On Double Danielewski Surfaces and the Cancellation Problem

Neena Gupta; Sourav Sen

REVIEW 3 major objections 3 minor 15 references

On Double Danielewski Surfaces and the Cancellation Problem

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A family of two-equation affine surfaces gives new counterexamples to the Cancellation Problem: non-isomorphic rings become isomorphic after adjoining one variable.

desk verdict A solid new family of cancellation counterexamples with complete proofs; one compressed step in the stable-isomorphism theorem needs a small fix but the argument goes through. read the letter →

arxiv 1908.03403 v1 pith:CEXDWRGZ submitted 2019-08-09 math.AC math.AG

Neena Gupta , Sourav Sen This is my paper · ORCID

classification math.ACmath.AG MSC 14R0514R1013A5013B2513A0214R20

keywords CancellationProblemDoubleDanielewskisurfacesMakar-LimanovinvariantExponentialmapsAutomorphismStableisomorphismAffine

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

The paper studies affine surfaces defined by two equations of the form $X^dY=P(X,Z)$ and $X^eT=Q(X,Y,Z)$, called double Danielewski surfaces. It shows that when $r=\deg_Z P\ge 2$ and $s=\deg_Y Q\ge 2$, or in two boundary cases, every nontrivial exponential map on the coordinate ring fixes exactly the polynomial subring $k[x]$, so the Makar-Limanov invariant is $k[x]$. It then classifies these rings up to isomorphism and describes their automorphisms as explicit polynomial substitutions. The main result is a stable-isomorphism theorem: under two coprimality conditions, $B_{d,e}[1]$ is isomorphic to $B_{d,e-1}[1]$, while $B_{d,e}$ and $B_{d,e+1}$ are not isomorphic; hence these surfaces are counterexamples to the Cancellation Problem. A curious reader should care because this gives a large, explicitly computable family of surfaces in which adjoining one variable erases a genuine difference.

What carries the argument

The load-bearing object is the double Danielewski ring $B_{d,e}$, and the engine is the Makar-Limanov invariant—the common fixed subring of all nontrivial exponential maps from $B$ to $B[U]$. The invariant computation runs through two associated graded rings: first $D=k[X,Y,Z,T]/(X^dY-P(0,Z),\,X^eT-Y^s)$, then $C=k[X,Y,Z,T]/(X^dY-Z^r,\,X^eT-Y^s)$; every exponential map on $B$ induces homogeneous exponential maps on $D$ and $C$, where the invariant is shown to lie in $k[x]$. For the cancellation result, the key step is an element $v$ inside $A=B_{d,e}[w]$ formed from $w$, auxiliary polynomials $a,f,g,h$, and $x$; the coprimality hypotheses make a certain product a unit modulo $x$, which forces the exponential map to send $v$ to $v-U$ and identifies $A$ with a polynomial ring in $v$ over an invariant subring isomorphic to $B_{d,e-1}$.

What would settle it

Take $d=e=2$, $P(0,Z)=Z^2+1$ and $Q(0,Y,Z)=Y^2+Z$ over a field of characteristic not $2$, then compute the Makar-Limanov invariant of $B_{2,2}$; if any exponential map fixes more than $k[x]$, Theorem 3.8 fails. Alternatively, construct the subring $E=k[x,f,g,h]$ inside $B_{2,2}[w]$ exactly as in the proof of Theorem 3.14 and check whether $E\cong B_{2,1}$; if not, the cancellation conclusion for these data fails.

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Extended reading notes

Core claim

The central claim is that, for a fixed field $k$ and integers $d,e\ge 1$, the coordinate ring $B_{d,e}=k[X,Y,Z,T]/(X^dY-P(X,Z),\,X^eT-Q(X,Y,Z))$, with $r=\deg_Z P\ge 2$ and $s=\deg_Y Q\ge 2$, carries a complete structural theory. The Makar-Limanov invariant—the common fixed ring of all exponential maps, the algebraic counterpart of one-parameter additive group actions—is exactly $k[x]$ whenever $(r,s,e)$ satisfies condition (4) of Lemma 3.6. Two such rings are isomorphic only if their parameter quadruples $(d,e,r,s)$ agree and the polynomials are related by the explicit substitutions of Theorem 3.10, so no double Danielewski surface is isomorphic to any one-equation Danielewski surface. Under the additional coprimality assumptions on $P(0,Z)$, $Q(0,Y,Z)$ and their derivatives, $B_{d,e}$ and $B_{d,e+1}$ are non-isomorphic while $B_{d,e}[1]$ and $B_{d,e+1}[1]$ are isomorphic, giving the claimed counterexamples to the Cancellation Problem.

Load-bearing premise

The proof assumes that after setting $X=0$, the polynomials $P(0,Z)$ and $P'(0,Z)$ together generate the whole ring $k[Z]$, and that $P(0,Z),Q(0,Y,Z),Q'(0,Y,Z)$ together generate $k[Y,Z]$; if that fails, the element $v$ used to prove the stable isomorphism is not known to exist and the argument collapses.

Editorial extensions

If this is right

For every admissible choice of $d,e,P,Q$, the rings $B_{d,e}$ and $B_{d,e+1}$ are non-isomorphic while their one-variable extensions are isomorphic, so the cancellation property fails for an infinite family of affine surfaces.
Iterating the stable isomorphism gives $B_{d,e}[1]\cong B_{d,e'}[1]$ for all $e,e'\ge 2$, so each fixed $d$ produces a chain of distinct rings that become the same after adjoining one variable.
Theorem 3.10 reduces isomorphism testing in this family to checking the parameter quadruple and solving polynomial substitution equations, making the classification explicit and algorithmic in spirit.
Corollary 3.11 shows the double surfaces are a genuinely new class: no member is isomorphic to any classical Danielewski surface.
Theorem 3.13 gives a short criterion for automorphisms: an endomorphism fixing $x$ and $k[x,z]$ is automatically an automorphism.

Reading between the lines

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

Because the proof is characteristic-free, the same construction yields dimension-two counterexamples to cancellation in positive characteristic, where the polynomial ring $k[3]$ itself is known not to cancel; this puts non-cancellation at the lowest possible dimension there.
The two-stage filtration through $D$ and $C$ looks like a general recipe for surfaces whose leading forms are monomial equations, so iterated versions of the construction with three or more equations might be analyzable by the same reduction.
The explicit isomorphism data in Theorem 3.10 can be read as an algorithm: search for the polynomial data $\lambda,\gamma,\delta,f,g,h$ and check the two displayed congruences, turning isomorphism testing in the family into a polynomial-system problem.
The element $v$ constructed in Theorem 3.14 might serve as an explicit witness for the cancellation, giving a direct formula for the $B_{d,e-1}$ factor inside $B_{d,e}[w]$ rather than an existence proof.

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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

Short version: this is a good paper that deserves a serious referee. It constructs a genuinely new family of affine surfaces (the double Danielewski surfaces), computes their ML-invariant, classifies isomorphisms, describes automorphisms, and proves a stable isomorphism theorem yielding new counterexamples to the Cancellation Problem. The main theorems are not in the cited literature, and the proofs are mostly direct, with no fitted parameters or circular reasoning. The ML-invariant computation chains two filtrations and uses the standard Derksen-Hadas-Makar-Limanov homogenization theorem; that part reads cleanly. I agree with the reader's assessment.

The soft spots are localized. Theorem 3.10's proof is compressed, especially the steps showing d1=d2 and e1=e2 from the ideal equalities; they are believable but a referee should ask for a few more words. The larger issue is in Theorem 3.14. The paper wants to show A^phi=E, reduces to injectivity of E/xE -> A/xA, and then asserts 'dimensions equal, hence kernel is xE.' That implication is not valid as a general dimension argument: a surjection between two 1-dimensional rings can have nonzero kernel if the source is not a domain or not equidimensional. However, the paper has already displayed enough to fix it: the induced map sends F to z~, G to y~, H to P'(0,z~)Q'(0,y~,z~)w~, and by (19) the multiplier is a unit in A/xA. That gives an explicit inverse and proves injectivity. So the gap is a missing sentence, not a missing theorem. I would not call it a fatal flaw.

The citation pattern is normal; the self-citations are to background lemmas. The paper is aimed at affine algebraic geometers working on cancellation and locally nilpotent derivations. It is a substantial application of established tools rather than a new framework, but that's fine. A careful referee could usefully tighten Theorem 3.10 and patch Theorem 3.14 as above.

My recommendation: send it to review. It is the kind of paper where a knowledgeable referee can verify everything in a couple of days, and the result is worth having in the record.

Referee Report

3 major / 3 minor

Summary. The paper studies double Danielewski surfaces B_{d,e}=k[X,Y,Z,T]/(X^dY-P(X,Z), X^eT-Q(X,Y,Z)), with deg_Z P=r>=2 and deg_Y Q=s>=2. It computes the Makar-Limanov invariant of these surfaces, showing that under the conditions of (4) one has ML(B)=k[x] (Theorem 3.8); it gives a complete isomorphism classification (Theorem 3.10); it characterizes automorphisms (Theorems 3.12-3.13); and it proves stable isomorphisms B_{d,e}[1] =~ B_{d,e-1}[1] under explicit coprimality conditions (Theorem 3.14), yielding new counterexamples to the Cancellation Problem (Corollary 3.15). The proofs rely on exponential maps, admissible filtrations, and the Derksen-Hadas-Makar-Limanov homogenization theorem.

Significance. If the results hold, the paper provides a new two-dimensional family of affine surfaces over arbitrary fields that fail the cancellation property, together with an explicit classification of isomorphisms and automorphisms. The stable-isomorphism construction is explicit and parameter-free, and the ML-invariant computation is a genuine extension of the classical Danielewski theory. The main gap identified below concerns a formally insufficient step in the proof of Theorem 3.14, but the theorem can be repaired by a direct argument; the overall contribution is substantial and publishable in a good journal after revision.

major comments (3)

[Theorem 3.14] The step 'dim(E/xE)=1=dim ι(E). Hence kernel of ι is xE' is not justified: for a surjective map of affine k-algebras of equal Krull dimension, a nonzero kernel supported on a single component need not lower the dimension. The intended conclusion follows directly from the explicit map: the induced map E/xE -> ι(E) sends F to z~, G to y~, and H to P'(0,z~)Q'(0,y~,z~)w~; by (19) the multiplier is a unit, so this map is an isomorphism. Please replace the dimension criterion with this explicit isomorphism, since the present argument is load-bearing for the proof that A^φ = E.
[Theorem 3.14] The assertion 'Since A[1/x] = E[1/x][w]' is used to deduce E[1/x] = A^φ[1/x] and to compute the dimension of E, but it is not immediate. It can be verified from the displayed identities: from f = x^{d+e-1}w+z one obtains z ∈ E[1/x][w]; then from the formulas for g and h one obtains y and t in E[1/x][w]. Please include this verification, as the equality is load-bearing for the proof that A^φ = E.
[Theorem 3.10] The proof uses without comment the equalities x^{d_1}_1 B ∩ k[x_1,z_1] = (x^{d_1}_1, P_1(x_1,z_1)) and the analogous equality for Q, which are not immediate from the definitions. These equalities can be derived from the normal form (2) established in Lemma 3.4, but the derivation should be supplied, since the matching of the parameters d_i and e_i in (9) and (13) rests on them.

minor comments (3)

[Lemmas 3.4 and 3.5] In the statements of Lemmas 3.4 and 3.5, the graded ring is written as ⊕(B_n/B_{n+1}) and ⊕(D_n/D_{n+1}); with the given increasing filtrations these quotients are zero, and the intended expression is B_n/B_{n-1} and D_n/D_{n-1}, as used in the proofs.
[Lemma 3.6] In the sentence describing a homogeneous element of C in R, the range '0 ≤ j < r' should read '0 ≤ j < s' to match the representation obtained from the relation X^eT = Y^s.
[Lemma 3.6] The citation to Lemma 2.1(iii) for the absence of nontrivial exponential maps on a non-normal one-dimensional ring appears to be a mismatched reference; the argument is better supported by Lemma 2.1(v) or by the standard fact that a polynomial ring over a field is normal.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the ML-invariant, isomorphism-class, and stable-isomorphism proofs are explicit computations from the defining equations and standard external theorems; the few self-citations are background lemmas, not load-bearing inputs.

full rationale

The derivation chain is self-contained. Theorem 3.8 computes ML(B)=k[x] by reducing to graded rings via the external homogenization theorem of Derksen-Hadas-Makar-Limanov (Theorem 2.4) and proving Lemma 3.6 directly from a degree argument and standard exponential-map properties (Lemma 2.1). Lemma 3.7 constructs an exponential map with invariant ring k[x] by explicit substitution, not by assuming the conclusion. Theorem 3.10 derives isomorphism invariants from ML(B_i)=k[x_i] and from monicity and degree comparisons; conversely, it builds an explicit isomorphism when conditions (I)-(II) hold. Theorem 3.14 constructs v, f, g, h explicitly, verifies the identities (17)-(18), and uses hypothesis (19) to prove v in A and A=A^phi[v], then identifies A^phi=E via the subring E=k[x,f,g,h]; the isomorphism E≅E1 is an explicit quotient map plus a dimension argument. No parameter is fitted to data, no claimed prediction is an input by construction, and the self-citations ([2], [6], [10], [12]) are used only for elementary lemmas or for externally attributed results, not as the source of the main conclusions. The one flagged gap is in Theorem 3.14, the paragraph beginning 'Clearly iota(f)=...': the inference 'dim(E/xE)=1=dim iota(E). Hence kernel of iota is xE' is formally insufficient because equal Krull dimensions do not imply injectivity of a surjective map; however this is a correctness gap, not circularity, since the desired equality xE = xA ∩ E is not assumed as an input. Weighted in the verdict, this concern does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

There are no fitted constants and no invented physical or algebraic entities. The paper's inputs are the polynomials P and Q and the integer parameters d, e, r, s; the extra assumptions used in Theorem 3.14 are stated explicitly as mathematical hypotheses, not hidden choices.

assumptions (5)

domain assumption The defining data: k is a field, P is monic in Z with deg_Z P = r, Q is monic in Y with deg_Y Q = s, and the parameter restrictions in (4) of Lemma 3.6 hold.
These define the double Danielewski family and the regime in which ML(B)=k[x]; Theorems 3.10 and 3.13 use this regime.
domain assumption The regularity hypotheses of Theorem 3.14: (P(0,Z),P'(0,Z))=k[Z] and (P(0,Z),Q(0,Y,Z),Q'(0,Y,Z))=k[Y,Z].
These make P'(0,z)Q'(0,y,z) a unit in B/xB, which is the key step in the explicit stable-isomorphism construction.
standard math Standard theory of exponential maps and Makar-Limanov invariants, including Lemma 2.1 and the homogenization theorem of Derksen, Hadas, and Makar-Limanov.
Imported from references [3], [5], [10], and [12]; used throughout Section 3.
domain assumption The localized ring A=k(T)[X,Y,Z]/(X^dY-Z^r, X^eT-Y^s) is a non-normal one-dimensional domain when one of the conditions in (4) holds.
Stated and used in Lemma 3.6 to force any exponential map on A to be trivial; this is a checkable algebraic fact but not proved in detail there.
standard math For a one-dimensional non-normal affine domain, a nontrivial exponential map would make the ring a polynomial ring over its invariant field, contradicting non-normality.
Invoked in Lemma 3.6 through Lemma 2.1(iii); it is part of the standard exponential-map toolkit.

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Pith. "Pith review of On Double Danielewski Surfaces and the Cancellation Problem." pith.science (2026). https://pith.science/paper/CEXDWRGZ

@misc{pith2026190803403,
  author       = {Pith},
  title        = {Pith review of: On Double Danielewski Surfaces and the Cancellation Problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CEXDWRGZ}},
  note         = {Machine review of arXiv:1908.03403}
}

read the original abstract

We study a two-dimensional family of affine surfaces which are counter-examples to the Cancellation Problem. We describe the Makar-Limanov invariant of these surfaces, determine their isomorphism classes and characterize the automorphisms of these surfaces.

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Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

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Pith tools

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

[1] [1]

Abhyankar, P

S. Abhyankar, P. Eakin, W. Heinzer, On the uniqueness of the coeﬃcient ring in a polynomial ring , J. Algebra 23 (1972) 310–342

work page 1972

[2] [2]

Bhatwadekar and N

S.M. Bhatwadekar and N. Gupta, A Note on the Cancellation Property of k[X,Y ], J. Algebra and its Applications, (Special issue in honour of Professor Shreeram S. Abhyankar), 14(9) (2015)1540007(1–5)

work page 2015

[3] [3]

Crachiola, The hypersurface x +x2y +z2 +t3 = 0 over a ﬁeld of arbitrary characteristic, Proc

A.J. Crachiola, The hypersurface x +x2y +z2 +t3 = 0 over a ﬁeld of arbitrary characteristic, Proc. Amer. Math. Soc. 134(5) (2005) 1289–1298

work page 2005

[4] [4]

Danielewski, On a cancellation problem and automorphism groups of aﬃne algebraic varieties , preprint 1989

W. Danielewski, On a cancellation problem and automorphism groups of aﬃne algebraic varieties , preprint 1989. 15

work page 1989

[5] [5]

Derksen, O

H. Derksen, O. Hadas and L. Makar-Limanov, Newton polytopes of invariants of additive group actions , J. Pure and Applied Algebra 156 (2001) 187–197

work page 2001

[6] [6]

Dutta, N

A.K. Dutta, N. Gupta and N. Onoda, Some patching results on algebras over two-dimensional factorial domains , J. Pure Appl. Algebra 216 (2012) 1667– 1679

work page 2012

[7] [7]

Fieseler, On complex aﬃne surfaces with C+-actions, Comment

K.H. Fieseler, On complex aﬃne surfaces with C+-actions, Comment. Math. Helvetici 69 (1994) 5–27

work page 1994

[8] [8]

Gene Freudenburg, Algebraic Theory of Locally Nilpotent Derivations(Second Edition), Encycl. Math. Sci., 136, Inv. Theory and Alg. Tr. Groups, VI I, Springer-Verlag, (2017)

work page 2017