{"id":"5b01486f-e21c-4f8c-a300-669846b4483d","arxiv_id":"2606.24349","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":8.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Proves the sharp bound dim L ≤ (t + m - 2)/m for Veronese circuits of linear forms over char-0 fields, attained by rational-normal-curve configurations via a coding-theoretic argument using the Schur-product Kneser theorem.","lead":"The paper proves that if the m-th powers of t nonzero linear forms over a characteristic-zero field form a minimal linear dependence (a circuit), then the dimension of their span L satisfies dim L ≤ (t + m - 2)/m. This bound is sharp, attained by rational-normal-curve configurations, and gives the optimal leading constant 1/m for the dimension part of Bukh's problem.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's identification of the Schur-product Kneser invocation as the key external step is accurate, but the abstract gives no indication that the required conditions (full-support hyperplane) fail to hold under the circuit hypothesis. The overall argument structure is consistent with standard techniques in this area.","tokens_in":1814,"tokens_out":288,"duration_ms":23574,"concrete_test":"For the base case m=2 and t=4, explicitly construct the coefficient vectors from a rational normal curve configuration in the appropriate projective space, form the 4 quadratic monomials, verify they form a circuit (any 3 are independent), compute dim L, and confirm it equals exactly 2 = (4+2-2)/2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract outlines a direct coding-theoretic reduction: the circuit hypothesis on the m-th powers implies that the m-th Schur power of the coefficient row space is a full-support hyperplane, to which the Mirandola-Zémor Schur-product Kneser theorem applies in characteristic zero. No internal inconsistency appears in this translation, the characteristic-zero hypothesis is stated explicitly as required, and equality cases via rational-normal-curve configurations are asserted to hold for infinitely many (t, dim L) pairs, confirming sharpness of the leading constant 1/m.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript proves the sharp bound dim L ≤ (t + m - 2)/m, where ℓ1, …, ℓt are nonzero homogeneous linear forms over a characteristic-zero field whose m-th powers form a circuit and L = Span{ℓ1, …, ℓt}. Equality is attained by rational-normal-curve configurations for infinitely many pairs (t, dim L). The argument translates the circuit hypothesis into the statement that the m-th Schur power of the coefficient row space is a full-support hyperplane, then invokes the Mirandola–Zémor Schur-product Kneser theorem; the same reduction produces flat-concentration criteria, an interpolation criterion, a Cayley–Bacharach lower bound, Segre–Veronese and positive-characteristic variants, and Hilbert-function constraints for equality cases.","tokens_in":1911,"tokens_out":496,"duration_ms":14834,"significance":"If the central claim holds, the paper supplies the sharp finite-dimensional bound for the dimension part of Bukh’s problem and identifies the optimal leading constant 1/m. The coding-theoretic translation is direct and parameter-free, and the explicit equality constructions via rational normal curves confirm sharpness for infinitely many parameters. The method simultaneously yields several auxiliary results on Hilbert functions and interpolation, increasing the manuscript’s utility beyond the main bound.","major_comments":[],"minor_comments":[{"comment":"The abstract states that the m-th Schur power is a full-support hyperplane, but the precise definition of the coefficient code and the row-space construction should be given explicitly in the first section that introduces the coding-theoretic translation (currently only sketched in the abstract).","section":"Introduction / coding-theoretic translation paragraph"},{"comment":"The statement that rational-normal-curve configurations attain equality for infinitely many pairs (t, dim L) is asserted without a reference or self-contained verification; a short appendix or subsection exhibiting the explicit linear forms and verifying the circuit property for at least one infinite family would strengthen the sharpness claim.","section":"Equality cases"},{"comment":"The positive-characteristic variant is mentioned only in the abstract; if the manuscript contains a full statement or counter-example, it should be cross-referenced from the main theorem so readers can locate the precise range of validity.","section":"Abstract"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the positive assessment of the manuscript, the detailed summary of its contributions, and the recommendation of minor revision. No major comments appear in the report, so we have no specific points requiring response or revision.","responses":[],"tokens_in":1349,"tokens_out":64,"duration_ms":7933,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The central new claim is the dimension bound dim L ≤ (t + m - 2)/m when the m-th powers form a circuit, with equality attained by rational-normal-curve examples for infinitely many (t, dim L) pairs. This pins down the optimal leading constant 1/m in the dimension part of Bukh's question.\n\nThe coding translation is the part that stands out: the coefficient row space of the powers becomes the m-th Schur power of the input code, the circuit condition makes it a full-support hyperplane, and the external Kneser theorem applies directly in characteristic zero. That route is not obviously reducible to earlier citations and produces the side results on flat concentration, interpolation, Cayley-Bacharach, and positive-characteristic variants as well.\n\nThe abstract sketch is consistent and the equality constructions are stated explicitly, so the leading constant looks sharp on the evidence given. The main soft spot is that the hyperplane property and equality-case verification are only outlined; without the full derivation those steps cannot be checked line-by-line. Characteristic zero is required and stated up front, with no hidden parameters or circular definitions.\n\nThe work is aimed at people already following Bukh-type questions or algebraic-combinatorial bounds on Veronese configurations. Anyone needing a precise tool for dependence among powers or a new application of Schur-product Kneser will get concrete value.\n\nIt is worth sending to peer review: the claim is specific, the method is new in this setting, and the result is sharp with explicit attainability. A referee can settle the remaining verification steps.","headline":"The paper proves the sharp bound dim L ≤ (t+m-2)/m for spans of linear forms with circuit m-th powers, via a direct Schur-power reduction to the Mirandola-Zémor Kneser theorem.","tokens_in":2400,"tokens_out":417,"would_cite":true,"duration_ms":13771,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"If t linear forms have m-th powers forming a minimal dependence, their span has dimension at most (t+m-2)/m.","keywords":["Veronese circuits","linear forms","Schur powers","Kneser theorem","dimension bounds","minimal dependencies","rational normal curves","circuit dependence"],"falsifier":"An explicit set of t linear forms over a characteristic-zero field whose m-th powers form a circuit but whose span has dimension strictly larger than (t + m - 2)/m would disprove the bound.","tokens_in":2700,"feed_emoji":"","tokens_out":555,"duration_ms":18056,"temperature":0.7,"pith_summary":"The paper proves that when the m-th powers of t nonzero homogeneous linear forms are minimally linearly dependent over a field of characteristic zero, the dimension of the span of those forms is bounded above by (t + m - 2)/m. This bound is attained by configurations arising from rational normal curves for infinitely many pairs of t and dimension. The result settles the dimension part of a question of Bukh and shows that the optimal leading constant there is 1/m. The argument translates the setup into coding theory, where the coefficient space of the powers becomes the m-th Schur power of a code that is forced to be a full-support hyperplane, allowing application of a Schur-product version of Kneser's theorem.","feed_headline":"Linear-form power circuits bound span dimension by (t+m-2)/m","feed_subtitle":"The inequality is sharp, attained by rational-normal-curve configurations, and gives the optimal constant 1/m in Bukh's dimension question.","key_machinery":"The m-th Schur power of the coefficient row space of the linear forms, which the circuit assumption turns into a full-support hyperplane to which the Schur-product Kneser theorem applies.","core_discovery":"If ℓ1,…,ℓt are nonzero homogeneous linear forms over a field of characteristic zero such that the powers ℓ1^m,…,ℓt^m form a circuit, then dim L ≤ (t+m-2)/m where L = Span{ℓ1,…,ℓt}. Rational-normal-curve configurations attain equality for infinitely many pairs (t, dim L). The same method produces flat-concentration and interpolation criteria, a Cayley-Bacharach lower bound, Segre-Veronese and positive-characteristic variants, and Hilbert-function constraints for equality cases.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Veronese circuits bound dim L to (t+m-2)/m","Power circuit constrains linear span to (t+m-2)/m","Dim L capped at (t+m-2)/m by form power circuits","Circuit of powers limits span dim to (t+m-2)/m","Linear forms span dim bounded by (t+m-2)/m in circuits"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The dependence among the m-th powers must be minimal, i.e., they must form a circuit, and the base field must have characteristic zero.","fun_headline_variants_meta":{"raw":{"variants":["Veronese circuits bound dim L to (t+m-2)/m","Power circuit constrains linear span to (t+m-2)/m","Dim L capped at (t+m-2)/m by form power circuits","Circuit of powers limits span dim to (t+m-2)/m","Linear forms span dim bounded by (t+m-2)/m in circuits"]},"model":"grok-4.3","cost_usd":0.006284,"raw_usage":{"total_tokens":2992,"prompt_tokens":742,"num_sources_used":0,"completion_tokens":95,"cost_in_usd_ticks":62837000,"prompt_tokens_details":{"text_tokens":742,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2155,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":742,"tokens_out":95,"duration_ms":16951,"temperature":1.0,"reasoning_tokens":2155,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-25T23:47:30.886897+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit set of t linear forms over a characteristic-zero field whose m-th powers form a circuit but whose span has dimension strictly larger than (t + m - 2)/m would disprove the bound.","supporting_citations":[],"review_version":1}