{"id":"cf5ccdea-8a8a-42a3-ac86-eaf0ae5338be","arxiv_id":"1908.04012","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For any element a in a commutative ring containing Q, polynomials that are a-strongly residual coordinates form a partial coordinate system.","lead":"The note proves a conjecture about partial coordinate systems: if n-1 polynomials work after dividing by an element a and after localizing at a, then they already form a coordinate system over the original ring. The proof reduces the statement to a recent theorem by Das and Dutta.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.1 rests on an unproved extension of Das–Dutta's Corollary 3.19 from Noetherian domains to arbitrary Noetherian rings containing Q; the reduction itself is sound.","rationale":"The reader's weakest assumption identifies the same point I regard as load-bearing: Remark 1.3 asserts, without proof, that Das–Dutta's domain-based theorem extends to all Noetherian rings containing Q, and Theorem 2.1 relies on exactly that extension at the step where the finitely generated subring S may be non-reduced. I do not see a separate flaw in the reduction: the construction of S by adjoining finitely many coefficients, the use of equations (1) and (2) to verify the residual-coordinate condition at each prime of S, and the final descent from S to R are all internally consistent. The paper is transparent about the gap, but the gap is the entire difference between the old non-zerodivisor theorem and the conjecture being solved. A complete proof or a precise citation of a theorem stated for arbitrary Noetherian rings would remove the obstacle. Since the reader already judged the paper CONDITIONAL for this reason, the appropriate outcome is unchanged.","tokens_in":2431,"tokens_out":14434,"duration_ms":159568,"concrete_test":"Obtain [3] and check whether Theorems 2.4 and 3.13 are formally stated for arbitrary Noetherian rings containing Q, then re-run the proof of Corollary 3.19 with R = Q[ε]/(ε^2), n = 2, and f = X + εY^2. The residual-coordinate condition holds in the single residue field, and the extension is valid only if the proof can still produce g in R[X,Y] with R[X,Y] = R[f,g] (for instance g = Y) without invoking injectivity of R into its total quotient ring or passage to the field of fractions. If the proof uses the domain property to complete f, then Remark 1.3 is false; if the proof succeeds, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central argument of Theorem 2.1 is a reduction: from an a-strongly partial residual coordinate system in R[X], the author constructs a finitely generated Q-subalgebra S of R and shows that f1,...,f_{n-1} satisfy the residual-coordinate condition at every prime of S. The final step then applies Theorem 1.2 to the Noetherian ring S. The reduction itself is coherent: equation (1) covers primes with a in p, equation (2) covers primes with a not in p, and finite generation of S is enough to make S Noetherian.\n\nThe load-bearing point is Theorem 1.2. It is quoted from [3, Corollary 3.19], which is stated only for Noetherian domains containing Q. Remark 1.3 asserts, without proof, that the proof works for every Noetherian ring containing Q because Theorem 3.13 and Theorem 2.4 of [3] hold in that generality. This asserted extension is exactly what is needed to solve the conjecture for arbitrary a: if a is nilpotent, the localization condition in the definition is degenerate and the non-domain nature of R is essential, and S can be non-reduced. If the Das–Dutta equivalence fails for rings with zerodivisors, the inference \"partial residual implies partial coordinate\" over S fails and Theorem 2.1 has no proof. No independent source or machine-checked verification for the extension is supplied, so the correctness risk is concentrated in this one unproved generalization.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper addresses a conjecture of Berson, Bikker, and van den Essen on partial coordinate systems in polynomial rings over rings containing Q. For a non-zerodivisor a, the earlier theorem says that if n-1 polynomials in R[X_1,...,X_n] form partial coordinate systems both modulo a and after localizing at a, then they form a partial coordinate system over R. The present note claims to prove the conjecture for arbitrary a, including zerodivisors and nilpotent elements. The proof is a reduction: from the assumed a-strongly partial residual coordinate system, the author constructs a finitely generated Q-subalgebra S of R and shows that the given polynomials form a partial residual coordinate system over S at every prime; a result of Das-Dutta, quoted as Theorem 1.2, is then invoked to conclude they form a partial coordinate system over S, and hence over R.","tokens_in":2687,"tokens_out":8141,"duration_ms":86981,"significance":"If the proof is correct, the paper resolves [1, Conjecture 4.4] in full generality, confirming a natural extension of the Berson-Bikker-van den Essen theorem. The reduction to a finitely generated Noetherian Q-algebra is elementary and elegant, and the paper makes a clear conceptual point: the residual-variable equivalence, if available for all Noetherian rings containing Q, subsumes the earlier non-zerodivisor theorem. However, the decisive step depends on an extension of [3, Corollary 3.19] from Noetherian domains to arbitrary Noetherian rings, an extension that is asserted in Remark 1.3 but not proved or explicitly referenced. The correctness risk is concentrated in that one unproved generalization.","major_comments":[{"comment":"Theorem 1.2 is quoted from [3, Corollary 3.19], which the author states is proved only for Noetherian domains containing Q. Remark 1.3 asserts, without proof, that the result extends to every Noetherian ring containing Q. This extension is load-bearing: the ring S constructed in the proof of Theorem 2.1 is a finitely generated Q-algebra but need not be a domain or reduced when R has zerodivisors, for instance when a is nilpotent. If the Das-Dutta equivalence fails for rings with zerodivisors, the inference from a partial residual coordinate system to a partial coordinate system over S collapses, and Theorem 2.1 has no proof. The author should supply a complete proof of the asserted extension or cite a published result that establishes it.","section":"Remark 1.3 and proof of Theorem 2.1"}],"minor_comments":[{"comment":"There are typographical errors in 'pr oved' (abstract) and 'Noetherain' (Remark 1.3) that should be corrected.","section":"Abstract and Remark 1.3"},{"comment":"As displayed, equation (2) has positive powers a^{k_i} with k_i >= 0 on the right, but an identity in the localization R_a generally requires denominators a^{-k_i} or a cleared-denominator form a^N X_i = ... in A. This appears to be a typographical issue, but the notation should be made precise, and the subsequent construction of S should be stated so that S does not need to contain inverses of a.","section":"Equation (2), Section 2"},{"comment":"After applying Theorem 1.2 to conclude that f_1,...,f_{n-1} form a partial coordinate system in B, the step 'and hence in A' is asserted without justification. It should be explained that A = R ⊗_S B, so the partial coordinate structure over S extends to R by base change; the argument is straightforward but should be included.","section":"Proof of Theorem 2.1, last lines"}],"recommendation":"major_revision","confidential_remarks":"The paper is concise and the main idea is attractive, but the pivotal generalization of Das-Dutta's Corollary 3.19 is asserted rather than proved. I recommend asking the author to supply a complete proof of that extension, or a precise published reference, before acceptance. The rest of the proof is routine and sound once the extension is available."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short take: this note resolves Berson-Bikker-van den Essen's Conjecture 4.4, and the proof is a clean reduction rather than new machinery. The author shows that the a-strongly partial residual condition can be witnessed by finitely many equations, so it descends to a finitely generated Q-subalgebra S of R; applying the residual-coordinate criterion over the Noetherian ring S and extending scalars back gives the result. I checked the two cases a∈p and a∉p in the proof of Theorem 2.1; both work directly from equations (1) and (2), and the surjectivity argument is enough because the target is a polynomial ring over a field.\n\nCredit where due: the observation is honest, the reduction is neat, and the paper is clearly written. The conjecture was genuinely open, and the note closes it if the quoted theorem is valid at the advertised level of generality.\n\nThe soft spot is Remark 1.3. Theorem 1.2 is quoted from Das-Dutta's Corollary 3.19, which is stated for Noetherian domains. The author asserts, without proof, that the proof works over every Noetherian ring containing Q. That assertion is load-bearing: if the equivalence 'partial residual ⇒ partial coordinate' fails for rings with zerodivisors, the proof over S collapses. I don't know a counterexample, and the n=2 case is known to hold in this generality (Bhatwadekar-Dutta), so the extension is plausible. But plausible is not proven, and the note does not even state which results in [3] are being extended. A referee can check this, but the manuscript as written has a gap. Also, the abstract says 'arbitrary element of A' where it should say R, but that is a typo.\n\nWho this is for: anyone working on coordinates, residual variables, or local-global questions for polynomial automorphisms. It is a small but real contribution. I would send it to review; the referee's job is to verify the Das-Dutta generalization, and if it is true, accept.","headline":"Clean reduction proves the BBvdE conjecture for arbitrary a; the one load-bearing gap is the unproved extension of Das-Dutta's criterion to non-domain Noetherian rings.","tokens_in":3165,"tokens_out":16090,"would_cite":true,"duration_ms":173901,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13B25","14R25"],"pacs":[],"model":"deepseek-v4-flash","headline":"For rings containing the rationals, partial residual coordinates force genuine partial coordinates.","keywords":["polynomial algebra","partial coordinate system","residual coordinate","zerodivisor","affine fibration","Noetherian ring","residual variables"],"falsifier":"Find a ring $R$ containing $\\mathbb{Q}$, a zero divisor $a\\in R$, and $f_1,\\ldots,f_{n-1}\\in R[X_1,\\ldots,X_n]$ whose images form a partial coordinate system in both $R/aR[X_1,\\ldots,X_n]$ and $R_a[X_1,\\ldots,X_n]$ yet which do not form a partial coordinate system over $R$; even one such family would refute Theorem 2.1. Alternatively, a Noetherian ring $S$ with zero divisors where the residual-coordinate equivalence of [3] fails would locate the breakdown in the proof's key step.","tokens_in":2232,"feed_emoji":"📐","tokens_out":9310,"duration_ms":87053,"temperature":0.7,"pith_summary":"The paper establishes a local-to-global principle for partial coordinate systems in polynomial rings. If $n-1$ polynomials in $R[X_1,\\ldots,X_n]$ become a partial coordinate system both after setting $a=0$ and after inverting $a$, for an arbitrary element $a$ of a ring $R$ containing the rationals, then they already form a partial coordinate system over $R$. This removes the non-zerodivisor hypothesis from an earlier theorem and settles a conjecture posed in the cited literature. The interest is that coordinate-like behaviour over the special fibre and over the localization at $a$ is enough to force genuine coordinates, even when $a$ has zero divisors.","feed_headline":"Partial coordinates survive even when the parameter is a zerodivisor","feed_subtitle":"Settles the coordinate-system conjecture for rings containing the rationals, including zerodivisor cases.","key_machinery":"The carrying device is the residual-coordinate equivalence: for a Noetherian ring $S$ containing $\\mathbb{Q}$, $n-1$ polynomials form a partial coordinate system if and only if they form a partial residual coordinate system, meaning that over every residue field $k(\\mathfrak p)$ of $S$ the fibre is a one-variable polynomial algebra over the images of the $f_i$. The proof builds a finite-type $\\mathbb{Q}$-subalgebra $S\\subseteq R$ generated by $a$ and by every coefficient appearing in the two coordinate representations supplied by the hypotheses; $S$ is Noetherian. For each prime $\\mathfrak p$ of $S$, the equation $X_i=G_i+aH_i$ handles the case $a\\in\\mathfrak p$, while the coordinate expression obtained after localizing at $a$ handles $a\\notin\\mathfrak p$; together they show the $f_i$ form a partial residual coordinate system over $S$. The equivalence then upgrades this to a genuine partial coordinate system over $S$, and hence over $A$.","core_discovery":"Theorem 2.1 asserts the following. Let $R$ be a ring containing $\\mathbb{Q}$, let $a\\in R$ be arbitrary, and set $A=R[X_1,\\ldots,X_n]$. If $n-1$ polynomials $f_1,\\ldots,f_{n-1}\\in A$ form an $a$-strongly partial residual coordinate system of colength 1 in $A$ (meaning their images form a partial coordinate system both in $A/aA$ and in $A_a$), then $f_1,\\ldots,f_{n-1}$ form a partial coordinate system in $A$, i.e. $A=R[f_1,\\ldots,f_{n-1}][1]$. This is precisely the zerodivisor case of the conjecture left open in [1], and the note proves it by reducing to a Noetherian subring and invoking the residual-variables equivalence of [3] in a form the author extends beyond domains.","pith_inferences":["The same coefficient-subalgebra reduction would plausibly push the result to colength greater than 1 if the corresponding residual-coordinate equivalence is available there.","If the extension in Remark 1.3 is correct, the argument yields a descent-style statement: coordinate data over residue fields of a finitely generated subring forces coordinates over the original ring, which may apply beyond the one-element localization setup.","A natural test case is to take $R=k[x,y]$, $a=xy$, and try to construct $f_1,\\ldots,f_{n-1}$ satisfying both fibre conditions but not forming a partial coordinate system; success or failure would map the exact boundary of the method."],"forward_implications":["The zerodivisor case of the conjecture from [1] is settled for every ring containing $\\mathbb{Q}$.","The earlier non-zerodivisor theorem becomes a special case, now covered by a uniform proof.","For colength 1, checking partial coordinates over all residue fields of a Noetherian ring is equivalent to having an actual partial coordinate system.","The proof gives a finite-generation principle: verifying the two fibre conditions over an arbitrary ring can be reduced to a Noetherian $\\mathbb{Q}$-subalgebra."],"supporting_citations":[{"why":"Proved the result for non-zerodivisors and posed the conjecture that the note settles for arbitrary $a$.","marker":"[1]"},{"why":"Introduced the residual-variables theory and supplied the $n=2$ case of the equivalence used in the proof.","marker":"[2]"},{"why":"Stated the equivalence between partial residual and partial coordinate systems for Noetherian domains; Remark 1.3 extends it to all Noetherian rings.","marker":"[3]"}],"fun_headline_variants":["Zerodivisor case of coordinate conjecture now proved","Partial coordinate systems survive arbitrary parameters","Residual variables settle coordinate conjecture for zerodivisors","Coordinate systems: from non-zerodivisors to any element","Residual variables generalize coordinate systems to every ring element"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central argument depends on the unsupported assertion in Remark 1.3 that the equivalence stated in [3] for Noetherian domains holds for arbitrary Noetherian rings containing $\\mathbb{Q}$; if that extension is false, the proof of Theorem 2.1 collapses.","fun_headline_variants_meta":{"raw":{"variants":["Zerodivisor case of coordinate conjecture now proved","Partial coordinate systems survive arbitrary parameters","Residual variables settle coordinate conjecture for zerodivisors","Coordinate systems: from non-zerodivisors to any element","Residual variables generalize coordinate systems to every ring element"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00078,"raw_usage":{"total_tokens":3407,"prompt_tokens":869,"completion_tokens":2538,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":485,"completion_tokens_details":{"reasoning_tokens":2461}},"tokens_in":485,"tokens_out":2538,"duration_ms":21121,"temperature":1.0,"reasoning_tokens":2461,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:54:27.913604+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a ring $R$ containing $\\mathbb{Q}$, a zero divisor $a\\in R$, and $f_1,\\ldots,f_{n-1}\\in R[X_1,\\ldots,X_n]$ whose images form a partial coordinate system in both $R/aR[X_1,\\ldots,X_n]$ and $R_a[X_1,\\ldots,X_n]$ yet which do not form a partial coordinate system over $R$; even one such family would refute Theorem 2.1. Alternatively, a Noetherian ring $S$ with zero divisors where the residual-coordinate equivalence of [3] fails would locate the breakdown in the proof's key step.","supporting_citations":[{"cited_title":"Berson, J.W","cited_arxiv_id":null,"evidence_quote":"Proved the result for non-zerodivisors and posed the conjecture that the note settles for arbitrary $a$."},{"cited_title":"Bhatwadekar and A.K","cited_arxiv_id":null,"evidence_quote":"Introduced the residual-variables theory and supplied the $n=2$ case of the equivalence used in the proof."},{"cited_title":"Das and A.K","cited_arxiv_id":null,"evidence_quote":"Stated the equivalence between partial residual and partial coordinate systems for Noetherian domains; Remark 1.3 extends it to all Noetherian rings."}],"review_version":1}