On Residual and Stable Coordinates

Amartya Kumar Dutta; Animesh Lahiri

On Residual and Stable Coordinates

T0 review · 0 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Residual coordinates over one-dimensional seminormal bases are one-stable, in all characteristics, and Noetherian stable-coordinate bounds hold without the rationals.

desk verdict Extends Kahoui-Ouali's residual-coordinate theorems to arbitrary characteristic, drops the Q-hypothesis, and sharpens the stability bound; solid, minor fixable slips only. read the letter →

arxiv 1908.03549 v1 pith:W22SRYMM submitted 2019-08-09 math.AC

Amartya Kumar Dutta , Animesh Lahiri This is my paper · ORCID

classification math.AC MSC 13B2514R2514R1013A50

keywords residualcoordinatestablepolynomialalgebraexponentialmapseminormalringNoetherianaffine

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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 $W$.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The 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.

What 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.

The 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.

I 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.

Referee Report

0 major / 6 minor

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.

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.

minor comments (6)

[Lemma 3.1, proof] 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.
[Theorem 3.7, proof, base case d = 1] 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.
[Theorem 3.4, proof] 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.
[Theorems 3.6 and 3.7, induction step] 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.
[Abstract and Introduction] 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.
[Lemma 3.1, proof] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the claimed theorems reduce to independent published results and honest inductions, not to their own conclusions.

full rationale

The derivation chain is non-circular. The main theorems are proved by reducing to established black boxes: Bhatwadekar–Dutta Theorem 2.7 (a residual coordinate in a two-variable polynomial ring is a coordinate when the base contains Q or its reduction is seminormal), Kahoui–Ouali Proposition 2.6, Berson–Bikker–van den Essen Theorem 2.8, Quillen local-global, Bass cancellation, and Bass–Connell–Wright/Suslin local-global results. Although Theorem 2.7 cites work involving one of the present authors, it is a prior published theorem with assumptions that do not include the target conclusion of this paper; the paper never assumes that residual coordinates are stable coordinates in order to prove that they are. The higher-dimensional cases are honest inductions: Theorem 3.6 uses induction on dimension after reducing modulo a non-zerodivisor, and each induction hypothesis applies only to a strictly lower-dimensional ring. Theorem 3.7 similarly sharpens the bound by induction with the one-dimensional case supplied by Theorem 3.2. Lemma 3.1's cancellation argument is justified by the symmetry algebra, Quillen's local-global theorem, and Bass's cancellation theorem; the n=2 determinant argument supplies the missing cancellation step. No fitted parameters, no renamed empirical patterns, and no ansatz is smuggled in via citation. The only potentially self-referential ingredient is the cited Bhatwadekar–Dutta theorem, but it is external support for a strictly weaker statement than what is proved here, so it does not create circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters or fitted constants appear: this is a purely algebraic proof. No new entities are introduced; the paper works entirely within polynomial rings and standard module constructions. The proof burden is a chain of published theorems in commutative algebra, and the main ones are listed above.

assumptions (7)

standard math Hilbert Nullstellensatz: for an affine algebra R over an algebraically closed field k, every maximal ideal has residue field k.
Used in Theorem 3.2 to identify R/m with k and to pass the coordinate property to k(Q).
standard math Bhatwadekar-Dutta theorem 2.7: residual coordinates in R[2] are coordinates when R contains Q or R_red is seminormal.
This is the key black box that upgrades the local residual property to a global coordinate in Theorem 3.2; all later theorems depend on it.
standard math Bass cancellation theorem 2.14: projective modules of rank at least dim R + 1 cancel from direct sums.
Used in Lemma 3.1 to conclude Q is free; for the n=2 edge case a determinant argument substitutes.
standard math Quillen local-global theorem 2.15: a module over R[X] that is extended locally is extended globally.
Used in Lemma 3.1 to lift the local splitting Q'_m to a global projective module Q over R.
standard math Kahoui-Ouali proposition 2.6: residual coordinates over Artinian rings are coordinates.
Used in the induction step to show that a residual coordinate becomes a coordinate over the total quotient ring S^{-1}R.
standard math Stable-coordinate lifting theorem 2.8: if the reduction of P is m-stable over R/aR, then aW + P is (2m + n - 1)-stable.
Used in Theorems 3.6 and 3.7 to propagate stability bounds down the induction.
standard math Sathaye's triviality of A^2-fibrations over discrete valuation rings containing Q, and Bhatwadekar-Dutta's linear-planes theorem 2.9.
Used only in Proposition 3.9 for the Dedekind-domain case of the open question.

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Pith. "Pith review of On Residual and Stable Coordinates." pith.science (2026). https://pith.science/paper/W22SRYMM

@misc{pith2026190803549,
  author       = {Pith},
  title        = {Pith review of: On Residual and Stable Coordinates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W22SRYMM}},
  note         = {Machine review of arXiv:1908.03549}
}

abstract

In a recent paper, M. E. Kahoui and M. Ouali have proved that over an algebraically closed field $k$ of characteristic zero, residual coordinates in $k[X][Z_1,\dots,Z_n]$ are one-stable coordinates. In this paper we extend their result to the case of an algebraically closed field $k$ of arbitrary characteristic. In fact, we show that the result holds when $k[X]$ is replaced by any one-dimensional seminormal domain $R$ which is affine over an algebraically closed field $k$. For our proof, we extend a result of S. Maubach giving a criterion for a polynomial of the form $a(X)W+P(X,Z_1,\dots,Z_n)$ to be a coordinate in $k[X][Z_1,\dots,Z_n,W]$. Kahoui and Ouali had also shown that over a Noetherian $d$-dimensional ring $R$ containing $Q$ any residual coordinate in $R[Z_1,\dots,Z_n]$ is an $r$-stable coordinate, where $r=(2^d-1)n$. We will give a sharper bound for $r$ when $R$ is affine over an algebraically closed field of characteristic zero.

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Reference graph

Works this paper leans on

14 extracted references · 13 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Bass, K-theory and stable algebra, Inst

H. Bass, K-theory and stable algebra, Inst. Hautes Etudes Sci. Publ. Math. 22 (1964) 5–60

work page 1964

[2] [2]

H. Bass, E. Connell and D. Wright, Locally polynomial algebras are symmetric algebras, Invent. Math. 38 (1977) 279–299

work page 1977

[3] [3]

Berson, J.W

J. Berson, J.W. Bikker and A. van den Essen, Adapting Coordinates, J. Pure Appl. Algebra 184(2–3) (2003) 165–174

work page 2003

[4] [4]

S. M. Bhatwadekar and A. K. Dutta, On residual variables and stably polynomial algebras, Comm. Algebra 21(2) (1993) 635–645

work page 1993

[5] [5]

S. M. Bhatwadekar and A. K. Dutta, Linear planes over a discrete valuation ring, J. Algebra 166(2) (1994) 393–405

work page 1994

[6] [6]

Das and A

P. Das and A. K. Dutta, A note on residual variables of an aﬃne ﬁbration, J. Pure Appl. Algebra 218(10) (2014) 1792–1799

work page 2014

[7] [7]

Eakin, W

P. Eakin, W. Heinzer, A cancellation problem for rings, Conference on Commutative Alge- bra (Univ. Kansas, Lawrence, Kan., 1972), 61–77. Lecture No tes in Math. 311, Springer, Berlin, 1973

work page 1972

[8] [8]

van den Essen, Polynomial automorphism and the Jacobian conjecture, Progress in Mathematics 190

A. van den Essen, Polynomial automorphism and the Jacobian conjecture, Progress in Mathematics 190. Birkhuser Verlag, Basel, 2000

work page 2000