REVIEW 1 major objections 3 minor 3 references
A note on partial coordinate system in a polynomial ring
T0 review · 1 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read For rings containing the rationals, partial residual coordinates force genuine partial coordinates.
desk verdict 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. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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 $A$.
What would settle it
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.
Extended reading notes
Core claim
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.
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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 (1)
- [Remark 1.3 and proof of Theorem 2.1] 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.
minor comments (3)
- [Abstract and Remark 1.3] There are typographical errors in 'pr oved' (abstract) and 'Noetherain' (Remark 1.3) that should be corrected.
- [Equation (2), Section 2] 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.
- [Proof of Theorem 2.1, last lines] 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.
Circularity Check
No circularity: Theorem 2.1 is a genuine reduction to an independent prior theorem (Das–Dutta), with no fitting, self-citation, or definitional equivalence.
full rationale
The paper's central claim, Theorem 2.1, is not circular. It assumes an a-strongly partial residual coordinate system in R[X_1,...,X_n] and constructs a finitely generated Q-subalgebra S of R so that the same polynomials form a partial residual coordinate system over S, checking the condition separately for primes containing a and primes not containing a via equations (1) and (2). It then appeals to Theorem 1.2, quoted from Das–Dutta's Corollary 3.19, which states that partial residual coordinate systems and partial coordinate systems are equivalent over Noetherian rings containing Q. This is a standard and legitimate use of an external prior result. The reduction is not a renaming: the hypotheses supply equations (1) and (2), and those equations are used to verify the residual-coordinate condition at every prime of S; the conclusion of a partial coordinate system over S then transfers to R by base change. No parameter is fitted, no prediction is renamed as an input, and no load-bearing step is justified by a self-citation. The only identified concern is Remark 1.3, which asserts without proof that Das–Dutta's result extends from Noetherian domains to arbitrary Noetherian rings containing Q because the cited theorems hold in that generality. This is a possible correctness gap, not circularity: the proof depends on an unverified generalization of an external theorem, but the argument does not assume its own conclusion. Accordingly, the circularity score is 0.
Assumptions & free parameters
assumptions (3)
- domain assumption Theorem 1.2 (Das-Dutta [3, Corollary 3.19]): For R Noetherian containing Q, partial residual coordinate system implies partial coordinate system.
- domain assumption R is a commutative ring containing Q with unity.
- standard math Standard commutative algebra facts: localization, tensor products, residue fields, finitely generated Q-algebras are Noetherian.
Cite this review
Pith. "Pith review of A note on partial coordinate system in a polynomial ring." pith.science (2026). https://pith.science/paper/HEGSIK62
@misc{pith2026190804012,
author = {Pith},
title = {Pith review of: A note on partial coordinate system in a polynomial ring},
year = {2026},
howpublished = {\url{https://pith.science/paper/HEGSIK62}},
note = {Machine review of arXiv:1908.04012}
}
abstract
J. Berson, J. W. Bikker and A. van den Essen proved that for a non-zerodivisor $a$ in a commutative ring $R$ containing $Q$ if the polynomials $f_1,\dots,f_{n-1}$ in $R[X_1,\dots,X_n]$ form a partial coordinate system over the rings $R_a$ and $\dfrac{R}{aR}$ then $f_1,\dots,f_{n-1}$ form a partial coordinate system over the ring $R$. In this note we show that the theory of residual variables of Bhatwadekar-Dutta and its recent extension by Das-Dutta, extends their result to the case when $a$ is an arbitrary element of $A$.
Reference graph
Works this paper leans on
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[3]
P. Das and A.K. Dutta, A note on residual variables of an affine fibration , J. Pure Appl. Algebra 218(10) (2014) 1792-1799. 3
work page 2014
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[1]
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
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[2]
S.M. Bhatwadekar and A.K. Dutta, On residual variables and stably polynomial algebras, Comm. Algebra 21(2) (1993) 635–645
work page 1993
Reviewed August 14, 2026 · model on record in the stance chip above.
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