Commutative algebra and differential operators

Nakai conjecture

Collaboration beta

Can every singular finitely generated commutative algebra in characteristic zero be detected by a differential operator that cannot be built from multiplication and derivations?

D_{A / k} = A ⟨ {Der}_{k} (A) ⟩ \Rightarrow A is regular, char (k) = 0.

Known results and sources

Illustration: Smooth and singular local geometry motivate the question; neither drawing decides the general algebraic implication.

Research problem

Exact mathematical statement

Let k be any field of characteristic zero and A any finitely generated commutative k-algebra. Let D^0_{A/k}=A act by multiplication, and define differential operators of order at most q by requiring every commutator with multiplication by an element of A to have order at most q−1. Set

D_{A/k}=\bigcup_{q\geq0}D^q_{A/k},\qquad \Delta_A=A\langle\operatorname{Der}_k(A)\rangle.

The target is

D_{A/k}=\Delta_A\quad\Longrightarrow\quad A\text{ is regular}.

The algebra Δ_A permits arbitrary finite words of derivations. There is no reducedness, domain, normality, isolatedness, hypersurface, locality, or dimension assumption. The source also permits completion of the research task by an actual counterexample.

Problem infographic

Problem at a glance

Illustration: Polynomial differentiation explains derivation words, while smooth and singular local curve views introduce the unresolved regularity question over a characteristic-zero field. — Illustration: Nakai’s conjecture concerns all finitely generated commutative algebras over characteristic-zero fields. A local curve example does not settle the unrestricted target.

Current mathematical picture

Where work on Nakai conjecture stands

Open conjecture

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

Useful failureTreating Jacobian-power detector silence as generatedness

The source records one nongenerated sextic order-two operator preserving every Jacobian power; it does not claim that all order-two operators on that germ are silent. A nonzero detector still supplies a sufficient separator, while a zero detector can require another ideal or a higher-order construction.

Route status · Narrowed route

Main reductionCurrent reduction

For the isolated hypersurface classes reached by the source, construct a compatible symbol, lift it to an actual operator, and separate it from every derivation word using a stable ideal or an actual-multiplier argument. This supplies scoped obstructions; no reduction of every reduced singular algebra to these classes is given.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeEstablish the missing filtered presentation for the actual logarithmic fields underlying the newest ordinary-boundary full-ring claims.Task status · Ready to work on

Work mapped so far

Nakai conjecture in numbers

1.2kretained lines of mathematical investigation1,226 in the current working snapshot

Argument development: 652 · 53%
Explored or eliminated routes: 174 · 14%
Computational analysis: 56 · 5%
Open obligations: 133 · 11%
Definitions and setup: 211 · 17%

8selected mapped statements2routes investigated4open questions4contribution-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.

Status checkedSep 7, 2026

Source recordedSep 7, 2026

Recommended next task

Establish the missing filtered presentation for the actual logarithmic fields underlying the newest ordinary-boundary full-ring claims.

Suggested move: Suggested next step: Reconstruct the pair-relation and binary full-word arguments with actual multipliers, enumerate intrinsic Koszul terms, and eliminate every surviving longer-word jet, including the stated septic boundary.

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

14 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 Jacobian-power detector silence as generatedness

Route status · Narrowed route

Narrowed routeExtending automatic order-two lifting to every order

The source records cubic higher-order failures and a family with arbitrarily large first escape order, so neither universal automatic lifting nor a universal fixed escape order follows. Order-two compatible lifts remain available in their proved source scope; higher-order candidates require their own compatibility and full-word separation arguments.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

4 featured tasks

Establish the missing filtered presentation for the actual logarithmic fields underlying the newest ordinary-boundary full-ring claims.Suggested move: Suggested next step: Reconstruct the pair-relation and binary full-word arguments with actual multipliers, enumerate intrinsic Koszul terms, and eliminate every surviving longer-word jet, including the stated septic boundary.

Ready to work on

Determine whether the finite Jacobian-power detector has a structural nonvanishing theorem beyond the already covered weighted and cyclic classes.Suggested move: Suggested next step: Choose an uncovered local isolated hypersurface, retain its local exponent certificate and both finite maps, and recover an exact lifted separator when the detector is nonzero; investigate another ideal or higher order when it is zero.

Ready to work on

Conditional ordinary-boundary conclusions

GAP-FILTERED-MULTIPLIER-01 leaves the ternary q=2d−6, degree-six pair-relation, binary nonsquare, binary tangent-square, binary full-word, broad ordinary q=2d−6, ordinary septic, and one-cubic extension interfaces conditional at their full-ring step. Their compatible constructions and separately proved older subcases must not be withdrawn merely because this transfer is missing.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.

Ready to work on

Bridge the constructive local classes to arbitrary reduced singular algebras, including nonisolated normal and higher-codimension cases.Suggested move: Suggested next step: Develop a full-word separator for a broader canonical-module, stable-ideal, or complete-intersection route while preserving field, localization, and codimension hypotheses at each step.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Sep 7, 2026

Current statusOpen conjecture

The general question remains open: for a finite-type algebra over a field of characteristic zero, does generation of its differential-operator algebra by operators of order at most one force regularity? Results for restricted singularity classes do not settle that general implication.

[1][2][3]

External progress

What the literature has established

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

2026-08-19
PreprintVersion 3 reports the Nakai case for singular coordinate rings A=k[x₁,…,xₙ]/(f) over an algebraically closed field of characteristic zero, where f is weighted homogeneous and has an isolated hypersurface singularity at the origin.[2]
2026
Peer reviewedSebag gives an effective proof for reduced homogeneous plane curves over fields of characteristic zero, without requiring irreducibility.[3]
2025-02-07
PreprintLi, Xiao and Zuo report the Nakai case for complex analytic local rings of isolated hypersurface germs of modality at most two.[1]

Mathematical neighborhood

Related results and reusable starting points

Current focusNakai's conjecture

Solved special caseReduced homogeneous plane curves in characteristic zero

This published singularity-class result is a special case of the regularity question, not a theorem for every finite-type algebra.

[3]

Detailed research inventory

Claims, milestones, and routes in the current map

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

6 standing statements2 proposed statements4 open questions2 narrowed routes

Statements by mathematical role8 selected mapped statements

theorem candidate1 of 81
reduction1 of 81
lemma2 of 82
special case3 of 83
equivalence1 of 81

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.24 displayed rows · 2 routes included

retained route statementDoes generation of every differential operator by derivations force regularity?
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementNonreduced algebrasintermediate
retained route statementA finite detector for one kind of escapeintermediate
retained route statementWeighted socle deformationsintermediate
retained route statementA uniform cyclic familyintermediate
retained route statementOrder-two lifting is separated from escapeintermediate
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 Jacobian-power detector silence as generatednessreported failure
Useful failureExtending automatic order-two lifting to every orderreported failure
Research targetEstablish the missing filtered presentation for the actual logarithmic fields underlying the newest ordinary-boundary full-ring claims.open
Research targetDetermine whether the finite Jacobian-power detector has a structural nonvanishing theorem beyond the already covered weighted and cyclic classes.open
Research targetBridge the constructive local classes to arbitrary reduced singular algebras, including nonisolated normal and higher-codimension cases.open
Research targetConditional ordinary-boundary conclusionsopen
ComputationThe source reports an end-to-end order-two Jacobian-power detector certificate for the local cusp y²−x³−x⁴, with retained matrices and an exact lift.The reported ranks are 67 and 69, with a nonzero first normal residue. This is one audit example of the finite detector; these computations were not independently replayed for this draft. · reported unreproduced
Narrowed routeTreating Jacobian-power detector silence as generatednessThe source records one nongenerated sextic order-two operator preserving every Jacobian power; it does not claim that all order-two operators on that germ are silent. A nonzero detector still supplies a sufficient separator, while a zero detector can require another ideal or a higher-order construction.
Narrowed routeExtending automatic order-two lifting to every orderThe source records cubic higher-order failures and a family with arbitrarily large first escape order, so neither universal automatic lifting nor a universal fixed escape order follows. Order-two compatible lifts remain available in their proved source scope; higher-order candidates require their own compatibility and full-word separation arguments.

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 bridgeEstablish the missing filtered presentation for the actual logarithmic fields underlying the newest ordinary-boundary full-ring claims.

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

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

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

Continue the mathematics

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Prepared starting pointEstablish the missing filtered presentation for the actual logarithmic fields underlying the newest ordinary-boundary full-ring claims.

Nakai conjecture · ready to start

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Research contextPrepared context for any AI agent

Can every singular finitely generated commutative algebra in characteristic zero be detected by a differential operator that cannot be built from multiplication and derivations?

Exact question and boundaries
Current routes and known obstacles
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Sources and references3 cited works · next context review by Dec 7, 2026

The mathematical context was checked on Sep 7, 2026. Status can be refreshed sooner after a material result or claim.

1
The Nakai Conjecture for isolated hypersurface singularities of modality ≤ 2preprint · Rui Li, Zida Xiao, Huaiqing Zuo · 2025-02-07 · ARXIV 2502.04672 · accessed Sep 7, 2026
2
Nakai conjectures for isolated weighted homogeneous hypersurface singularitiespreprint · Stephen S.-T. Yau, Qiwei Zhu, Huaiqing Zuo · 2026-08-19 · ARXIV 2604.24508v3 · accessed Sep 7, 2026
3
Differential operators on homogeneous plane curve singularitiespeer reviewed result · Julien Sebag · Journal of Symbolic Computation · 2026 · DOI 10.1016/j.jsc.2026.102556 · accessed Sep 7, 2026

Important qualifications

The bounded research did not establish an original proposal date or original-publication binding; proposer and year fields remain unknown.
The inspected 2026 manuscript is version 3 and a preprint. Its weighted homogeneous hypersurface result is not a resolution for arbitrary finite-type characteristic-zero algebras.
The Sebag publisher page returned HTTP 403 on direct opening; its indexed publisher abstract and bibliographic metadata were available and inspected.
No exact formalization or computation was established or executed. This private draft grants no mathematical acceptance, rights, authorship, contributor credit, or publication authority.

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