{"id":"7f433ea6-7bae-4331-b2f7-343f81f271b6","arxiv_id":"1908.08896","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The 3x3 determinant has Waring rank at least 15, improving the known lower bound from 14, and the cactus rank of the 3x3 permanent is at least 14.","lead":"The paper proves the 3x3 determinant cannot be written as a sum of 14 cubes, so its Waring rank is at least 15. This improves a long-standing lower bound and introduces syzygy-counting as a tool for rank lower bounds, with a new cactus rank bound for the 3x3 permanent as well.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof's decisive step, Proposition 15, rests on an unscripted Macaulay2 computation of the Betti table of the universal ideal H~ (β5,6 = 135); if that computation is incorrect, the claimed bound rk(det3) ≥ 15 is unsupported.","rationale":"The reader's weakest assumption identifies exactly the load-bearing step: an unscripted Macaulay2 computation of the Betti table for the universal ideal H~, plus the flatness argument that specializes it. My reading of the paper confirms this is the point where the proof is least secure. The logical architecture around it is sound: Proposition 6 gives a clean, verifiable bound for ideals of degree 13, Proposition 3 transfers it to apolar ideals, and the reduction in §3.3 to three normal forms is correct. But the rank-3 case, which is the genuine new content, rests entirely on the asserted Betti number β5,6 = 135 for H~. Without reproducible code or an independent certificate, a referee or reader cannot confirm the inequality β5,6 < 140 that drives Theorem 17. The flatness argument is plausible but also depends on the constancy of the Hilbert function, which in turn relies on Proposition 14; the latter contains a terse 'direct computation' that is not spelled out. I find no internal inconsistency or mathematical error in the surrounding reasoning, and the computations, if correct, do imply the stated bounds. Therefore the appropriate verdict is CONDITIONAL, matching the reader: the results stand conditional on independent confirmation of the Macaulay2 computations.","tokens_in":17444,"tokens_out":38562,"duration_ms":354692,"concrete_test":"Recompute the graded Betti table of T~/H~ in Macaulay2, where T~ = QQ[y_1..y_9, µ, λ] and H~ is generated by the 36 quadrics listed in Proposition 14, treating µ and λ as degree-0 variables. Confirm that β_{5,6} = 135. As a cross-check independent of the universal family, compute β_{5,6}(T/(det3−(x1+x5+x9)^3)^⊥) directly for a random specialization such as µ=1, λ=2 and verify that it is < 140.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central new claim is Theorem 17, rk(det3) ≥ 15. The proof reduces to showing β5,6(T/(det3−ℓ^3)^⊥) < 140 for every linear form ℓ (§3.4). For the full-rank case this is exactly Proposition 15's assertion that β5,6(T/F(µ,λ)^⊥) < 140 for all µ≠0, λ. The assertion is established by (i) identifying F^⊥ with the ideal H generated by the 36 quadrics of Proposition 14, and (ii) computing in Macaulay2 a free resolution of the universal ideal H~ in T~ = T[µ,λ], obtaining β5,6 = 135, then arguing by flatness that specialization preserves exactness, hence β5,6(T/F^⊥) ≤ 135. The computation is not accompanied by code or a certificate; if the true value of β5,6 for H~ were, say, at least 140, the inequality would fail and the lower bound would collapse. The flatness step additionally relies on the Hilbert function of the family being constantly (1,9,9,1), which depends on Proposition 14; that proposition's proof contains its own unshown 'direct computation' about the span of the first 35 quadrics. Thus the proof is structurally sound but is not self-contained at the decisive numerical step.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that the Waring rank of the 3x3 determinant is at least 15, improving the previously known lower bound of 14. The method is based on syzygies of apolar ideals, which the authors introduce for this purpose. They show that any one-dimensional saturated homogeneous ideal of degree 13 containing no linear form has Betti number β5,6 ≥ 140 (Proposition 6), while the apolar ideal of det3 satisfies β5,6 = 100 and for every linear form ℓ the apolar ideal of det3 − ℓ^3 satisfies β5,6 < 140 (Propositions 13 and 15). This rules out a 13-term decomposition of det3 − ℓ^3, giving rk(det3 − ℓ^3) ≥ 14 for all ℓ and hence rk(det3) ≥ 15. The same argument proves the symmetric cactus rank of the 3x3 permanent is at least 14. The paper also gives a new proof of the known bound rk(det3) ≥ 14 and reviews prior bounds.","tokens_in":17708,"tokens_out":7404,"duration_ms":67706,"significance":"If the computational steps are correct, this is a meaningful advance on a longstanding problem: it is the first improvement over 14 for the Waring rank of det3. The syzygy-of-apolar-ideal method is a novel tool in this area and is likely to be applicable to other forms. The paper is clearly written and gives a careful reduction of the linear-form perturbation to three normal forms via the action of SL3 × SL3. Proposition 6 uses Peeva's consecutive cancellation theorem and a neat h-vector enumeration, and the authors include Macaulay2 code for the lex-segment computations in that proposition. However, the decisive Betti table in Proposition 15 is not accompanied by a reproducible script, and Proposition 14 contains an unexpanded 'direct computation.' These gaps prevent the reader from fully verifying the main theorem from the manuscript alone.","major_comments":[{"comment":"The central inequality β5,6(T/F^⊥) < 140 is established by a Macaulay2 computation of the Betti table of the universal ideal H~ in T~ = T[µ,λ], reporting β5,6 = 135. No code, log, or certificate is provided, so this decisive numerical assertion is not independently checkable from the manuscript. The authors should include the Macaulay2 script (or a certified free resolution) used to obtain the table, and ideally an independent verification (for example via a different algorithm or a Betti number certificate). Without this, the proof of Theorem 17 is incomplete.","section":"§3.4, Proposition 15"},{"comment":"The proof that H = F^⊥ depends on the statement: 'One computes directly that the first 35 generators of H generate a codimension 2 space of cubics, while (F^⊥)_3 has codimension 1.' This is a load-bearing assertion: it is what allows the conclusion that H agrees with F^⊥ in degree 3 and hence that the listed quadrics generate F^⊥. The computation is not shown, and it is not a trivial consequence of the listed generators. Please provide the actual computation, for example a basis of (F^⊥)_3 and the dimension of the span of the products T_1 · (span of the first 35 quadrics), or a reproducible code snippet.","section":"§3.4, Proposition 14"},{"comment":"The flatness/specialization argument is too terse. The sentence 'We can choose a constant monomial cobasis for the family of ideals F(µ,λ)^⊥ ... so that the family is locally free' requires justification: one must show that a single set of monomials maps to a basis of every fiber algebra T/F(µ,λ)^⊥, and that this yields a locally free sheaf over the chosen base Spec k[µ±1,λ]. While the constant Hilbert function (1,9,9,1) is a standard starting point, the authors should spell out the argument or cite a precise theorem for why the family is flat, and why exactness of the specialized complex follows. This step is essential for transferring the computed β5,6 = 135 to arbitrary µ,λ with µ ≠ 0.","section":"§3.4, Proposition 15"}],"minor_comments":[{"comment":"The sentence 'Multiply the ith row of A by a^{-1}_{i,i} and the dth row of A by a_{i,i}, for i from 1 to k or d-1, whichever is less' is unclear and likely contains a typo; please rephrase the normalization step.","section":"§3.3, proof of Proposition 12"},{"comment":"The Betti tables would be easier to read if the convention (β_{i,j} appears in column i, row j−i) were stated explicitly before the first table.","section":"§3.1, Betti tables"},{"comment":"The abstract uses 'symmetric cactus rank' while the body defines 'cactus rank' and then states that all ranks are symmetric; please unify the terminology.","section":"Abstract and §2.2"},{"comment":"The two equalities β5,6 = 100 in Proposition 13 are stated as 'a direct computation in Macaulay2' without code; please include the script for these computations as well, or at least specify the exact input.","section":"§3.4, Proposition 13"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely correct, but the decisive numerical steps (Proposition 15 and part of Proposition 14) are not fully reproducible from the manuscript. I recommend requesting that the authors supply the Macaulay2 scripts and fill the gap in the proof of Proposition 14. The method is novel and the result is significant if the computations are verified. No concerns about originality or scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe headline is simple: this paper improves the lower bound for the Waring rank of the 3x3 determinant from 14 to 15, and it does so with a method—syzygies of the apolar ideal—that is genuinely new to rank lower bounds. The same machinery gives cactus rank at least 14 for the 3x3 permanent, up from 10. If the computation at the core of Proposition 15 checks out, this is a clean, real result.\n\nThe paper deserves credit for the structure it builds. The reduction of the problem to a finite list of cases by normalizing the subtracted cube is elegant. The bound β5,6 ≥ 140 for degree-13 ideals via Peeva's consecutive cancellations is a nice argument, cleanly presented. The explicit description of the apolar ideal for µ det3 − λ(tr)^3 in Proposition 14 is a concrete piece of work that appears correct as far as I can see, and the Gorenstein symmetry argument tying the Hilbert function to (1,9,9,1) is sound.\n\nThe soft spot is exactly where the stress test points. Proposition 15 rests on an unscripted Macaulay2 computation of the Betti table of the universal ideal H~, giving β5,6 = 135. That single integer is what makes β5,6 < 140, and the whole theorem depends on it. The flatness argument is sketched but not fully detailed, and it leans on the constancy of the Hilbert function, which itself relies on a 'direct computation' in Proposition 14. None of this is disqualifying—computations this size are standard in commutative algebra and the authors are experienced enough that I'd trust them—but the paper as written is not reproducible at its most crucial step. The Macaulay2 code should be included or linked; without it, the referee can only re-run the computation or take it on faith.\n\nThe literature review is thorough and accurate, and the limitations are honestly acknowledged in the remarks on trying to push to 16. This is not a case of overclaiming; the authors know exactly where the method stops.\n\nWho should read it: anyone working on Waring rank, apolarity, or the complexity of the determinant. The method is likely to be useful beyond this instance.\n\nMy recommendation: send it to peer review. The central argument is sound, the bound is a genuine improvement, and the computational gap is fixable with the code attached. I'd want the scripts before accepting, but this is exactly the kind of paper a good referee should see.","headline":"A real improvement on a benchmark rank bound via a genuinely new syzygy method; the proof hangs on one uncertified Macaulay2 computation that should be shipped before publication.","tokens_in":18270,"tokens_out":3472,"would_cite":true,"duration_ms":35707,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["15A21","15A69","14N15","13D02"],"pacs":[],"model":"deepseek-v4-flash","headline":"Waring rank of the 3x3 determinant is at least 15","keywords":["Waring rank","determinant","permanent","apolar ideal","syzygies","Betti numbers","cactus rank","lower bound"],"falsifier":"Recompute, in an independent computer algebra system or by a hand-checked Gröbner basis, the minimal free resolution of $T/(\\mu\\det_3 - \\lambda(x_1+x_5+x_9)^3)^\\perp$ for one explicit pair (e.g., $\\mu=\\lambda=1$) and check whether $\\beta_{5,6}$ is 140 or larger; a value $\\ge 140$ would invalidate the proof of the main theorem.","tokens_in":17223,"feed_emoji":"🧮","tokens_out":7492,"duration_ms":59020,"temperature":0.7,"pith_summary":"The paper proves that the Waring rank of the $3\\times 3$ determinant is at least 15, improving the known lower bound from 14 to 15 while the best upper bound remains 18. The proof is carried by a new technique: instead of studying apolar ideals directly, it counts syzygies of those ideals, specifically the graded Betti number $\\beta_{5,6}$. A general result shows any zero-dimensional scheme of degree 13 in $\\mathbb{P}^8$ with no linear forms has $\\beta_{5,6} \\ge 140$, while for every linear form $\\ell$ the apolar ideal of $\\det_3 - \\ell^3$ has $\\beta_{5,6} < 140$; the two inequalities force every such $\\det_3 - \\ell^3$ to have rank at least 14, and hence $\\det_3$ itself cannot be a sum of 14 cubes. The same argument establishes that the symmetric cactus rank of the $3\\times 3$ permanent is at least 14, up from the previously known 10.","feed_headline":"Waring rank of 3x3 determinant at least 15","feed_subtitle":"New syzygy-counting method breaks the old lower bound of 14 and lifts permanent's cactus rank to 14.","key_machinery":"The load-bearing object is the graded Betti number $\\beta_{5,6}(T/I)$, the number of minimal generators in homological degree 5 and internal degree 6 of the minimal free resolution of $T/I$. The argument rests on three inputs. First, the Apolarity Lemma equates Waring rank $r$ with the existence of a reduced zero-dimensional scheme of degree $r$ whose ideal is apolar to the form. Second, a low-degree-strand Betti inequality shows that if $I \\subseteq J$ and $J$ contains no linear form, then $\\beta_{i,i+1}(T/I) \\le \\beta_{i,i+1}(T/J)$ for all $i$; this lets small Betti numbers of an apolar ideal obstruct the existence of small apolar point ideals. Third, a consecutive-cancellations result gives the universal lower bound $\\beta_{5,6}(T/I) \\ge 140$ for every saturated ideal $I$ of degree 13 without linear forms. For the shifted determinant, the apolar ideal is described by 36 explicit quadratic generators in a universal family $\\mu\\det_3 - \\lambda(x_1+x_5+x_9)^3$, and a computed free resolution of the universal ideal has $\\beta_{5,6}=135$, which survives specialization by a flatness argument. Thus the mechanism is: universal lower bound 140 for degree-13 ideals versus universal upper bound 135 for apolar ideals of $\\det_3-\\ell^3$.","core_discovery":"The central claim is that $\\operatorname{rk}(\\det_3) \\ge 15$: the $3\\times 3$ determinant cannot be written as a sum of 14 or fewer cubes of linear forms. The proof proceeds by contradiction. For any linear form $\\ell$, the apolar ideal $(\\det_3 - \\ell^3)^\\perp$ has graded Betti number $\\beta_{5,6} < 140$, and by the Apolarity Lemma a rank-13 expression of $\\det_3 - \\ell^3$ would produce an apolar ideal of a 13-point scheme, whose $\\beta_{5,6}$ is at least 140. Hence $\\operatorname{rk}(\\det_3 - \\ell^3) \\ge 14$ for every $\\ell$, so if $\\det_3$ had rank 14, removing one cube would leave a rank-13 polynomial, a contradiction. The same comparison shows $\\operatorname{crk}(\\operatorname{per}_3) \\ge 14$. The paper also gives a new proof of the previously known bounds $\\operatorname{rk}(\\det_3) \\ge 14$, $\\operatorname{crk}(\\det_3) \\ge 14$, and $\\operatorname{rk}(\\operatorname{per}_3) \\ge 14$. The novelty is the use of syzygies of the apolar ideal, rather than the ideal's generators or Hilbert function, to bound rank.","pith_inferences":["A testable extension is to apply the same $\\beta_{5,6}$ comparison to other forms with known apolar ideals, such as larger determinants or permanents, where the required Betti-number computations may still be feasible.","Because the universal lower bound 140 comes from a consecutive-cancellations result, the method depends on the Hilbert function of the points; varying the degree or the ambient dimension should yield similar thresholds, potentially giving new rank bounds for other forms.","If the flatness step is verified independently (or fails), the method's scope changes: the syzygy obstruction would then only apply to smoothable schemes, and non-smoothable Gorenstein schemes would need separate treatment.","The paper's remark that $\\beta_{5,6}$ for the universal family is 135 rather than 100 suggests the actual value for specializations may often be 100; if that is true, a sharper analysis of the family could yield $\\operatorname{rk}(\\det_3) \\ge 16$ from the same framework."],"forward_implications":["The Waring rank of $\\det_3$ lies between 15 and 18, narrowing the previously known interval by one.","The symmetric cactus rank of $\\operatorname{per}_3$ is at least 14, improving the prior lower bound of 10.","Syzygies of apolar ideals are an effective lower-bound tool for Waring and cactus rank, yielding new proofs of the rank-14 bounds for $\\det_3$ and $\\operatorname{per}_3$.","Any polynomial of the form $\\det_3 - \\ell^3$ has Waring rank at least 14 for every linear form $\\ell$."],"supporting_citations":[{"why":"Supplies the explicit quadratic generators of the apolar ideals of the determinant and permanent, the starting point for the Betti computations.","marker":"[46]"},{"why":"Provides the consecutive-cancellations result used to establish the universal lower bound $\\beta_{5,6}(T/I) \\ge 140$ for degree-13 ideals.","marker":"[41]"},{"why":"Contains the low-degree-strand Betti inequality (Proposition 3) that lets an inclusion of ideals transfer the $\\beta_{5,6}$ obstruction.","marker":"[24]"},{"why":"Provides the computer algebra system used to compute the Betti tables of the apolar ideals and the universal family.","marker":"[29]"},{"why":"Supplies the flatness criterion used to show that the computed resolution remains exact after specializing $\\mu$ and $\\lambda$.","marker":"[30]"},{"why":"Gives the indecomposability and conciseness results for determinant and permanent used to show $\\det_3 - \\ell^3$ is concise.","marker":"[13]"}],"fun_headline_variants":["Syzygy trick raises det_3 Waring rank lower bound to 15","3x3 determinant needs ≥15 cubes: new syzygy proof","Waring rank of 3×3 determinant: new bound 15 via syzygies","Syzygies lift det rank lower bound and permanent cactus rank"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire bound rests on a computer-generated free resolution of a universal apolar ideal (reported with $\\beta_{5,6}=135$) and on the flatness argument that this resolution stays exact after substituting any nonzero $\\mu$ and any $\\lambda$; if either the computation or the flatness is wrong, the rank bound could fail.","fun_headline_variants_meta":{"raw":{"variants":["Syzygy trick raises det_3 Waring rank lower bound to 15","3x3 determinant needs ≥15 cubes: new syzygy proof","Waring rank of 3×3 determinant: new bound 15 via syzygies","Syzygies lift det rank lower bound and permanent cactus rank"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000586,"raw_usage":{"total_tokens":2729,"prompt_tokens":897,"completion_tokens":1832,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":513,"completion_tokens_details":{"reasoning_tokens":1749}},"tokens_in":513,"tokens_out":1832,"duration_ms":13819,"temperature":1.0,"reasoning_tokens":1749,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:26:37.717995+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute, in an independent computer algebra system or by a hand-checked Gröbner basis, the minimal free resolution of $T/(\\mu\\det_3 - \\lambda(x_1+x_5+x_9)^3)^\\perp$ for one explicit pair (e.g., $\\mu=\\lambda=1$) and check whether $\\beta_{5,6}$ is 140 or larger; a value $\\ge 140$ would invalidate the proof of the main theorem.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the explicit quadratic generators of the apolar ideals of the determinant and permanent, the starting point for the Betti computations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the consecutive-cancellations result used to establish the universal lower bound $\\beta_{5,6}(T/I) \\ge 140$ for degree-13 ideals."},{"cited_title":"229, Springer-Verlag , New York, 2005, A second course in commutative algebra and al gebraic geometry","cited_arxiv_id":null,"evidence_quote":"Contains the low-degree-strand Betti inequality (Proposition 3) that lets an inclusion of ideals transfer the $\\beta_{5,6}$ obstruction."},{"cited_title":"Grayson and Michael E","cited_arxiv_id":null,"evidence_quote":"Provides the computer algebra system used to compute the Betti tables of the apolar ideals and the universal family."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the flatness criterion used to show that the computed resolution remains exact after specializing $\\mu$ and $\\lambda$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the indecomposability and conciseness results for determinant and permanent used to show $\\det_3 - \\ell^3$ is concise."}],"review_version":1}