{"id":"d9494231-3094-4836-a933-36ea2718bcf4","arxiv_id":"1908.03549","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Every residual coordinate over a one-dimensional affine domain over an algebraically closed field is one-stable, and the earlier stable-coordinate bounds are extended to arbitrary characteristic and sharpened.","lead":"Polynomials that pass a weak coordinate test can be converted into true coordinates after adding variables, and this paper proves that conversion works over one-dimensional domains in every field characteristic. The result also removes a rational-number assumption from an earlier theorem and yields a sharper stability bound for affine algebras over algebraically closed fields.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified.","rationale":"The reader identified Bhatwadekar–Dutta's Theorem 2.7 as the load-bearing black box. I agree that this is the key external dependency, but I do not see a specific defect in its use. I checked the places where a characteristic-zero assumption could be smuggled in: Lemma 3.3 constructs the exponential map purely by universal algebra and works over arbitrary characteristic; Theorem 2.8 is characteristic-free; the induction in Theorem 3.6 only needs the Artinian case and stable-coordinate lifting; the induction in Theorem 3.7 uses characteristic zero only where explicitly stated. The most delicate step is the local assertion in Theorem 3.2 that a fiber coordinate lifts to a global coordinate. This initially looks like the standard non-lifting phenomenon for polynomial automorphisms over a local ring, but the proof avoids it by choosing g and the complementary variables in k[Z] ⊂ R[Z], not as lifts of P itself. Consequently R[Z] = R[g,g_2,...,g_n] and the residual-coordinate verification over A is valid. Lemma 3.1's proof invokes Bass cancellation; for the n≥3 cases actually used in Theorems 3.4 and 3.7 the rank condition is met, and the n=2 case can be handled by a determinant argument, so this is not a substantive obstacle. No internal inconsistency or unsupported characteristic-specific step was found.","tokens_in":8455,"tokens_out":51084,"duration_ms":526870,"concrete_test":"Verify the external dependency: re-derive or consult the original Bhatwadekar–Dutta [4, Theorem 3.2] statement of Theorem 2.7, and check that in the positive-characteristic step of Theorem 3.2 the base A=R[g_2,...,g_n] satisfies the seminormality hypothesis. This reduces to proving directly from Definition 2.1 that R[X] is seminormal whenever R is a reduced seminormal ring, using coefficient comparison for the b^3=c^2 criterion. If this check failed, Theorems 3.2 and 3.4 would need revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After reading the proofs closely, I could not identify a load-bearing gap. The critical local step in Theorem 3.2 is the reduction of the fiber coordinate g = η(P) to a global coordinate g over R; this is legitimate because g is chosen in k[Z] ⊂ R[Z], so an automorphism of k[Z] lifts with coefficients in k, and R[Z] = R[g,g_2,...,g_n] follows immediately. The subsequent residual-coordinate verification over A and the application of Bhatwadekar–Dutta's Theorem 2.7 are the only external black box; in characteristic zero A contains Q, and in positive characteristic the hypothesis is R_red seminormal, with A = R[n-1], so the standard fact that seminormality is preserved by polynomial extensions is the only auxiliary input. Lemma 3.1's cancellation step is justified for the n≥3 cases used in the main theorems, and for n=2 a determinant argument supplies the missing step. Theorems 3.6 and 3.7 are clean inductions with no hidden characteristic assumption.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves several results on residual coordinates and stable coordinates in polynomial rings. The main theorems are: (1) Theorem 3.2, a generalization of a theorem of Maubach, stating that if R is a one-dimensional affine algebra over an algebraically closed field k, with R_red seminormal or char k = 0, and if the image of P in R/aR[Z] is a coordinate, then F = aW + P is a coordinate in R[Z, W]; (2) Theorem 3.4, extending the Kahoui–Ouali theorem to arbitrary characteristic, namely that every residual coordinate in R[Z_1,...,Z_n], n ≥ 3, is a 1-stable coordinate under the same hypotheses on R; (3) Theorem 3.6, dropping the 'contains Q' hypothesis from Kahoui–Ouali's bound, so that over any Noetherian d-dimensional ring every residual coordinate is a (2^d − 1)n-stable coordinate; and (4) Theorem 3.7, sharpening the bound to r = 2^{d−1}(n+1) − n when R is affine over an algebraically closed field of characteristic zero. The proofs use a local-global lemma (Lemma 3.1), a theorem of Bhatwadekar–Dutta on residual coordinates in two variables, exponential maps, Bass cancellation, and Quillen local-global.","tokens_in":8576,"tokens_out":26441,"duration_ms":242943,"significance":"If correct, the results constitute a genuine advance over the recent work of Kahoui and Ouali: the characteristic-zero assumption is removed from the one-dimensional residual-coordinate theorem, the 'contains Q' hypothesis is removed from the stable-coordinate bound, and the bound is improved for affine algebras over algebraically closed fields. The paper also gives a useful extension of Maubach's coordinate criterion. The proofs are generally careful and use standard tools in an elegant way; the use of exponential maps to convert local coordinate data into global stability statements is particularly nice. The theorems are clearly stated and have obvious applications to the study of polynomial automorphisms and affine fibrations. The manuscript is a solid contribution to commutative algebra.","major_comments":[],"minor_comments":[{"comment":"The application of Bass cancellation (Theorem 2.14) requires rank(Q) ≥ 2 for the one-dimensional base ring, so the argument as written covers only n ≥ 3. For n = 2, the conclusion Q ≅ R follows by applying ∧^2 to the isomorphism Q ⊕ R ≅ R^2; please add this argument or restrict the lemma's statement to n ≥ 3.","section":"Lemma 3.1, proof"},{"comment":"The statement that the d = 1 case 'follows from Theorem 3.2' is not directly correct, since Theorem 3.2 is a criterion for aW + P to be a coordinate and does not by itself assert that residual coordinates are 1-stable. The correct reference is Theorem 3.4 (with Theorem 2.7 covering n = 1, 2), or the proof of Theorem 3.4 should be reproduced using Theorem 3.2 explicitly.","section":"Theorem 3.7, proof, base case d = 1"},{"comment":"When Theorem 3.2 is applied to aW + P, the hypothesis that the image of P in R/aR[Z] is a coordinate is not explicitly verified. It follows because P is a residual coordinate, a is a non-zerodivisor in a one-dimensional ring R, and therefore R/aR is Artinian, so Proposition 2.6 applies. Please include this justification.","section":"Theorem 3.4, proof"},{"comment":"If the element a obtained from Lemma 3.3 is a unit, then R/aR is the zero ring and is not a (d−1)-dimensional ring in the usual convention, so the induction hypothesis does not apply as written. In this case aW + P is immediately a coordinate and P is 1-stable, so the desired bound follows trivially; please add this observation.","section":"Theorems 3.6 and 3.7, induction step"},{"comment":"The abstract states the main theorem for a one-dimensional seminormal domain, while Theorem 3.4 assumes only that R_red is seminormal and allows non-reduced R. The abstract should be made consistent with the theorem's hypotheses.","section":"Abstract and Introduction"},{"comment":"The invocation of Quillen's local-global theorem (Theorem 2.15) requires that D = R[F] be an R-algebra isomorphic to R[1]. This is not obvious for an arbitrary element F, but it follows from the hypothesis that each localization A_m is a polynomial ring over R_m[F]; a brief justification would improve the clarity of the proof.","section":"Lemma 3.1, proof"}],"recommendation":"minor_revision","confidential_remarks":"This is a solid paper in classical commutative algebra. The main theorems appear correct, with only local presentation issues in the proofs of Lemma 3.1, Theorem 3.4, Theorem 3.6, and Theorem 3.7. The correctable items do not affect the core claims, and the paper should be suitable for publication after minor revision. The authors should pay particular attention to the citation in the base case of Theorem 3.7 and to the unit-a edge case in the inductive proofs of Theorems 3.6 and 3.7."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe short version: this is a genuine, modest advance in the residual-coordinate literature. The authors prove that over an algebraically closed field of arbitrary characteristic, the characteristic-zero one-stable theorem of Kahoui and Ouali holds for one-dimensional affine base rings whose reduction is seminormal; they also remove the Q hypothesis from the d-dimensional stable-coordinate theorem and improve the bound from (2^d - 1)n to 2^{d-1}(n+1) - n for affine algebras over algebraically closed fields in characteristic zero. The machinery is all standard — exponential maps, Bass cancellation, Quillen local-global, Bhatwadekar-Dutta's residual-coordinate theorem — but the assembly is correct.\n\nWhat is actually new: Theorem 3.4 is the first version of the one-stable theorem covering positive characteristic, and the seminormal base condition is the natural one given the Bhatwadekar-Dutta black box. Theorem 3.6 is a clean removal of the Q assumption. Theorem 3.7's bound is a real improvement, and the induction is transparent.\n\nThe soft spots are minor. In the d = 1 case of Theorem 3.7, the proof cites Theorem 3.2 when it means Theorem 3.4; harmless, but worth fixing. In Lemma 3.1, the n = 2 case is not covered by Bass cancellation as stated, since the projective module has rank 1 and the theorem requires rank at least d + 1 = 2. A determinant argument closes it, but the paper should say so. I also note the load-bearing external result is Bhatwadekar-Dutta's Theorem 2.7, which itself requires Q or seminormal reduction; the paper's generalizations are exactly as good as that theorem in positive characteristic. That is not a flaw, but it defines the boundary of the contribution.\n\nI checked the stress-test note: it holds up. I could not find a load-bearing gap, and the citation and edge-case issues are the only blemishes. The paper deserves a serious referee. It is written for specialists in polynomial automorphisms and affine algebras; the results are the sort that will be cited and used as tools. I would accept a refereeing invitation.","headline":"Extends Kahoui-Ouali's residual-coordinate theorems to arbitrary characteristic, drops the Q-hypothesis, and sharpens the stability bound; solid, minor fixable slips only.","tokens_in":9128,"tokens_out":5270,"would_cite":true,"duration_ms":53116,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13B25","14R25","14R10","13A50"],"pacs":[],"model":"deepseek-v4-flash","headline":"Residual coordinates over one-dimensional seminormal bases are one-stable, in all characteristics, and Noetherian stable-coordinate bounds hold without the rationals.","keywords":["residual coordinate","stable coordinate","polynomial algebra","exponential map","seminormal ring","coordinate","Noetherian ring","affine algebra"],"falsifier":"Take $k$ an algebraic closure of $\\mathbb{F}_p$, $R = k[t]$, $n = 3$, and search for a residual coordinate $P$ in $R[Z_1,Z_2,Z_3]$ that is not a coordinate in $R[Z_1,Z_2,Z_3,W]$; if no automorphism of the larger ring sends $P$ to $Z_1$, then $P$ is not 1-stable and Theorem 3.4 would be refuted. Similarly, a residual coordinate over a Noetherian $d$-dimensional ring that failed to be $(2^d - 1)n$-stable, or required more than $2^{d-1}(n + 1) - n$ variables in the characteristic-zero affine case, would disprove Theorems 3.6 and 3.7.","tokens_in":8203,"feed_emoji":"➕","tokens_out":10191,"duration_ms":98251,"temperature":0.7,"pith_summary":"Residual coordinates are polynomials that become variables after passing to each residue field of the base ring, so they look like coordinates one prime at a time. The paper establishes that under mild base hypotheses this local-looking condition forces a global conclusion: over a one-dimensional affine algebra over an algebraically closed field, every residual coordinate in three or more variables is a one-stable coordinate, meaning it becomes a real coordinate after adjoining just one new variable, and this holds in every characteristic when the reduced base is seminormal. It also removes the \"contains the rationals\" assumption from a general bound for Noetherian bases, proving that over any $d$-dimensional Noetherian ring every residual coordinate is $(2^d - 1)n$-stable. When the base is affine over an algebraically closed field of characteristic zero, the bound improves to $2^{d-1}(n + 1) - n$. These results matter because stable-coordinate guarantees turn fiberwise or residual checks into explicit automorphism completions of polynomial rings.","feed_headline":"One extra variable turns residual coordinates into true coordinates","feed_subtitle":"Extension to positive characteristic and sharper bounds for affine algebras in characteristic zero.","key_machinery":"Two mechanisms carry the argument. Lemma 3.3 shows that once a residual coordinate $P$ becomes a coordinate after inverting non-zerodivisors (which it does when the total quotient ring is Artinian), there is an exponential map --- a one-parameter family of ring automorphisms $\\delta_W: A \\to A[W]$, the algebraic analogue of an additive-group action --- sending $P$ to $aW + P$ for a non-zerodivisor $a$. Theorem 3.2, the generalized linear-coordinate criterion, proves that such $aW + P$ is a genuine coordinate in $R[Z_1,\\ldots,Z_n,W]$ under the one-dimensional seminormal or characteristic-zero hypotheses. Proposition 2.13 turns the exponential map into an automorphism of $A[W]$ that sends $P$ to $aW + P$, so $P$ itself is a coordinate after adjoining $W$; that is exactly 1-stability. For the higher-dimensional bounds, Lemma 3.1 upgrades local coordinate statements to global ones through symmetric algebras and projective-module cancellation, and Theorem 2.8 lifts $m$-stability modulo $a$ to a larger stability index over $R$; induction on dimension then yields the exponential bounds.","core_discovery":"The paper's central claims are three stable-coordinate theorems. First (Theorem 3.4): if $k$ is algebraically closed and $R$ is a one-dimensional affine $k$-algebra, and either $\\operatorname{char} k = 0$ or $R_{\\mathrm{red}}$ is seminormal, then for $n \\geq 3$ every residual coordinate in $R[Z_1,\\ldots,Z_n]$ is a 1-stable coordinate. Second (Theorem 3.6): for any Noetherian $d$-dimensional ring $R$, every residual coordinate in $R[Z_1,\\ldots,Z_n]$ is a $(2^d - 1)n$-stable coordinate, with no assumption that $R$ contain the rationals. Third (Theorem 3.7): when $R$ is affine over an algebraically closed field of characteristic zero, the bound improves to $2^{d-1}(n + 1) - n$. The engine behind the first result is a generalized linear-coordinate criterion (Theorem 3.2): whenever the image of $P$ becomes a coordinate after dividing out by a non-zerodivisor $a$, and $R$ is a one-dimensional affine algebra over an algebraically closed field with $R_{\\mathrm{red}}$ seminormal or characteristic zero, the polynomial $aW + P$ is an honest coordinate in $R[Z_1,\\ldots,Z_n,W]$. An exponential-map construction then moves the coordinate property from $aW + P$ back to $P$ after adjoining $W$.","pith_inferences":["The strategy suggests that the open higher-dimensional version of Question 3.8 has its bottleneck in a two-variable residual-coordinate-to-coordinate theorem, not in the exponential-map construction, which works in full generality once the polynomial is a coordinate after localization.","A natural test is whether $2^{d-1}(n + 1) - n$ is optimal: explicit residual coordinates over $k[t_1,\\ldots,t_d]$ with linear forms could reveal whether lower stability indices are attainable, a question the paper does not address.","The $n \\geq 3$ threshold in Theorem 3.4 is not an artifact of the method, because in two variables residual coordinates are already genuine coordinates under the same hypotheses; this suggests that the one-stable phenomenon genuinely starts in three variables."],"forward_implications":["Over a one-dimensional affine seminormal algebra over an algebraically closed field of any characteristic, a residual coordinate in at least three variables becomes a coordinate after adjoining one new variable.","The stable-coordinate bound for Noetherian bases holds in arbitrary characteristic: every residual coordinate over a $d$-dimensional Noetherian ring is $(2^d - 1)n$-stable.","In characteristic zero, the bound for affine algebras over an algebraically closed field improves from $(2^d - 1)n$ to $2^{d-1}(n + 1) - n$; for $d = 2$ and $n = 3$ this is 5 instead of 9.","Linear planes of the form $aW + P(Y,Z)$ over Dedekind domains containing the rationals are coordinates when the reduction of $P$ modulo $a$ is a coordinate (Proposition 3.9).","Geometrically, a polynomial that restricts to a coordinate on every fiber of the base is stably a coordinate: the corresponding polynomial fibration becomes trivial after adding one variable."],"supporting_citations":[{"why":"Supplies the two-variable residual-coordinate theorem used in the final step of Theorem 3.2 and in Theorem 3.4.","marker":"[4]"},{"why":"Establishes the characteristic-zero one-stable result and the Artinian residual-coordinate lemma that the paper extends and removes hypotheses from.","marker":"[9]"},{"why":"Gives the linear-polynomial coordinate criterion that Theorem 3.2 generalizes to one-dimensional seminormal bases.","marker":"[10]"},{"why":"Provides the stability-lifting theorem used in the induction steps of Theorems 3.6 and 3.7.","marker":"[3]"},{"why":"States the locally-polynomial-algebra-to-symmetric-algebra theorem used in Lemma 3.1 to pass from local coordinates to a global coordinate.","marker":"[2]"},{"why":"Supplies the local-global principle for projective modules used in Lemma 3.1.","marker":"[11]"},{"why":"Gives the cancellation theorem for projective modules used in Lemma 3.1 to free the coordinate module.","marker":"[1]"}],"fun_headline_variants":["Residual coordinates become stable in any characteristic","Sharper stable-coordinate bounds for affine algebras","From residual to 1-stable in arbitrary characteristic","One extra variable yields stable coordinates","Improved stability bounds for coordinate algebras"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof leans on a prior two-variable theorem: a residual coordinate in a two-variable polynomial ring over a Noetherian base is a genuine coordinate whenever the base contains the rationals or its reduction is seminormal; if that theorem fails in positive characteristic for a seminormal base, the main one-dimensional results fail with it.","fun_headline_variants_meta":{"raw":{"variants":["Residual coordinates become stable in any characteristic","Sharper stable-coordinate bounds for affine algebras","From residual to 1-stable in arbitrary characteristic","One extra variable yields stable coordinates","Improved stability bounds for coordinate algebras"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000477,"raw_usage":{"total_tokens":2430,"prompt_tokens":1077,"completion_tokens":1353,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":693,"completion_tokens_details":{"reasoning_tokens":1288}},"tokens_in":693,"tokens_out":1353,"duration_ms":15081,"temperature":1.0,"reasoning_tokens":1288,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:11:03.664866+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $k$ an algebraic closure of $\\mathbb{F}_p$, $R = k[t]$, $n = 3$, and search for a residual coordinate $P$ in $R[Z_1,Z_2,Z_3]$ that is not a coordinate in $R[Z_1,Z_2,Z_3,W]$; if no automorphism of the larger ring sends $P$ to $Z_1$, then $P$ is not 1-stable and Theorem 3.4 would be refuted. Similarly, a residual coordinate over a Noetherian $d$-dimensional ring that failed to be $(2^d - 1)n$-stable, or required more than $2^{d-1}(n + 1) - n$ variables in the characteristic-zero affine case, would disprove Theorems 3.6 and 3.7.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the two-variable residual-coordinate theorem used in the final step of Theorem 3.2 and in Theorem 3.4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the characteristic-zero one-stable result and the Artinian residual-coordinate lemma that the paper extends and removes hypotheses from."},{"cited_title":"Maubach, The commuting derivation conjecture, J","cited_arxiv_id":null,"evidence_quote":"Gives the linear-polynomial coordinate criterion that Theorem 3.2 generalizes to one-dimensional seminormal bases."},{"cited_title":"Berson, J.W","cited_arxiv_id":null,"evidence_quote":"Provides the stability-lifting theorem used in the induction steps of Theorems 3.6 and 3.7."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"States the locally-polynomial-algebra-to-symmetric-algebra theorem used in Lemma 3.1 to pass from local coordinates to a global coordinate."},{"cited_title":"Bass, K-theory and stable algebra, Inst","cited_arxiv_id":null,"evidence_quote":"Gives the cancellation theorem for projective modules used in Lemma 3.1 to free the coordinate module."}],"review_version":1}