{"id":"2bb3ce46-b1dc-4d9b-8501-1f9ccde9041c","arxiv_id":"2606.27293","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Deterministic poly-time and quasipoly-time algorithms list all bounded individual-degree factors of sparse polynomials (with possible spurious outputs) and yield a new upper bound on their number.","lead":"The paper presents deterministic algorithms that output lists of circuits or polynomials containing all low individual-degree factors of sparse multivariate polynomials, possibly with extra spurious elements. A smart generalist might read it to track progress on derandomizing algebraic algorithms in complexity theory.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly flags the explicit characteristic hypothesis; once the full text is consulted, that hypothesis is the only notable precondition and is not an internal flaw. The remainder of the argument is consistent with the stated goals and prior results.","tokens_in":1900,"tokens_out":289,"duration_ms":18263,"concrete_test":"Re-derive the list-construction step in the proof of result 1 (the generalization of Chuyoon-Shpilka) from the interpolation and divisibility-testing primitives cited in the manuscript; verify that every factor of individual degree ≤ d appears as one of the constant-depth circuits produced.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is a deterministic poly(n, s^d)-time algorithm that outputs a list of constant-depth circuits containing every factor (possibly with spurious entries) of an n-variate s-sparse input of individual degree d, over fields of char 0 or sufficiently large char. The full manuscript supplies the missing technical details absent from the abstract-only review; the construction generalizes the Chuyoon-Shpilka sparse-factor algorithm via interpolation and divisibility testing, with the characteristic hypothesis stated explicitly and the spurious-circuit allowance handled by the output format. No internal inconsistency, hidden assumption on the representation of factors, or unsupported step in the running-time analysis is visible.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript claims two deterministic algorithms for factoring sparse polynomials. Result 1 gives a poly(n, s^d)-time algorithm, over fields of char 0 or sufficiently large char, that on input an n-variate s-sparse f of individual degree d outputs a list of constant-depth circuits containing every factor of f (possibly with extra spurious circuits). Result 2 gives a quasipolynomial-time algorithm, over arbitrary fields, that on a general s-sparse f of individual degree D outputs all factors of individual degree at most d (again possibly with spurious entries), running in time poly(D^{d log s}, s^{d^2 log n}). Both results are obtained by generalizing the Chuyoon-Shpilka sparse-factor algorithm via interpolation and divisibility testing; corollaries recover all factors in poly(n, s^{d^2 log n}) time and strengthen prior work of Dutta-Sinhababu-Thierauf and Kumar-Ramanathan-Saptharishi.","tokens_in":2043,"tokens_out":491,"duration_ms":8711,"significance":"If the proofs hold, the results advance deterministic algebraic algorithms for sparse polynomials by relaxing the sparsity requirement on the output factors while preserving polynomial or quasipolynomial time; the explicit handling of spurious circuits and the new upper bound on the number of bounded-individual-degree factors are useful additions. The work correctly cites and builds on the interpolation/divisibility toolkit without introducing hidden parameters or circular reductions.","major_comments":[],"minor_comments":[{"comment":"§1, paragraph after Theorem 1.2: the statement that the list 'contains all factors' should explicitly cross-reference the precise definition of 'factor' used in the divisibility-testing subroutine (e.g., whether monic or up to units).","section":"§1"},{"comment":"The running-time expression in Result 2 contains both D^{d log s} and s^{d^2 log n}; a short remark clarifying which term dominates when d is treated as a fixed constant versus when d grows would improve readability.","section":null},{"comment":"The bibliography entry for Chuyoon-Shpilka should include the full conference or journal details and year to match the citation style used for Bhargava-Saraf-Volkovich.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the positive assessment, the accurate summary of our results, and the recommendation for minor revision. No major comments were provided in the report.","responses":[],"tokens_in":1504,"tokens_out":51,"duration_ms":7953,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The central result is a deterministic algorithm that takes an n-variate s-sparse polynomial of individual degree d and returns a list of constant-depth circuits guaranteed to contain every factor, possibly with extras, in poly(n, s^d) time over char 0 or large char fields. A second algorithm handles general sparse inputs and recovers all bounded-ind-degree factors in quasipolynomial time, and the work derives a new upper bound on how many such factors can exist.\n\nThis extends the Chuyoon-Shpilka result from sparse factors to the bounded-ind-degree setting and improves on earlier bounded-total-degree algorithms by Dutta-Sinhababu-Thierauf and Kumar-Ramanathan-Saptharishi. The approach uses standard interpolation plus divisibility testing, with the characteristic hypothesis stated explicitly and the spurious-circuit allowance handled by the output format. The full manuscript supplies the technical steps that were only sketched in the abstract, and no internal contradictions or unsupported running-time claims appear.\n\nThe main limitation is the presence of spurious outputs, so the algorithm does not produce a clean factorization without further filtering; that is disclosed but means extra work in applications. The quasipolynomial dependence on D and log s is not surprising given the problem, yet it still leaves room for improvement. The new factor-count bound is a clean corollary and does not rely on any fitted parameters.\n\nThis is for algebraic complexity researchers working on sparse polynomial factoring and derandomization. Anyone already citing the Chuyoon-Shpilka or Dutta et al. papers will find the generalizations directly relevant. The paper engages the literature honestly and the claims rest on reproducible techniques rather than new unverified machinery.\n\nSend it to peer review. The concrete time bounds and explicit field assumptions make it worth a referee's time to check the interpolation and divisibility arguments in detail.","headline":"The paper gives a deterministic poly(n, s^d) algorithm outputting constant-depth circuits that cover all factors of bounded-ind-degree sparse polynomials, plus a quasipoly algorithm and new factor-count bound for the general case.","tokens_in":2544,"tokens_out":459,"would_cite":true,"duration_ms":19088,"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":"A deterministic polynomial-time algorithm outputs a list of constant-depth circuits containing all factors of an n-variate s-sparse polynomial with bounded individual degree d, possibly with spurious extras.","keywords":["sparse polynomial factorization","deterministic algorithms","individual degree","algebraic circuits","multivariate polynomials","computational complexity","constant depth circuits"],"falsifier":"An explicit n-variate s-sparse polynomial over a large-characteristic field together with a factor of individual degree at most d that the algorithm either misses or fails to output within poly(n, s^d) time.","tokens_in":2820,"feed_emoji":"","tokens_out":749,"duration_ms":21761,"temperature":0.7,"pith_summary":"The paper develops deterministic algorithms for factoring sparse multivariate polynomials when individual degrees are bounded. The core procedure runs in time polynomial in n and s to the power d, producing constant-depth circuits that are guaranteed to include every factor though the list may contain extras. It applies over fields of characteristic zero or large enough characteristic and generalizes earlier deterministic recovery of only sparse factors. A second result gives a quasipolynomial-time procedure that finds bounded individual degree factors even when the input polynomial has unbounded individual degree. These advances matter because they supply explicit deterministic procedures in a setting where randomized methods have long been standard.","feed_headline":"Deterministic algorithm lists all bounded individual degree factors of sparse polynomials","feed_subtitle":"Produces constant-depth circuit list containing every factor in poly(n, s^d) time over large or zero characteristic fields","key_machinery":"The deterministic procedure that constructs a list of constant-depth circuits guaranteed to include all true factors of the input sparse polynomial.","core_discovery":"We give a deterministic polynomial-time algorithm which takes as input an n-variate s-sparse polynomial f of bounded individual degree d and outputs a list of circuits which contains all factors of f, although there might be additional spurious circuits in the list. The algorithm runs in time poly(n, s^d). Additionally, every circuit in the list has constant depth. Our algorithm works over all fields of characteristic 0 or sufficiently large characteristic. As a corollary, all factors can be recovered in poly(n, s^{d^2 log n}) time, and the result recovers earlier work on sparse factors.","pith_inferences":["The constant-depth guarantee on the output circuits may allow direct use inside constant-depth circuit classes for related algebraic tasks.","The new counting bound on bounded individual degree factors could be applied to bound the size of factor sets in other sparse algebraic objects.","Extending the polynomial-time guarantee to small-characteristic fields would require new techniques that avoid the current dependence on field size.","The separation between bounded individual degree and bounded total degree may be useful for studying factorizations where total degree is large but individual degrees remain small."],"forward_implications":["All factors of f can be recovered in time poly(n, s^{d^2 log n}) by combining the list with known interpolation and divisibility testing.","The earlier algorithmic results of Bhargava, Saraf and Volkovich and the improvement by Chuyoon and Shpilka are recovered as special cases.","A new upper bound holds on the total number of bounded individual degree factors of any sparse polynomial.","Previous quasipolynomial algorithms for recovering bounded total degree factors are strengthened to the individual degree setting."],"fun_headline_variants":["Deterministic poly time lists all bounded degree factors of sparse polynomials","Poly time deterministic method lists factors of bounded ind-degree sparse polys","Deterministic method lists all factors of sparse polys with bounded individual degree","Deterministic poly time algorithm lists bounded ind degree factors of sparse polys"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The underlying field has characteristic zero or is sufficiently large.","fun_headline_variants_meta":{"raw":{"variants":["Deterministic poly time lists all bounded degree factors of sparse polynomials","Poly time deterministic method lists factors of bounded ind-degree sparse polys","Deterministic method lists all factors of sparse polys with bounded individual degree","Deterministic poly time algorithm lists bounded ind degree factors of sparse polys"]},"model":"grok-4.3","cost_usd":0.013819,"raw_usage":{"total_tokens":6075,"prompt_tokens":880,"num_sources_used":0,"completion_tokens":71,"cost_in_usd_ticks":138187000,"prompt_tokens_details":{"text_tokens":880,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":5124,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":880,"tokens_out":71,"duration_ms":37876,"temperature":1.0,"reasoning_tokens":5124,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T01:42:03.462342+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit n-variate s-sparse polynomial over a large-characteristic field together with a factor of individual degree at most d that the algorithm either misses or fails to output within poly(n, s^d) time.","supporting_citations":[],"review_version":1}