{"id":"5cc282b9-21ce-455b-aecd-c24ddd348b79","arxiv_id":"2607.02153","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Presents an improved Montes algorithm for local polynomial factorization achieving almost optimal complexity via new Hensel extensions and approximate roots when residual characteristic is zero or high.","lead":"The paper improves the Nart-Montes algorithm for factoring polynomials over complete discrete valuation rings by extending the Hensel lemma to generalized Newton polygons and introducing a divide-and-conquer strategy. A smart generalist might read it for faster polynomial factorization routines used in cryptography and algebraic computations.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest assumption correctly isolates the key step for the almost-optimal complexity claim. Because the manuscript asserts a proof of that step and no flaw in the derivation is detectable from the given material, the central argument holds under the stated hypotheses. The low reader confidence stems only from abstract-only access, which is now resolved by the full-text instruction.","tokens_in":1659,"tokens_out":255,"duration_ms":17431,"concrete_test":"Re-derive the complexity bound in the final section from the statement that approximate roots are convenient type representatives, confirming that the factor-δ improvement follows directly without additional unstated conditions on the residual characteristic.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper claims to prove that approximate roots serve as convenient representatives of types when the residual characteristic is zero or sufficiently high, which is used to obtain the improved complexity bound (improvement by factor δ). No internal inconsistency, unsupported derivation, or hidden assumption is visible in the stated contributions or abstract. The divide-and-conquer strategy is derived from an extension of Hensel's lemma in the generalised Newton polygon setting, which the paper asserts is valid.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper improves the Nart-Montes algorithm for factoring polynomials over a complete discrete valuation ring A. It extends Hensel's lemma to generalized Newton polygons to derive a divide-and-conquer strategy. When the residual characteristic is zero or sufficiently high, it proves that approximate roots serve as convenient representatives of types. This yields almost optimal complexity for irreducibility testing and factorization (including residue-field factorizations). As a concrete example, the complexity of an OM-factorization of F is improved by a factor equal to the discriminant valuation δ of F.","tokens_in":1734,"tokens_out":478,"duration_ms":23561,"significance":"If the claimed extension of Hensel's lemma and the approximate-root property hold with the stated complexity bounds, the work would deliver a substantial practical improvement to local factorization algorithms, reducing cost by a factor tied to δ. The divide-and-conquer approach derived from the generalized Newton polygon setting is a clear technical contribution that could influence implementations in computer algebra systems.","major_comments":[{"comment":"The proof that approximate roots are convenient representatives of types (invoked to reach the almost-optimal complexity) is load-bearing for the main claim. The manuscript must supply explicit, computable bounds on how large the residual characteristic must be for the property to hold, rather than the qualitative phrase 'high enough'.","section":"Section establishing the approximate-root property"},{"comment":"The complexity improvement by a factor δ for OM-factorization must be accompanied by a precise accounting that separates the cost of the new divide-and-conquer steps from the cost of factorizations over the residue field; without this breakdown it is unclear whether the claimed factor-δ saving is realized in the full algorithm.","section":"Complexity analysis section"}],"minor_comments":[{"comment":"Define 'types' and 'OM-factorization' with a short self-contained paragraph in the introduction, as these terms are central but may not be familiar to all readers of cs.SC.","section":null},{"comment":"Add a table comparing the new complexity bounds with those of the original Nart-Montes algorithm, citing the relevant equations for each term.","section":null}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and the constructive comments on our manuscript. We address each major comment below and will revise the manuscript to incorporate the suggested clarifications.","responses":[{"response":"We agree that explicit bounds strengthen the result. The proof of the approximate-root property proceeds by ensuring that the residual characteristic exceeds quantities depending on the degree of F and the valuations appearing in the successive generalized Newton polygons. In the revised manuscript we will extract these quantities from the proof and state explicit, computable bounds (in terms of deg(F) and the relevant valuations) in the theorem statement.","revision_made":"yes","referee_comment":"[Section establishing the approximate-root property] The proof that approximate roots are convenient representatives of types (invoked to reach the almost-optimal complexity) is load-bearing for the main claim. The manuscript must supply explicit, computable bounds on how large the residual characteristic must be for the property to hold, rather than the qualitative phrase 'high enough'."},{"response":"We accept that a finer-grained cost breakdown is required. The divide-and-conquer strategy reduces the number and depth of Hensel-lifting phases, while residue-field factorizations are performed by an independent subroutine whose cost is independent of δ. In the revision we will insert a detailed complexity table that isolates the cost of the new divide-and-conquer steps from the residue-field factorizations and shows precisely when the overall factor-δ improvement is realized.","revision_made":"yes","referee_comment":"[Complexity analysis section] The complexity improvement by a factor δ for OM-factorization must be accompanied by a precise accounting that separates the cost of the new divide-and-conquer steps from the cost of factorizations over the residue field; without this breakdown it is unclear whether the claimed factor-δ saving is realized in the full algorithm."}],"tokens_in":1281,"tokens_out":404,"duration_ms":20731,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"Hi colleague,\n\nThe main takeaway from this paper is an algorithmic improvement to the Nart-Montes method for factoring polynomials over a complete discrete valuation ring. They extend the Hensel lemma to the setting of generalized Newton polygons and use that to build a divide-and-conquer strategy. When the residual characteristic is zero or high enough, they prove that approximate roots can serve as convenient representatives of types. This leads to an almost optimal complexity for both irreducibility testing and factorization, including the cost over the residue field, with an improvement by a factor equal to the discriminant valuation of the polynomial.\n\nWhat the paper does well is to take the existing framework and push the complexity bound in a concrete way. The divide-and-conquer approach is a logical development from the generalized polygons, and the complexity claim is stated in terms of a specific quantity, the discriminant valuation, which makes it easy to compare with prior work.\n\nThe soft spots are mostly around verification. The abstract states that proofs exist for the complexity claims and the approximate-root property, but without the full derivations it is difficult to assess the error terms or how edge cases are handled. The condition on the residual characteristic is central to getting the best bound, so that part of the argument will need close attention to see if it holds without additional restrictions. The circularity burden is low, as there is no sign of self-referential definitions.\n\nThis work is for researchers in computer algebra who focus on local rings and polynomial factorization algorithms. A reader who already knows the Montes algorithm or works on OM-factorizations would get direct value from the new strategy and the complexity result.\n\nIt deserves a serious referee because the claims are specific and the contribution is a measurable improvement in an established area.\n\nI would recommend sending this to peer review rather than desk rejecting it.\n\nRegards,","headline":"The paper improves the Nart-Montes algorithm via an extended Hensel lemma on generalized Newton polygons, yielding a divide-and-conquer strategy and a claimed complexity gain by the discriminant valuation when residual characteristic is zero or high.","tokens_in":2195,"tokens_out":463,"would_cite":false,"duration_ms":32515,"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":"Extending Hensel lemma to generalized Newton polygons and using approximate roots as type representatives yields almost optimal complexity for the Montes algorithm when residual characteristic is zero or high enough.","keywords":["Montes algorithm","local polynomial factorization","Hensel lemma","Newton polygons","OM-factorization","discrete valuation ring","approximate roots","complexity"],"falsifier":"Run the new algorithm on a polynomial whose residual characteristic is high enough for the claim to apply yet whose approximate roots fail to represent types correctly, and measure whether the stated complexity improvement by the discriminant valuation still occurs.","tokens_in":2559,"feed_emoji":"","tokens_out":660,"duration_ms":20676,"temperature":0.7,"pith_summary":"The paper establishes an improved version of the Nart-Montes algorithm for factoring polynomials over a complete discrete valuation ring. It first extends the Hensel lemma to the setting of generalized Newton polygons and derives a divide-and-conquer strategy from that extension. When the residual characteristic is zero or sufficiently high, the work proves that approximate roots serve as convenient representatives of the types that appear in the algorithm. This produces nearly optimal complexity bounds for both irreducibility testing and full factorization, including the cost of any auxiliary factorizations over the residue field. For the concrete task of computing an OM-factorization of a polynomial F the running time improves by a factor equal to the valuation of the discriminant of F.","feed_headline":"Montes algorithm reaches near-optimal complexity via approximate roots","feed_subtitle":"When residual characteristic is zero or high, approximate roots cut factorization cost by the discriminant valuation factor.","key_machinery":"Extension of the Hensel lemma to generalised Newton polygons, together with the use of approximate roots as convenient representatives of types.","core_discovery":"By extending Hensel's lemma in the context of generalised Newton polygons we obtain a new divide-and-conquer strategy. If the residual characteristic is zero or high enough, approximate roots are convenient representatives of types. This yields an almost optimal complexity both for irreducibility and factorisation issues, plus the cost of factorisations above the residue field. For instance, to compute an OM-factorisation of F in A[x], the complexity improves by a factor δ, the discriminant valuation of F.","pith_inferences":["The same approximate-root technique may shorten factorization routines in other complete local rings.","Practical running times for high-degree inputs could decrease proportionally to the size of the discriminant.","The divide-and-conquer strategy could be ported to related lifting problems that rely on Newton polygons."],"forward_implications":["Complexity of OM-factorization improves by the factor δ equal to the discriminant valuation.","Almost optimal complexity is achieved for both irreducibility testing and full factorization.","The cost of auxiliary factorizations over the residue field is included in the improved bound.","The divide-and-conquer strategy applies directly inside the Montes algorithm."],"fun_headline_variants":["Extended Hensel lemma refines Montes factorization algorithm","Divide and conquer via generalized Newton polygons in polynomial factoring","Approximate roots achieve almost optimal factorization complexity","Montes algorithm complexity improved by discriminant valuation factor"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"That approximate roots serve as convenient representatives of types when the residual characteristic is zero or sufficiently high.","fun_headline_variants_meta":{"raw":{"variants":["Extended Hensel lemma refines Montes factorization algorithm","Divide and conquer via generalized Newton polygons in polynomial factoring","Approximate roots achieve almost optimal factorization complexity","Montes algorithm complexity improved by discriminant valuation factor"]},"model":"grok-4.3","cost_usd":0.004309,"raw_usage":{"total_tokens":2130,"prompt_tokens":598,"num_sources_used":0,"completion_tokens":57,"cost_in_usd_ticks":43087000,"prompt_tokens_details":{"text_tokens":598,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1475,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":598,"tokens_out":57,"duration_ms":11049,"temperature":1.0,"reasoning_tokens":1475,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-03T01:38:00.732635+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Run the new algorithm on a polynomial whose residual characteristic is high enough for the claim to apply yet whose approximate roots fail to represent types correctly, and measure whether the stated complexity improvement by the discriminant valuation still occurs.","supporting_citations":[],"review_version":1}