Affine algebraic geometry · polynomial cancellation · locally nilpotent derivations · additive group actions

Zariski Cancellation for Affine Three-Space in Characteristic Zero

Collaboration beta

Over an algebraically closed field of characteristic zero, if a finitely generated three-dimensional affine domain becomes four-dimensional affine space after adjoining one variable, was it already affine three-space?

k algebraically closed, char (k) = 0, A [t] ≅ k [x_{1,} x_{2,} x_{3,} x_{4}] \Rightarrow ? A ≅ k [x_{1,} x_{2,} x_{3}]

Known results and sources

Over an algebraically closed field of characteristic zero, cancellation asks whether a finitely generated threefold that becomes affine four-space after adding one line was already affine three-space; a rank-trichotomy route studies the question through commuting additive directions.

Research problem

Exact mathematical statement

Let k be an algebraically closed field of characteristic zero and let A be a finitely generated k-domain. If adjoining one indeterminate makes A a polynomial ring in four variables,

A[t] \cong k[x_1,x_2,x_3,x_4],

must A already be a polynomial ring in three variables,

A \cong k[x_1,x_2,x_3]?

This characteristic-zero affine-three-space cancellation problem remains open. A proposed route studies three commuting locally nilpotent derivations and branches according to their generic rank. Its rank-three conclusion, rank-two structure, and invariant-lifting obstruction are research claims awaiting independent mathematical review; the rank-two and rank-one closing branches remain open.

Problem infographic

Problem at a glance

Wide mathematical problem explainer over an algebraically closed field of characteristic zero: a four-dimensional polynomial cylinder contracts to an unknown finitely generated threefold, then branches into generic ranks three, two, and one; rank three is marked source-reported, while rank two ends at torsion and rank one at hidden lower-degree directions, both visibly open. — Over an algebraically closed field of characteristic zero, the cancellation question branches by the generic rank of descended locally nilpotent derivations. The proposed rank-three branch closes, while invariant-lifting torsion and filtered rank raising remain open; the full cancellation problem is unresolved.

Current mathematical picture

Where work on Zariski Cancellation for Affine Three-Space in Characteristic Zero stands

Open problem

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureTreating a root of the collapse polynomial as evidence of an exotic threefold

In A = k[u,x,y], the commuting locally nilpotent derivations D = partial_y and E = u partial_x + x partial_y have collapse form -u du, yet (E - xD)/u = partial_x removes the collapse inside ordinary affine three-space. The full horizontal matrix and its unit-generating maximal minors remain available to constrain rank-one collapse, while the exact invariant-lifting torsion sequence gives a concrete target in rank two. Either route requires new independently reviewed…

Route status · Narrowed route

Main reductionCurrent reduction

The proposed leading-term route reduces any still-unresolved case to generic rank at most two for a commuting triple of locally nilpotent derivations, subject to independent validation of the descent and rank-three arguments.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeProve the leading-term descent lemma with explicit filtration hypotheses before relying on the rank trichotomy.Task status · Ready to work on

Research-record correctionResearch-record correction

We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Zariski Cancellation for Affine Three-Space in Characteristic Zero in numbers

648retained lines of mathematical investigation648 in the current working snapshot

Argument development: 515 · 79%
Explored or eliminated routes: 35 · 5%
Computational analysis: 3 · 0%
Open obligations: 24 · 4%
Definitions and setup: 71 · 11%

8selected mapped statements2routes investigated3open questions3contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byOscar Zariski (historical namesake; the modern affine-space formulation developed later)Status checkedAug 13, 2026

Source recordedAug 13, 2026

Recommended next task

Prove the leading-term descent lemma with explicit filtration hypotheses before relying on the rank trichotomy.

Suggested move: Write and independently audit the associated-graded locally nilpotent derivation lemma, including the filtration hypotheses and the descent from coordinate translations to at least three commuting nonzero derivations on A.

What would count as progress

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

13 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

2 recorded

Narrowed routeTreating a root of the collapse polynomial as evidence of an exotic threefold

Route status · Narrowed route

Narrowed routeTreating common zeros of plinth-like coefficients as intrinsic obstructions

In A = k[u,y,z], the commuting locally nilpotent derivations D = u partial_y + y partial_z and E = partial_z have two transverse coefficients vanishing at u = 0, yet (D - yE)/u = partial_y recovers the missing direction. The live rank-two question is whether coordinate-frame structure forces the needed invariant lifting and finite triangular division in every unresolved case.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Prove the leading-term descent lemma with explicit filtration hypotheses before relying on the rank trichotomy.Suggested move: Write and independently audit the associated-graded locally nilpotent derivation lemma, including the filtration hypotheses and the descent from coordinate translations to at least three commuting nonzero derivations on A.

Ready to work on

Close or refute the source-reported constant-wedge generic-rank-two branch without relying on an unverified surface-classification shortcut.Suggested move: Turn the constant-wedge fiberwise orbit calculation into an exact finite-presentation torsor-recognition proof and verify the cited surface-cancellation hypotheses.

Ready to work on

Resolve at least one of the two live bottlenecks—persistent invariant-lifting torsion in rank two or collapse of all leading horizontal directions to rank one—without relying on uncertified surface-classification or G-theory claims.Suggested move: Use the ambient cylinder and full coordinate frame to test whether persistent prime-fiber torsion can survive triangular division, while separately formulating the rank-one problem through a Rees module or Fitting ideal.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 13, 2026

Current statusOpen problem

For an algebraically closed field of characteristic zero, it remains open whether A[t] isomorphic to a polynomial ring in four variables forces A to be a polynomial ring in three variables. Cancellation is known in lower dimensions, while positive characteristic has counterexamples in dimensions above two. Recent characteristic-zero A1-contractible families and locally nilpotent derivation characterizations sharpen the surrounding landscape but do not solve this exact affine-three-space cylinder question.

[1][4]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2025
PreprintDubouloz and Ghosh constructed characteristic-zero families of A1-contractible affine varieties of dimensions at least four that are not affine spaces and described their relation to potential cancellation…[4]
2023
PreprintSai Krishna gave a characteristic-free algebraic characterization of affine three-space using Makar–Limanov-type invariants and recovered cancellation for the affine plane; the result is a neighboring…[3]
2022
Authoritative summaryGupta’s ICM survey reviewed the positive-characteristic solution and retained characteristic zero as the setting of the unresolved cancellation program, emphasizing affine fibrations and locally nilpotent…[1]
2014
Peer reviewedGupta proved that affine n-space is not cancellative over positive-characteristic fields for every n greater than two; this result does not resolve the characteristic-zero question.[2]

Mathematical neighborhood

Related results and reusable starting points

Current focusZariski cancellation problem for affine three-space in characteristic zero

Solved special caseCancellation for affine two-space

If B[t] is a polynomial ring in three variables under the standard domain hypotheses, then B is a polynomial ring in two variables. This lower-dimensional theorem does not automatically extend to A3.

[1][3]

Related problemPositive-characteristic noncancellation

Positive characteristic admits counterexamples in every affine-space dimension above two. Those examples establish the importance of the characteristic hypothesis but do not decide characteristic zero.

[2][1]

Dependency or reductionLocally nilpotent derivation and Makar–Limanov characterizations

Makar–Limanov and slice-sensitive variants characterize affine three-space under additional factoriality and nonrigidity hypotheses, making locally nilpotent derivations a major route to recognizing polynomial threefolds without equating the characterization with cancellation itself.

[1][3]

Related problemA1-contractible affine varieties

A smooth affine variety whose cylinder is affine space is A1-contractible, but A1-contractibility alone is weaker than being affine space. Higher-dimensional families therefore probe the boundary without solving the exact A3 cylinder question.

[4]

Equivalent formulationPolynomial-ring cancellation formulation

The geometric cylinder question X times A1 isomorphic to A4 versus X isomorphic to A3 is the variety-side form of cancellation for the three-variable polynomial ring under the stated affine-domain hypotheses.

[1]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

Formalization targetA proof-assistant development of finitely generated affine domains, Krull dimension, polynomial extensions, spectra, and isomorphisms of affine schemes at the required generality.
Formalization targetFormal locally nilpotent derivations, slices, additive-group actions, invariant rings, and Makar–Limanov-type invariants.
Formalization targetFormal algebraic geometry of smooth factorial affine threefolds, Kähler differentials, algebraic de Rham cohomology, and volume forms.
Formalization targetFormal orbit and torsor arguments for Ga-actions and exact hypotheses for lower-dimensional cancellation theorems used in reductions.
Formalization targetA formal statement separating the universal cancellation target from conditional branch theorems and source-reported research reductions.

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

3 standing statements5 proposed statements3 open questions2 narrowed routes

Statements by mathematical role8 selected mapped statements

theorem candidate1 of 81
reduction3 of 83
lemma2 of 82
equivalence1 of 81
negative result1 of 81

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.23 displayed rows · 2 routes included

retained route statementDoes one polynomial cylinder determine affine three-space?
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementStable cylinder questionintermediate
retained route statementLeading LND rank trichotomyintermediate
retained route statementUnimodular horizontal matrixintermediate
retained route statementGeneric-rank-three branchintermediate
retained route statementInvariant-lifting torsionintermediate
Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
Useful failureTreating a root of the collapse polynomial as evidence of an exotic threefoldreported failure
Useful failureTreating common zeros of plinth-like coefficients as intrinsic obstructionsreported failure
Research targetProve the leading-term descent lemma with explicit filtration hypotheses before relying on the rank trichotomy.open
Research targetClose or refute the source-reported constant-wedge generic-rank-two branch without relying on an unverified surface-classification shortcut.open
Research targetResolve at least one of the two live bottlenecks—persistent invariant-lifting torsion in rank two or collapse of all leading horizontal directions to rank one—without relying on uncertified surface-classification or G-theory claims.open
Research targetFiltered rank raisingsuperseded
Narrowed routeTreating a root of the collapse polynomial as evidence of an exotic threefoldIn A = k[u,x,y], the commuting locally nilpotent derivations D = partial_y and E = u partial_x + x partial_y have collapse form -u du, yet (E - xD)/u = partial_x removes the collapse inside ordinary affine three-space. The full horizontal matrix and its unit-generating maximal minors remain available to constrain rank-one collapse, while the exact invariant-lifting torsion sequence gives a concrete target in rank two. Either route requires new independently reviewed mathematics.
Narrowed routeTreating common zeros of plinth-like coefficients as intrinsic obstructionsIn A = k[u,y,z], the commuting locally nilpotent derivations D = u partial_y + y partial_z and E = partial_z have two transverse coefficients vanishing at u = 0, yet (D - yE)/u = partial_y recovers the missing direction. The live rank-two question is whether coordinate-frame structure forces the needed invariant lifting and finite triangular division in every unresolved case.

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeProve the leading-term descent lemma with explicit filtration hypotheses before relying on the rank trichotomy.

2 approaches have already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

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Prepared starting pointProve the leading-term descent lemma with explicit filtration hypotheses before relying on the rank trichotomy.

Zariski Cancellation for Affine Three-Space in Characteristic Zero · ready to start

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Sources and references6 cited works · next context review by Nov 13, 2026

The mathematical context was checked on Aug 13, 2026. Status can be refreshed sooner after a material result or claim.

1
The Zariski Cancellation Problem and related problems in Affine Algebraic Geometrysurvey or monograph · Neena Gupta · Proceedings of the International Congress of Mathematicians / EMS Press · 2022 · ARXIV 2208.14736 · DOI 10.4171/ICM2022/151 · accessed Aug 13, 2026
2
On Zariski's Cancellation Problem in positive characteristicpeer reviewed result · Neena Gupta · Advances in Mathematics · 2014 · ARXIV 1309.1368 · DOI 10.1016/j.aim.2014.07.012 · accessed Aug 13, 2026
3
An Algebraic characterization of the affine three space in arbitrary characteristicpreprint · P. M. S. Sai Krishna · arXiv · 2023 · ARXIV 2308.05424 · accessed Aug 13, 2026
4
Algebraic families of higher dimensional A1-contractible affine varieties non-isomorphic to affine spacespreprint · Adrien Dubouloz, Parnashree Ghosh · arXiv · 2025 · ARXIV 2501.09613 · accessed Aug 13, 2026
5
Mathlib documentation indexformalization · Lean community · accessed Aug 13, 2026
6
Formal Conjecturesformalization · Google DeepMind · GitHub · accessed Aug 13, 2026

Important qualifications

The status review was scoped to the exact affine-three-space cancellation question over an algebraically closed characteristic-zero field and representative neighboring results; it is not an exhaustive bibliography of cancellation theory.
The standard name ‘Zariski cancellation problem’ covers historically related field, ring, and variety formulations. This record does not claim that Oscar Zariski wrote the modern affine-space ring formulation verbatim in 1949, so proposedYear is left unresolved.
The 2025 A1-contractible families are higher-dimensional neighboring candidates and are not counterexamples to the exact affine-three-space cylinder statement recorded here.
The 2023 affine-three-space characterization is a structural theorem and reformulation tool, not a solution of characteristic-zero cancellation for A3.
The scoped formalization search checked current public Mathlib documentation and Google DeepMind’s Formal Conjectures repository. Finding no exact problem-level formalization there does not establish nonexistence in every proof assistant or private project.
No canonical computation, dataset, or certificate can decide the general isomorphism statement, and none was identified in the scoped search.
No unreviewed source material, packet theorem, source-reported label, contributor estimate, or embedded bibliography was used as external-status authority. This metadata has no proof, novelty, review, acceptance, visibility, publication, or deployment authority.

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