{"id":"5c7cbc6d-9e33-4e21-99ad-680cbdca5bb1","arxiv_id":"2502.10007","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For fixed degree d, strength and partition rank over any field are bounded by O(r^{d-1}) (plus a log factor on finite fields) in terms of their border rank analogues.","lead":"This paper proves that the strength of a polynomial and the partition rank of a tensor are bounded by a polynomial function of their border analogues for any fixed degree. This gives the first general-degree polynomial control of how these complexity measures can jump in a limit or drop under field extension.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The finite-field branch rests on Proposition 4.0.1's degree bound, but the binomial counting as written yields a sufficient condition for log D that grows linearly in m, so the claimed bound log D ≪_d log(r+m) is not established.","rationale":"The reader's verdict is CONDITIONAL, and I keep it CONDITIONAL: the concern below does not overturn the main de-bordering idea, but it identifies a specific unproved step in the finite-field branch. The strongest claim — polynomial-in-border-rank control of strength and partition rank over the algebraic closure, with a log factor over finite fields — requires Proposition 4.0.1 to hold uniformly in m. The proof of that proposition uses only crude binomial counts and explicitly discards structure. Re-doing the algebra shows that the displayed sufficient condition yields log D at least about 4m log(4e) + log(m n^d), not a constant multiple of log(r+m). Thus the proof does not currently justify the m-uniform logarithmic degree bound on which the finite-field theorems rest. This is not an objection to the theorem's truth; it is a precise gap in the written argument. I agree partially with the reader's weakest-assumption diagnosis: the reader flags the same Section 4 compressed claim, but the primary stated concern is the imported Theorem 3.0.1. My strongest concrete objection is the internal counting problem in Proposition 4.0.1, which is more readily checkable. No ad hominem is intended; the issue is purely mathematical and local to the proof.","tokens_in":14191,"tokens_out":30048,"duration_ms":294258,"concrete_test":"Recompute the logarithmic consequence of the displayed inequalities in Proposition 4.0.1 with a = (m-1/4)n^d and n = 4(r+m^2/d), simplifying log D ≥ 4m log(m n^d) + 4(m-1/4) log(4e/((m-1/4)n^d)). If the simplification yields log D ≥ log(m n^d) + 4(m-1/4)log(4e) + O(1), then the proof as written does not establish the claimed bound log D ≪_d log(r+m); a corrected proof must either remove the m-linear term using the monomial structure, or the finite-field theorem needs a weaker m-dependence.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.7.2 for finite K — and hence the finite-field cases of Theorems 1.5.3, 1.2.3, and 1.1.2 — depends on Proposition 4.0.1's bound log(D) ≪_d log(r+m). The proof compares the space of degree-≤D polynomials on the m n^d ambient coordinates with an upper bound A = binom(a+3D, a) for the pullback monomials, where a = (m-1/4)n^d. Accepting the paper's displayed bounds, the sufficient condition is (1/4)n^d log D ≥ m n^d log(m n^d) + a log(4e/a). Substituting a = (m-1/4)n^d and n = 4(r+m^2/d), this simplifies to log D ≥ log(m n^d) + 4(m-1/4)log(4e) + O(1), which contains a term linear in m and is not bounded by a constant multiple of log(r+m) as m grows. The proof explicitly says 'these monomials have more structure, but we ignore this' (Section 4); ignoring that structure is precisely what prevents the m-uniform logarithmic degree bound from following. The stated theorem may still be true, but the written argument for the finite-field branch has a concrete gap at Proposition 4.0.1.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies how the strength and partition rank of homogeneous polynomials and tensors behave under field extensions and under passage to limits (border rank). The main theorems (Theorems 1.2.3, 1.6.3, 1.7.2) assert that for fixed degree d, the ordinary strength or partition rank is bounded by a polynomial in the corresponding border rank, uniformly in the ground field under mild characteristic assumptions, with an additional logarithmic factor over finite fields. The proofs use polynomial functor machinery, import a reconstruction theorem from the authors' earlier work [BDE19], and proceed by dimension counts and, in the finite-field case, by a degree bound for equations vanishing on the relevant parameterized varieties. The paper also derives consequences for the drop of ranks under field extensions and for jumps under limits.","tokens_in":14515,"tokens_out":14712,"duration_ms":124855,"significance":"If the theorems are correct, they constitute a significant advance: they give the first quantitative, field-uniform de-bordering results for strength and partition rank in fixed degree, subsuming and strengthening earlier work of Lampert–Ziegler and others. The infinite-field part of the proof is coherent and appears sound, and the paper has useful corollaries (for example, control of rank drops under field extensions and control of jumps in limits). The manuscript is clearly written and the framework of polynomial functors is well suited to the problem. However, the finite-field branch rests on a delicate degree estimate (Proposition 4.0.1) that is not established by the written argument; this gap affects several of the main stated results.","major_comments":[{"comment":"The argument that log(D) ≪_d log(r+m) does not follow from the displayed binomial counting. After dividing the inequality (1/4)n^d log D ≥ m n^d log(m n^d) + (m-1/4)n^d log(4e/((m-1/4)n^d)) by (1/4)n^d, one obtains log D ≥ 4m log(m n^d) + 4(m-1/4) log(4e/((m-1/4)n^d)). With n = 4(r + m^2/d), the first term is of order m log(r+m), which is not O_d(log(r+m)). The subsequent \"sufficient condition\" displayed in the paper, log D ≥ 4 log(m n^d) + 4(m-1/4) log(3e/((m-1/3)n^d)), is not an algebraic consequence of the prior inequality; it appears to drop the factor m from the first term. Thus the claimed bound is not proven, and because Proposition 4.0.1 is the basis for the extension-degree estimate [L:K] ≪_d log(r+m) in the finite-field proof of Theorem 1.7.2, the finite-field cases of Theorems 1.7.2, 1.6.3, 1.5.3, 1.2.3, and 1.1.2 are not established.","section":"Section 4"},{"comment":"The theorem is introduced as a summary of results from [BDE19, §4] \"but generalised from tensors to m-tuples of tensors\". No proof or precise reference for the m-tuple generalization is given. This theorem is load-bearing: Corollary 3.0.2 and the inductive argument in the proof of Theorem 1.7.2 depend on it. The authors should either prove the m-tuple version or indicate explicitly where in [BDE19] the m-tuple case is proved.","section":"Section 3"},{"comment":"The statement that the minimal n satisfying (4) is \"linear in r + m2/d\" is ambiguous and, if read as r + m^2/d, is false (for d=3 the minimal n is O(r + m^{2/3})). The proof of the infinite-field case of Theorem 1.7.2 uses this to conclude that the expression in (6) is ≪_d m^3 r^{d-1}. The authors should clarify the exponent and confirm that (6) is indeed bounded as claimed for their choice of n. If the intended bound is n = O(r + m^{2/d}), the argument goes through; if n = 4(r + m^2/d) is used, the bound (6) becomes O(m r^{d-1} + m^{2d-1}), which is not the claimed m^3 r^{d-1} in all regimes.","section":"Section 3"}],"minor_comments":[{"comment":"In Proposition 4.0.1, the notation \"n = 4(r +m2/d)\" is ambiguous; please write either 4(r + m^2/d) or 4(r + m^{2/d}) consistently throughout the paper.","section":"Section 4"},{"comment":"In the derivation within Proposition 4.0.1, the transition from the logarithmic inequality to the displayed \"sufficient condition\" is algebraically incorrect; please correct the equations and re-derive the bound.","section":"Section 4"},{"comment":"The reference for Theorem 3.0.1 cites [BDE19, §4] but does not give a theorem number or page; adding one would help the reader verify the m-tuple generalization.","section":"Section 3"},{"comment":"There are several typographical errors (e.g., \"P ARTITION\" in the title, \"th e\" and \"anal ogue\" in the abstract) that should be corrected.","section":"Title and Abstract"}],"recommendation":"major_revision","confidential_remarks":"The manuscript draws heavily on the authors' earlier paper [BDE19], and while the dependence is not circular, the m-tuple generalization of Theorem 3.0.1 needs explicit justification. The central issue is the finite-field degree bound: the binomial counting in Proposition 4.0.1 appears to give only log D ≪ m log(r+m), not log D ≪ log(r+m), so the stated finite-field theorems are not currently proven. If this bound cannot be repaired, the paper would still be viable with the infinite-field results alone, but the finite-field claims would need to be revised or removed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a serious paper that substantially extends the BDE19 program to prove polynomial de-bordering for strength and partition rank in general degree, including collective (tuple) versions. The infinite-field proof is clean and the dimension counting checks out. The finite-field branch, however, has a concrete gap in Proposition 4.0.1: the binomial estimate as written gives a sufficient condition for log D that grows at least linearly in m, not O_d(log(r+m)). The paper's own displayed inequality seems to have a typo (the factor m on the first log term), and even after correcting it the bound is not uniform in m. The sentence 'these monomials have more structure, but we ignore this' is precisely the place where the uniform logarithmic degree bound is lost. This is load-bearing for all the finite-field theorems (1.1.2, 1.2.3, 1.5.3, 1.6.3, 1.7.2). I suspect the theorems are true and the gap is repairable, but as written the finite-field results are not proved.\n\nWhat is genuinely new: first polynomial-in-border-rank bounds for strength and partition rank for all fixed degrees (previous work covered cubics, quartics, or special fields, or gave a non-explicit function). The m-tuple collective versions are also new. The proof architecture is not a new framework—it follows BDE19—but the extension to general degree and tuples requires real work, and the induction over subvarieties with a degree-minimal polynomial is executed carefully. The dependence on BDE19's Theorem 3.0.1 is self-citation, but it is a prior published result whose assumptions are met; I don't see circularity.\n\nThe citation pattern looks reasonable: they compare with LZ24, BL24, AKZ21, KP23, BDS24a. No invented entities, no hidden parameters beyond the auxiliary space dimension n.\n\nWho should read it: anyone working on Schmidt/strength/partition/analytic rank, especially quantitative bounds over finite fields. It deserves a serious referee. The referee should focus on Proposition 4.0.1 and ask for a corrected proof or a different treatment of the finite-field case. I would not desk reject; I would send it out and expect a major revision or a replacement of Section 4.","headline":"Strong paper with real new results on de-bordering for strength and partition rank, but the finite-field branch rests on a degree estimate whose written proof doesn't deliver the claimed log bound.","tokens_in":15031,"tokens_out":8634,"would_cite":true,"duration_ms":73484,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["15A69","14L30"],"pacs":[],"model":"deepseek-v4-flash","headline":"Border rank polynomially bounds strength and partition rank.","keywords":["strength","partition rank","border rank","de-bordering","field extensions","polynomial functors","homogeneous polynomials","tensors"],"falsifier":"Exhibit, for some fixed degree $d\\ge 3$ and some field $K$ with $\\mathrm{char}(K)=0$ or $>d$, a sequence of forms $f_N$ with border strength $s(f_N)\\le r_N$ but honest strength $s_K(f_N)$ growing faster than $C_d r_N^{d-1}$ (over infinite $K$) or faster than $C_d r_N^{d-1}\\log r_N$ (over finite $K$), for every constant $C_d$; an analogous counterexample for tensors would disprove Theorem 1.6.3.","tokens_in":14014,"feed_emoji":"📐","tokens_out":19210,"duration_ms":155504,"temperature":0.7,"pith_summary":"Strength is the minimal number of products of lower-degree homogeneous polynomials needed to express a form; partition rank is the analogous measure for multilinear tensors. For fixed degree $d$, the paper proves that the border versions of these ranks, defined via Zariski closures, control the actual ranks over the ground field by a polynomial bound: if the border strength is $r$, then the strength over $K$ is at most $O_d(r^{d-1})$ for infinite $K$ and $O_d(r^{d-1}\\log r)$ for finite $K$, and the analogous statements hold for partition rank (with no characteristic restriction) and for tuples. This makes the previously non-explicit or field-dependent control of rank drops under field extensions and rank jumps in limits explicit and uniform. A by-product shows that a low-border-strength form lies in a subalgebra generated by few elements of its derivative space.","feed_headline":"Border rank polynomially bounds strength and partition rank","feed_subtitle":"A low border rank keeps strength and partition rank polynomially small, uniformly in the field.","key_machinery":"The load-bearing mechanism is a reconstruction theorem (Theorem 3.0.1), imported from [BDE19, Section 4] and generalized to $m$-tuples of tensors. It says that if a closed subvariety $X$ of $m$-tuples of tensors is defined by polynomials with coefficients in the prime field, and a nonzero partial derivative $h=\\partial f/\\partial x_1$ of a defining polynomial does not vanish at a tuple, then the $m$-th tensor can be reconstructed from the other $m-1$ tensors and from smaller tensor factors, so the collective partition rank is bounded by the number of small-tensor summands in a shift of $T_{[d]}^m$. The proof then runs an induction on the minimal degree of a vanishing polynomial for such subvarieties: either the tuple lands in a smaller subvariety where the induction hypothesis applies, or the reconstruction theorem bounds its partition rank. For finite fields, an additional degree estimate (Proposition 4.0.1) shows the minimal degree of a vanishing polynomial is $O_d(\\log(r+m))$, and a finite-field-extension trick (Proposition 2.2.1) multiplies the partition rank bound by the extension degree, producing the $\\log$-factor.","core_discovery":"The central discovery is a de-bordering theorem: border rank is not just a topological relaxation but a true upper bound on actual rank, up to a polynomial in fixed degree. For strength, Theorem 1.2.3 establishes that if a degree-$d$ form $f$ has border strength $s(f)=r$ and $\\mathrm{char}(K)=0$ or $>d$, then $s_K(f)\\ll_d r^{d-1}$ for infinite $K$ and $s_K(f)\\ll_d r^{d-1}\\log r$ for finite $K$. For partition rank, Theorem 1.6.3 gives the same polynomial control with no characteristic condition, and the collective versions (Theorems 1.5.3 and 1.7.2) extend the bounds to $m$-tuples with an extra factor $m^3$ and, in the finite-field case, a factor $\\log(r+m)$. The paper also derives Theorem 1.3.1: such a form lies in a subalgebra generated by $\\ll_d r^d$ (or $\\ll_d r^d\\log r$ over finite fields) elements of the space $D(f)$ of its partial derivatives. The paper thus establishes polynomial bounds that are explicit in $d$ and uniform across all fields satisfying the stated characteristic assumptions.","pith_inferences":["The proof's constants depend only on $d$, but the paper does not compute them; tracking them through the induction and the degree estimates could yield explicit (not just existential) bounds, which would make the de-bordering algorithmic.","The log factor for finite fields comes from the degree estimate in Proposition 4.0.1; if that estimate could be improved to $O_d(1)$, the finite-field bounds would become polynomial without a log term, matching the infinite-field rate.","The same reconstruction-from-derivative scheme could apply to other rank models (for instance, Waring rank or analytic rank) whenever an analogous defining-polynomial and derivative pair exists, potentially extending de-bordering beyond strength and partition rank."],"forward_implications":["For any field $K$ with $\\mathrm{char}(K)=0$ or $>d$, a form of border strength $r$ has honest strength at most $C_d r^{d-1}$; hence ranks cannot jump by more than a fixed polynomial in a limit.","Partition rank of tensors obeys the same polynomial control with no characteristic restriction, giving a field-independent de-bordering result for tensors.","The collective versions imply that for $m$-tuples of forms or tensors, bounded border collective rank bounds the actual collective rank by $O_d(m^3 r^{d-1})$ (with an extra $\\log(r+m)$ over finite fields).","Because strength over a field extension can only drop, the bounds control the drop: passing to the algebraic closure can reduce strength by at most the same polynomial factor.","Theorem 1.3.1 yields a structural statement: low-border-strength forms are generated as a polynomial algebra by at most $O_d(r^d)$ (or $O_d(r^d\\log r)$ over finite fields) elements of their derivative space."],"supporting_citations":[{"why":"Supplies Theorem 3.0.1, the reconstruction-from-partial-derivative statement that the entire de-bordering proof iterates, and the paper also draws on its section on covariants.","marker":"[BDE19]"},{"why":"Proves quantitative strength bounds under field extensions for fields with characteristic 0 or >d, which Theorem 1.1.2 extends to all such fields and to partition rank.","marker":"[LZ24]"},{"why":"Establishes a non-explicit bound for strength over semi-perfect fields, which the present paper makes explicit and polynomial under its characteristic assumptions.","marker":"[BDS24a]"}],"fun_headline_variants":["De-bordering: border rank controls actual rank polynomially","Field-uniform polynomial bound: border rank caps strength and partition rank","Border strength polynomial in border rank, uniformly over fields","Fixed-degree polynomial bounds from border rank for strength and partition rank","Border rank tames limit jumps and field drops polynomially"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on the reconstruction theorem (Theorem 3.0.1), imported from earlier work, that a single nonzero partial derivative of a defining polynomial lets one rebuild the last tensor from the others; if that theorem or its $m$-tuple generalization fails, the polynomial bounds collapse.","fun_headline_variants_meta":{"raw":{"variants":["De-bordering: border rank controls actual rank polynomially","Field-uniform polynomial bound: border rank caps strength and partition rank","Border strength polynomial in border rank, uniformly over fields","Fixed-degree polynomial bounds from border rank for strength and partition rank","Border rank tames limit jumps and field drops polynomially"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000891,"raw_usage":{"total_tokens":3815,"prompt_tokens":890,"completion_tokens":2925,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":506,"completion_tokens_details":{"reasoning_tokens":2841}},"tokens_in":506,"tokens_out":2925,"duration_ms":21675,"temperature":1.0,"reasoning_tokens":2841,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T19:43:52.934987+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit, for some fixed degree $d\\ge 3$ and some field $K$ with $\\mathrm{char}(K)=0$ or $>d$, a sequence of forms $f_N$ with border strength $s(f_N)\\le r_N$ but honest strength $s_K(f_N)$ growing faster than $C_d r_N^{d-1}$ (over infinite $K$) or faster than $C_d r_N^{d-1}\\log r_N$ (over finite $K$), for every constant $C_d$; an analogous counterexample for tensors would disprove Theorem 1.6.3.","supporting_citations":[],"review_version":1}