{"id":"ebec9f01-a4e9-4d5a-9d8c-628f49633fe0","arxiv_id":"2607.21015","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The determinant has strength p for prime p and partition rank n minus o(n), proved by a new Chern class obstruction.","lead":"For every prime p, the determinant polynomial of size p is shown to have exact strength p, meaning it needs p products of smaller polynomials; its partition rank is asymptotically n. The proof introduces a new Chern-class obstruction on the space of invertible matrices, which is of independent interest in algebraic geometry.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the Chow-ring input is standard and the obstruction is valid, though independent verification would be worthwhile.","rationale":"The paper's mathematical argument is coherent. Lemma 4.1 correctly converts a strength decomposition into a nowhere-vanishing section of a split vector bundle on P^N\\V_+(f), and Lemma 2.8 is valid for smooth varieties: a nowhere-vanishing section locally gives a free rank-one direct summand, so the top Chern class vanishes. The application to det_p in Theorem 4.2 is internally consistent: the bundle has rank p-1, its top Chern class is (prod d_i) x^{p-1}, and in the asserted ring Z[x]/(p x,x^p) this class is nonzero because each d_i is in {1,...,p-1}. The Hasse-derivative monotonicity lemma is carefully argued, including the treatment of positive characteristic and the absence of constant terms in the resulting decomposition. The prime-gap step and the partition-rank monotonicity argument are also sound. The reader's weakest assumption, the Chow-ring computation of PGL_p, is a legitimate external dependency, and the paper would be strengthened by a fuller proof or a more citable source for Lemma 3.2. However, the theorem cited is a standard result in intersection theory, the special case needed is stated correctly, and the localization computation in Lemma 3.3 checks the degree-one part independently. I therefore do not find a load-bearing correctness concern, and the reader's ACCEPT verdict remains appropriate. The requested independent verification of the Chow-ring presentation would be a useful but non-blocking check.","tokens_in":13911,"tokens_out":33727,"duration_ms":325425,"concrete_test":"Independently verify the needed part of Lemma 3.2 by recomputing CH^*(PGL_p) from the Gysin/localization sequence for the complement of the determinant hypersurface in P(Mat_p) for at least p=3 and p=5, and by directly checking that c_1(L_1)^{p-1} is nonzero in CH^{p-1}(PGL_p). If the recomputation matches Z[x]/(p x,x^p) with x^{p-1} nonzero, the Theorem 4.2 obstruction stands; if it produces an additional relation killing x^{p-1}, the exact identity would need revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified. The central claim str(det_p)=p rests on the Chow-ring computation CH^*(PGL_p)=Z[x]/(p x,x^p) in Lemma 3.2, which is imported from Grothendieck's theorem for split semisimple groups and presented partly through a blog-post citation. This is the single external dependency: if the ring had extra relations killing x^{p-1}, the contradiction in Theorem 4.2 would fail. I find no internal inconsistency, and the application of the theorem to PGL_p via the Borel character lattice is standard and appears correctly executed in Lemmas 3.2 and 3.3. The localization sequence in Lemma 3.3 independently confirms CH^1(PGL_p)=Z/p and that c_1(L_1) generates it, which is consistent with the asserted ring; the remaining higher-degree structure is imported but is a well-established result. Thus this is a verification gap rather than a demonstrated flaw.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":14209,"tokens_out":34916,"duration_ms":310140,"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":[{"comment":"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.","section":"Theorem 4.2 (§4)"},{"comment":"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.","section":"Lemma 5.1 (§5)"}],"minor_comments":[{"comment":"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.","section":"Lemma 3.2 (§3)"},{"comment":"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.","section":"Lemma 4.6 (§4)"},{"comment":"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.","section":"Lemma 5.1 (§5)"},{"comment":"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}\".","section":"Lemma 4.1 (§4)"}],"recommendation":"major_revision","confidential_remarks":"The results appear correct and the two proof gaps identified above are readily fixable. The paper is well within the journal's scope and makes a strong contribution. I would encourage the editor to ask the authors to replace the blog-post reference for the Chow ring of PGL_n with a standard citation, and to fix the two local proof gaps before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here is my read on arXiv:2607.21015. The headline result is real: for every prime p, str(det_p)=p, and the partition rank of det_n is n-o(n). The proof of the exact prime case is the genuine novelty—if det_p were a sum of fewer than p products, the corresponding section of a split vector bundle on PGL_p would be nowhere vanishing, but its top Chern class is (∏ d_i) x^{p-1} in CH^*(PGL_p)=Z[x]/(px,x^p), which is nonzero. I checked the steps and they hold together. This also gives the first explicit family showing strength can be much larger than a bounded Birch rank, confirms Lampert–Moshkovitz's conjecture, and answers their problem on partition rank.\n\nThe main external input is Grothendieck's computation of CH^*(PGL_p), imported through Lemmas 3.2–3.3 and cited partly to a blog post. That is the one place I would want a referee to double-check, but it is a standard theorem, and Lemma 3.3 independently verifies the degree-1 generator via localization, so I do not see a real possibility that the obstruction fails. The k-step monotonicity lemma (4.6) is a bit compressed in the dimension count, but the logic is sound. There are minor typos, and the private communication [25] is just context.\n\nThis is a well-written paper, honest about its dependencies, with no sign of post-hoc fitting. It will be cited. I would bring it to reading group and send it to a serious referee. The referee should verify the Chow ring presentation and the Hasse-derivative argument, but I do not expect surprises.","headline":"Settles the asymptotic partition rank of the determinant, gives exact strength for prime sizes, and the new intersection-theoretic method is sound.","tokens_in":14558,"tokens_out":20808,"would_cite":true,"duration_ms":177853,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14C15","14L35","14M12","15A69","11N05"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["strength of polynomial","partition rank","determinant","Chow ring","Chern class obstruction","PGL_n","prime gaps","singular locus codimension"],"falsifier":"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.","tokens_in":13736,"feed_emoji":"🧮","tokens_out":16275,"duration_ms":128174,"temperature":0.7,"pith_summary":"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.","feed_headline":"For prime-sized determinants, no shortcut below p terms","feed_subtitle":"The same argument shows the determinant's shortest decomposition has n - o(n) terms, nearly maximal.","key_machinery":"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.","core_discovery":"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 $n-n^{0.525}\\le \\operatorname{prk}(\\det_n)\\le n$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"supplies the Chow-ring computation for split semisimple groups from which the ring for PGL_p is derived.","marker":"[16]"},{"why":"gives the presentation of that computation used in Lemma 3.2.","marker":"[22]"},{"why":"supplies the Chow-ring description of the complete flag variety used as the starting point.","marker":"[13]"},{"why":"provides the compatibility of restriction with tensor products that turns a strength decomposition into a bundle section.","marker":"[32]"},{"why":"gives the prime-gap bound converting the prime-size result into the n^{0.475} lower bound.","marker":"[5]"}],"fun_headline_variants":["Det strength equals p for primes: no shortcut below p","Determinant's shortest decomposition is n - o(n) terms","Prime-sized determinants need p terms, exact lower bound","No cheap tricks: determinant strength near maximal n^n"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Det strength equals p for primes: no shortcut below p","Determinant's shortest decomposition is n - o(n) terms","Prime-sized determinants need p terms, exact lower bound","No cheap tricks: determinant strength near maximal n^n"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00017,"raw_usage":{"total_tokens":1354,"prompt_tokens":1120,"completion_tokens":234,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":736,"completion_tokens_details":{"reasoning_tokens":167}},"tokens_in":736,"tokens_out":234,"duration_ms":3058,"temperature":1.0,"reasoning_tokens":167,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:33:21.811941+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Grothendieck","cited_arxiv_id":null,"evidence_quote":"supplies the Chow-ring computation for split semisimple groups from which the ring for PGL_p is derived."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gives the presentation of that computation used in Lemma 3.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the Chow-ring description of the complete flag variety used as the starting point."},{"cited_title":"Stacks project authors","cited_arxiv_id":null,"evidence_quote":"provides the compatibility of restriction with tensor products that turns a strength decomposition into a bundle section."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gives the prime-gap bound converting the prime-size result into the n^{0.475} lower bound."}],"review_version":2}