorname{prk}(\det_n)=n-o(n)$.
  • Over every finite field, the ratios of partition rank to analytic or geometric rank for determinant tensors are at least $d/2-o(d)$, giving a super-logarithmic separation in the tensor order.

Reading between the lines

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

  • The Chern-class obstruction is a general template: for any homogeneous polynomial whose projective complement admits a known Chow ring, the same argument yields strength lower bounds without derivative or combinatorial methods.
  • The equality of strength and partition rank seen in small cases suggests the two quantities may coincide for more n; computing $\operatorname{str}(\det_6)$, $\operatorname{str}(\det_7)$, and $\operatorname{str}(\det_8)$ would test whether the prime pattern is an artifact of the obstruction or a genuine rigidity.
  • The slice-rank equality proved here gives a bridge: any upper or lower bound on the polynomial slice rank of the block-multilinear form transfers verbatim to the tensor slice rank, and hence to partition rank, of the original tensor.
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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

2 major / 4 minor

Summary. The paper proves new lower bounds for the strength and partition rank of the determinant over algebraically closed fields. The main results are: the exact identity str(det_p)=p for every prime p; a polynomial lower bound str(det_n) ≥ (1-o(1)) n^{0.475} for large n, obtained from the prime case by a weak monotonicity lemma and the Baker-Harman-Pintz prime-gap theorem; and the asymptotic formula prk(det_n)=n-o(n), with the quantitative bound n-n^{0.525} ≤ prk(det_n) ≤ n for large n. The new technique is an intersection-theoretic obstruction: a strength decomposition produces a nowhere-vanishing section of a split vector bundle on the complement P(Mat_p)\V(det_p) ≅ PGL_p, while a nonzero top Chern class in CH*(PGL_p)≅Z[x]/(px,x^p) rules out such a section. The paper also proves an equality between polynomial slice rank and tensor slice rank and discusses the relation between strength and partition rank.

Significance. If the results are correct, they are significant. The exact value str(det_p)=p and the resulting asymptotic prk(det_n)=n-o(n) resolve open problems of Lampert and Moshkovitz, and the determinant family provides the first explicit examples showing that the degree dependence in strength-versus-Birch-rank inequalities is unavoidable. The Chern-class obstruction is a genuinely new method for strength lower bounds, distinct from restriction, derivative, and combinatorial tools. The paper is largely self-contained beyond standard intersection theory and the quoted Chow-ring computation for PGL_n, and the main derivations are exact rather than parameter-dependent. The two proof gaps I identify below are local and fixable, so the central claims appear sound.

major comments (2)
  1. [Theorem 4.2 (§4)] The proof of the lower bound str(det_p)≥p only addresses a hypothetical strength decomposition with exactly p−1 summands. If str(det_p)≤p−1 with r<p−1 summands, Lemma 4.1 gives a nowhere-vanishing section of a rank-r bundle, not of the rank-(p−1) bundle whose Chern class is then computed. The argument is readily repaired: for every r≤p−1, the top Chern class of ⊕_{i=1}^r L_i is (∏ d_i)x^r, which is nonzero in CH*(PGL_p)≅Z[x]/(px,x^p); I recommend rewriting the proof in this form.
  2. [Lemma 5.1 (§5)] The step "Without loss of generality, we may assume c_{n+1}≠0" is not automatic. The solution space of g_1(c)=...=g_r(c)=0 has dimension at least n+1−r≥2, but it could be contained in the coordinate hyperplane {c_{n+1}=0}. Since that solution space is a nonzero subspace, one can choose a linear functional not vanishing on it and then apply a linear change of coordinates (or a column permutation) so that the chosen coordinate becomes the last one; this justification needs to be added for the proof to be complete.
minor comments (4)
  1. [Lemma 3.2 (§3)] The proof delegates the key structural fact to Grothendieck's seminar and to a blog post [22]. Because this Chow-ring computation is load-bearing for the main theorem, please cite a standard textbook treatment of the Chow ring of a split reductive group, or state Grothendieck's theorem in full, so that verification does not depend on a non-peer-reviewed source.
  2. [Lemma 4.6 (§4)] The scalar det(PQ) introduced by the change Y=PXQ is silently dropped when the Hasse derivative is applied. The argument should state that the constant can be absorbed into one of the factors, and that after restricting to the lower-right n×n block one should take the degree-n part of the identity so that every nonzero summand is a valid strength term. The undefined notation str_Y should also be replaced by str.
  3. [Lemma 5.1 (§5)] After the change of coordinates, the proof should explicitly note that the entries of the n×n matrix X in det_{n+1}(v_1,...,v_n,c)=c_{n+1} det(X) are independent, because the top n×n block of P·(v_1,...,v_n) ranges over all of M_n as v_1,...,v_n vary.
  4. [Lemma 4.1 (§4)] The displayed formula for the direct-sum bundle has a typographical error: the expression "O_{P^n}(d_i)|_{P^n\V+(f)})⊗di" should read "⊕_{i=1}^r (O_{P^n}(1)|_{P^n\V+(f)})^{⊗d_i}".

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the central derivation rests on external standard results, not on its own conclusions.

full rationale

The derivation of str(det_p)=p is not circular. Theorem 4.2 combines Lemma 4.1, which converts a hypothetical strength decomposition into a nowhere-vanishing global section of a split vector bundle on PGL_p, with the standard Chern-class obstruction of Lemma 2.8 and the Chow-ring computation CH*(PGL_p)=Z[x]/(px,x^p). That Chow ring is imported from Grothendieck's computation for split semisimple groups, an external theorem, and Lemma 3.2 derives it from that theorem rather than from the conclusion it is used to prove. The paper also independently checks the degree-one generator using the localization sequence in Lemma 3.3. The blog-post citation [22] is only a presentation of Grothendieck's result, not the load-bearing authority. The lower bounds for general n follow from the exact prime case, the monotonicity lemmas (proved from the definitions), and the external Baker-Harman-Pintz prime-gap theorem; there are no fitted parameters and no quantity is renamed as a prediction after being used as an input. The self-citations [6,7] appear only in the concluding discussion of unrelated rank comparisons and do not support the main theorem. The only notable external dependency is the standard Chow-ring computation; if that ring had extra relations, the contradiction in Theorem 4.2 would fail, but that is a correctness or verification risk, not a circularity. No step exhibits an output equation identical to an input by construction, so the appropriate finding is no significant circularity.

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

The central claims rest on standard theorems in algebraic geometry and number theory plus exact algebraic identities. No free parameters are fitted to data and no new entities are postulated. The paper's contribution is a new obstruction argument, not a fit or an ad hoc model.

assumptions (4)
  • standard math Grothendieck's solution: CH^*(G) is CH^*(G/B) modulo the ideal generated by the degree-1 characteristic homomorphism image, for split semisimple G.
    Used in Lemma 3.2 to compute CH^*(PGL_n); this is the load-bearing external theorem for the Chern class obstruction.
  • standard math Baker-Harman-Pintz prime gap bound: for large x, there is a prime in [x-x^0.525, x].
    Used in Theorem 4.8 to convert the exact prime-case bound into a polynomial lower bound for all large n.
  • standard math Hilbert Nullstellensatz over algebraically closed fields.
    Used in Lemma 4.1 to turn ideal membership into existence of a common zero, linking a strength decomposition to a nowhere-vanishing section.
  • standard math Krull's principal ideal theorem and dimension formulas for determinantal varieties.
    Used in Lemma 4.6 to find a rank-k matrix where all low-degree factors vanish.

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Pith. "Pith review of Lower bounds on the strength of the determinant." pith.science (2026). https://pith.science/paper/E6WFQOOA

@misc{pith2026260721015,
  author       = {Pith},
  title        = {Pith review of: Lower bounds on the strength of the determinant},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E6WFQOOA}},
  note         = {Machine review of arXiv:2607.21015}
}
abstract

We establish new lower bounds for the strength and partition rank of the determinant. For every prime $p$, we prove the exact identity \[ \operatorname{str}(\mathrm{det}_p)=p. \] A weak monotonicity argument, combined with a bound for gaps between consecutive primes, then gives $\operatorname{str}(\mathrm{det}_n)\ge (1-o(1))n^{0.475}$ for sufficiently large $n$. Since the Birch rank of $\mathrm{det}_n$ is always $4$, this gives the first explicit family showing that the dependence on the degree in bounds for strength in terms of Birch rank is unavoidable. Viewing $\mathrm{det}_n$ as an $n$-linear form in its columns, we also prove that its partition rank is at least the largest prime not exceeding $n$. Consequently, \[ n-n^{0.525}\le \operatorname{prk}(\mathrm{det}_n)\le n \] for all sufficiently large $n$, and hence the partition rank of the determinant is $n-o(n)$. The proof introduces an intersection-theoretic method for lower-bounding strength: a short strength decomposition produces a nowhere-vanishing section of a split vector bundle on the complement of the determinantal hypersurface, while a nonzero top Chern class in the Chow ring of $\mathrm{PGL}_n$ obstructs such a section.

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