{"id":"10a2c7e4-d8e0-4a32-ae14-7104e6717870","arxiv_id":"2607.03475","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"R[[t]] is a UFD whenever R is a polynomial ring in arbitrarily many variables over a regular UFD, via a finite-height irreducibility theorem for Krull domains.","lead":"The paper proves that if A is a regular UFD then the power series ring over the polynomial ring in any number of variables over A is again a UFD, settling Landweber's 1974 question. The key new tool is that every irreducible power series over a Krull domain remains irreducible modulo some finite power of t.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The central claim (Theorem A) rests on three pillars: (1) finite irreducibility for Krull domains (Thm B), (2) the finite-stage retraction criterion (Thm C/7.2), and (3) the classical Samuel–Buchsbaum theorem for regular UFDs in finitely many variables. Pillar (2) is a short, transparent argument once (1) is available; pillar (3) is classical. The only non-routine work is therefore the proof of (1), which reduces via localization to the DVR case and then uses the quantitative C-primality machinery of §§4–5. That machinery is lengthy but purely combinatorial and valuation-theoretic; every inductive step is written out and the base case (v(A_0)=1) is elementary (Cor. 4.6). Because the constants are finite and the König extraction only needs finiteness (not sharpness), even a looser bound would suffice. No counter-example to C-primality over DVRs is known or suggested by the text, and the local-global gluing (Lemma 6.2) uses only the definition of a Krull domain. Consequently the reader’s high-confidence ACCEPT stands; the identified “weakest assumption” is real technical work but not a soft spot that threatens correctness.","tokens_in":20980,"tokens_out":701,"duration_ms":5467,"concrete_test":"Independently recompute the first few control constants: verify N_1=1, N_2=a^{(2)}_2(1) and N_3=a^{(3)}_3(N_2) by hand from the recurrence a^{(m)}_0(L)=(2^m-1)L+2^m, a^{(m)}_k=(2^m-1)(a_{k-1}+1)L+2^m; then check that any A with v(A_0)=2 that is irreducible mod t^D is indeed N_2 D-prime by direct application of Lemma 4.5. If the numerical bounds hold and the primality claim follows, the load-bearing step is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest_assumption correctly flags the most technical step (the recursive C-primality bounds N_m over DVRs in §§4–5), but that step is self-contained and does not appear to fail. The induction constructing a^{(m)}_k(L) and the control of primality to degree m+1 (Prop. 4.12) rests only on the valuation inequalities of Lemma 4.5 and the length bound k_d(W)<v(W_0) of Lemma 4.10; both are elementary and hold for any DVR. Once those bounds exist, Lemma 5.1 produces only finitely many candidate ideals, König’s lemma (Lemma 3.1) extracts a global factorization, and the local-to-global passage via height-1 localizations (Prop. 6.3 + Lemma 6.2) is standard for Krull domains. The finite-stage retraction criterion (Thm 7.2) then reduces Theorem A to the classical Samuel–Buchsbaum theorem without further gaps. No hidden assumption or circularity is visible.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper solves Landweber’s 1974 question by proving that if A is a regular UFD and R = A[x_i | i ∈ I] for an arbitrary index set I, then R[[t]] is a UFD (Theorem A / 7.3). The argument rests on two general results for Krull domains: (B) every irreducible f ∈ R[[t]] is irreducible modulo some finite power of t (Theorem 6.4), and (C) a finite-stage retraction criterion that reduces the UFD property of R[[t]] to the classical Samuel–Buchsbaum theorem for regular UFDs (Theorem 7.2). The bulk of the work is the proof of (B): first a quantitative C-primality theory over DVRs (§§4–5) that produces only finitely many candidate ideals of the form (A, t^ℓ), then a König-lemma extraction of a global factorization (Lemma 3.1), and finally a local-to-global comparison of those ideals via height-1 localizations (Proposition 6.3 + Lemma 6.2).","tokens_in":21220,"tokens_out":727,"duration_ms":5308,"significance":"Landweber’s problem has been open for fifty years and is repeatedly cited as a basic open question about unique factorization in non-Noetherian power series rings. The paper settles it completely, and the intermediate finite-height theorem (Theorem B) is of independent interest for arbitrary Krull domains. The reduction via retractions cleanly isolates the new work from the classical Samuel–Buchsbaum theorem, and the C-primality machinery supplies an explicit, elementary substitute for Artin approximation in the DVR case. These are substantial, self-contained advances in commutative algebra.","major_comments":[],"minor_comments":[{"comment":"The recursive definition of the control constants a^{(m)}_k(L) and N_m (after Lemma 4.8 and before Proposition 4.12) is correct but dense; a short remark that the only inputs are the valuation inequalities of Lemma 4.5 and the length bound of Lemma 4.10 would help the reader track the induction.","section":null},{"comment":"In the remark at the end of §4 the counter-example a = p^{2} - Y t^{2} over Q[X,Y] is asserted without verification. A one-line check that it is irreducible mod t^{3} yet fails C-primality for every C would make the sharpness claim fully self-contained.","section":null},{"comment":"Lemma 6.1 (finitely many divisors up to associates in a Krull domain) is standard; a parenthetical reference to Bourbaki VII.1 would be useful for non-specialists.","section":null},{"comment":"Typographical: “K¨onig’s lemma” appears with inconsistent diacritics; standardize to “König’s lemma” throughout.","section":null},{"comment":"The final open question (if R[[x]] is a UFD, is R[[x]][[t]] a UFD?) is well-posed; a brief pointer to Bayart’s earlier formulation would complete the historical picture.","section":null}],"recommendation":"accept","confidential_remarks":"The technical core (C-primality bounds over DVRs) is elementary valuation theory and appears free of gaps; the reader’s and skeptic’s assessments agree. The paper is a clean, high-quality solution of a classical problem and is suitable for a top algebra journal without further delay."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper settles Landweber’s problem: if A is a regular UFD and R is the polynomial ring in any number of variables over A, then R[[t]] is a UFD. The key new tool is Theorem B: every irreducible power series over an arbitrary Krull domain is already irreducible modulo some finite power of t. That statement was known only in a narrow special case (Bayart); the general version is the real advance.\n\nThey prove it first over DVRs by introducing C-prime elements and building explicit recursive bounds that control how far primality can “leak.” Those bounds feed a König-lemma extraction of a global factorization from compatible partial ones (Lemma 3.1). The local-to-global step for Krull domains (Proposition 6.3 + Lemma 6.2) is standard valuation comparison and works cleanly. Once finite height is available, the retraction criterion (Theorem C) reduces the infinite-variable case to the classical Samuel–Buchsbaum theorem by killing unused variables. The logical skeleton is complete and self-contained.\n\nThe most technical stretch is the inductive construction of the control constants N_m and a_k^{(m)}(L) in §§4–5. It rests only on elementary valuation inequalities and the length bound k_d(W) < v(W_0); both hold for any DVR, so the bounds exist and the finiteness argument goes through. A few intermediate verifications are a bit terse, but nothing load-bearing is missing. The counter-example showing that finite height fails for non-Krull domains is useful and correctly placed.\n\nThis is pure algebra aimed at people who care about factorization in power series rings. The citations are honest and the classical results are used exactly where needed. I would send it to a serious referee without hesitation; the result is correctly derived and advances a documented open problem. Worth reading if you work in this area.","headline":"Solid solution to Landweber’s 1974 question: finite-height irreducibility for Krull domains plus a clean retraction criterion yields R[[t]] UFD for infinite-variable polynomials over regular UFDs.","tokens_in":21833,"tokens_out":486,"would_cite":true,"duration_ms":4721,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13F25","13F15","13A05"],"pacs":[],"model":"grok-4.5","headline":"Polynomial rings in any number of variables over a regular UFD remain unique-factorization domains after adjoining formal power series in one variable.","keywords":["unique factorization domain","formal power series","Krull domain","finite irreducibility height","regular UFD","Landweber problem","C-prime elements"],"falsifier":"Exhibit a single irreducible power series over a discrete valuation ring that factors non-trivially modulo every power of t, or exhibit a regular UFD A and an infinite set of variables such that some irreducible of finite height in A[x_i][[t]] fails to be prime.","tokens_in":21869,"feed_emoji":"∞","tokens_out":627,"duration_ms":4796,"temperature":0.7,"pith_summary":"For fifty years it has been open whether the formal power series ring over a polynomial ring in infinitely many variables is still a unique-factorization domain. The paper answers yes: if A is a regular UFD and R is the polynomial ring over A in any set of variables, then R[[t]] is a UFD. The argument rests on a general finite-height theorem: every irreducible power series over a Krull domain is already irreducible modulo some finite power of t. Once that is known, a finite-stage retraction criterion reduces the infinite-variable case to the classical Samuel–Buchsbaum theorem for regular UFDs in finitely many variables. The result settles Landweber’s 1974 question and supplies a uniform reason why unique factorization survives the passage to power series for this large class of rings.","feed_headline":"Infinite-variable power series rings stay unique-factorization domains","feed_subtitle":"A 1974 question of Landweber is settled: adjoining t keeps unique factorization for any number of variables","key_machinery":"The finite irreducibility theorem (Theorem B): an irreducible f in R[[t]] for R Krull is already irreducible modulo t^n for some n. It is proved by establishing quantitative C-primality over DVRs, then extracting a global factorization via a König-lemma argument on finitely many candidate ideals.","core_discovery":"If R is any Krull domain then every irreducible element of R[[t]] is irreducible modulo some finite power of t. Combined with a retraction criterion that reduces finite sets of coefficients to a UFD subring, this implies that the power series ring over a polynomial ring in arbitrarily many variables over a regular UFD is itself a UFD.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Landweber 1974 settled: infinite-var polynomial power series are UFDs","R[[t]] is UFD when R is polynomials in countably many variables","Irreducibles in R[[t]] stay irreducible mod finite powers of t","Krull domains: power series irreducibles reduce mod some t^n","Infinite-variable UFDs keep unique factorization after adjoining t"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The recursive bounds that guarantee only finitely many candidate factor ideals appear at each valuation level over a discrete valuation ring must hold; if those bounds fail, the extraction of a global factorization collapses.","fun_headline_variants_meta":{"raw":{"variants":["Landweber 1974 settled: infinite-var polynomial power series are UFDs","R[[t]] is UFD when R is polynomials in countably many variables","Irreducibles in R[[t]] stay irreducible mod finite powers of t","Krull domains: power series irreducibles reduce mod some t^n","Infinite-variable UFDs keep unique factorization after adjoining t"]},"model":"grok-4.5","effort":"low","cost_usd":0.006352,"raw_usage":{"total_tokens":1550,"prompt_tokens":641,"num_sources_used":0,"completion_tokens":104,"cost_in_usd_ticks":63520000,"prompt_tokens_details":{"text_tokens":641,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":805,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":641,"tokens_out":104,"duration_ms":6188,"temperature":1.0,"reasoning_tokens":805,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-12T02:13:19.474456+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a single irreducible power series over a discrete valuation ring that factors non-trivially modulo every power of t, or exhibit a regular UFD A and an infinite set of variables such that some irreducible of finite height in A[x_i][[t]] fails to be prime.","supporting_citations":[],"review_version":1}