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ShowingImportancePrizesTopicJacobian conjecture
A conjecture in algebraic geometry, due to Ott-Heinrich Keller, concerning whether a polynomial map from ℂⁿ to ℂⁿ with a non-zero constant Jacobian determinant has a polynomial inverse. Claims about its truth, falsity, or proof belong here.
No known construction converts the three-dimensional Jacobian counterexamples into a counterexample over the complex plane
VerifiedThe claim traces to reliable primary sources through a clear chain of evidence.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Algebraic geometryimportance · minorImportance 0.30, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
Gao's tangent-sweep construction produces constant-Jacobian counterexamples of arbitrarily large geometric degree in every dimension greater than two
SupportedEvidence favors the claim, but the chain is incomplete or the sources are secondary.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Algebraic geometryGao tangent-sweep constructionimportance · minorImportance 0.30, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The Hessian conjecture in dimension 2n implies the Jacobian conjecture in dimension n
VerifiedThe claim traces to reliable primary sources through a clear chain of evidence.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Hessian conjectureAlgebraic geometryimportance · minorImportance 0.30, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The 2026 Jacobian conjecture counterexample closely parallels Vitushkin's 1999 rational construction lifted to three dimensions
VerifiedThe claim traces to reliable primary sources through a clear chain of evidence.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Algebraic geometryVitushkin rational constructionAlpöge's ℂ³ mapimportance · minorImportance 0.30, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The separable Jacobian conjecture is false in dimension three in characteristic two.
VerifiedThe claim traces to reliable primary sources through a clear chain of evidence.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Algebraic geometryimportance · minorImportance 0.25, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
Claude Fable 5 generated the polynomial map used as the 2026 Jacobian conjecture counterexample
SupportedEvidence favors the claim, but the chain is incomplete or the sources are secondary.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →AI mathematical discoveryAI discovery creditimportance · minorImportance 0.35, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
An AI system produced work leading to a counterexample to the Jacobian conjecture.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · verifiableA factual claim that could be checked directly against observation or primary records.constitution →AI mathematical discoveryimportance · notableImportance 0.45, from 0 to 1 · notable: a contested point in a live debate (also the default before judging). Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The 2026 Jacobian conjecture counterexample could have been obtained from Claude Fable 5 with a single simple prompt
ContestedCredible evidence or argument exists on multiple sides.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →AI mathematical discoveryimportance · minorImportance 0.30, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
Levent Alpöge defined and steered the 2026 Jacobian counterexample search, with Claude Fable 5 generating candidates
ContestedCredible evidence or argument exists on multiple sides.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →AI mathematical discoveryAI discovery creditimportance · minorImportance 0.40, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The 2026 counterexample to the Jacobian conjecture was discovered primarily by an AI system
ContestedCredible evidence or argument exists on multiple sides.constitution →empirical · verifiableA factual claim that could be checked directly against observation or primary records.constitution →Algebraic geometryAI mathematical discoveryAI discovery creditimportance · notableImportance 0.55, from 0 to 1 · notable: a contested point in a live debate (also the default before judging). Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
An internal OpenAI Codex model independently rediscovered the 2026 Jacobian conjecture counterexample.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →2026 Jacobian counterexample constructionAI mathematical discoveryOpenAIimportance · minorImportance 0.30, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
Stable equivalence does not directly transfer a dimension-3 Jacobian counterexample to any Dixmier rank.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Dixmier conjecturePoisson conjectureimportance · minorImportance 0.40, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The Jacobian conjecture in dimension 2n implies the Dixmier conjecture in rank n.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Dixmier conjectureAlgebraic geometryimportance · minorImportance 0.35, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
In positive characteristic the Jacobian conjecture is standardly formulated as: étale polynomial self-maps of affine space with generic degree prime to the characteristic are automorphisms.
VerifiedThe claim traces to reliable primary sources through a clear chain of evidence.constitution →definitionalA claim about what a term means. It earns a node only when the definition itself is disputed.constitution →Algebraic geometrySeparable Jacobian conjectureimportance · settledImportance 0.20, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
Adjamagbo's separable Jacobian conjecture in positive characteristic implies the classical Jacobian conjecture in characteristic zero.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Separable Jacobian conjectureAlgebraic geometryimportance · settledImportance 0.20, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
Positive-characteristic formulations of the Jacobian conjecture other than Adjamagbo's degree-prime-to-p version have been proposed.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · verifiableA factual claim that could be checked directly against observation or primary records.constitution →Algebraic geometryimportance · settledImportance 0.20, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The naive Jacobian conjecture fails in positive characteristic: the polynomial x − x^p has derivative 1 but is not injective.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Algebraic geometrySeparable Jacobian conjectureimportance · settledImportance 0.10, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The 2026 Jacobian conjecture counterexample has been formally verified in a proof assistant
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Formal proof verification2026 Jacobian counterexample constructionAlgebraic geometryimportance · settledImportance 0.15, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
Alpöge's map (a,b,c) on ℂ³ has constant Jacobian determinant −2 yet is generically three-to-one
VerifiedThe claim traces to reliable primary sources through a clear chain of evidence.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Alpöge's ℂ³ mapAlgebraic geometryimportance · minorImportance 0.40, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The Alpöge–Fable polynomial map F:ℂ³→ℂ³ has constant Jacobian determinant −2 yet sends three distinct points to a common image.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Alpöge's ℂ³ mapAlgebraic geometryimportance · majorImportance 0.70, from 0 to 1 · major: real consequence within a domain, actively argued. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The Jacobian conjecture is false in every dimension n ≥ 3
VerifiedThe claim traces to reliable primary sources through a clear chain of evidence.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Algebraic geometryimportance · notableImportance 0.60, from 0 to 1 · notable: a contested point in a live debate (also the default before judging). Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The explicit three-dimensional polynomial map with constant nonzero Jacobian determinant has no polynomial inverse.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Algebraic geometryAlpöge's ℂ³ mapimportance · centralImportance 0.85, from 0 to 1 · central: consequential and genuinely contested: a live crux. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
Every polynomial map of ℂⁿ with nonzero constant Jacobian determinant has a polynomial inverse
ContradictedAvailable evidence weighs against the claim.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Algebraic geometryimportance · notableImportance 0.60, from 0 to 1 · notable: a contested point in a live debate (also the default before judging). Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The Jacobian, Dixmier, and Poisson conjectures are stably equivalent.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Dixmier conjecturePoisson conjectureAlgebraic geometryimportance · minorImportance 0.40, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
A machine-checked Lean 4 formalization confirms the determinant and collision facts of the Alpöge–Fable counterexample.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Formal proof verificationAlpöge's ℂ³ mapAlgebraic geometryimportance · minorImportance 0.35, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The Alpöge–Fable map sends three distinct points to a common image, so it is not injective.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Algebraic geometryAlpöge's ℂ³ mapimportance · minorImportance 0.40, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The Alpöge–Fable map F:ℂ³→ℂ³ has Jacobian determinant identically equal to −2.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Alpöge's ℂ³ mapAlgebraic geometryimportance · minorImportance 0.40, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
Vitushkin's construction method faces an obstruction in two dimensions that the three-dimensional polynomial counterexample circumvents
SupportedEvidence favors the claim, but the chain is incomplete or the sources are secondary.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Algebraic geometryVitushkin rational constructionAlpöge's ℂ³ mapimportance · minorImportance 0.30, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
Mondello's counterexample to the two-dimensional separable Jacobian conjecture in characteristic two is correct.
VerifiedThe claim traces to reliable primary sources through a clear chain of evidence.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Separable Jacobian conjectureAlgebraic geometryimportance · minorImportance 0.25, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
Every polynomial map of ℂ² with nonzero constant Jacobian determinant has a polynomial inverse
ContestedCredible evidence or argument exists on multiple sides.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Algebraic geometryimportance · majorImportance 0.65, from 0 to 1 · major: real consequence within a domain, actively argued. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The two-dimensional Jacobian conjecture remains open
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Algebraic geometryimportance · notableImportance 0.50, from 0 to 1 · notable: a contested point in a live debate (also the default before judging). Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The separable Jacobian conjecture is false in dimension two in characteristic two.
VerifiedThe claim traces to reliable primary sources through a clear chain of evidence.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Algebraic geometryimportance · minorImportance 0.30, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
When Abhyankar and Moh introduced the term 'Jacobian conjecture' in the 1970s-80s, they referred only to the two-dimensional case.
SupportedEvidence favors the claim, but the chain is incomplete or the sources are secondary.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Algebraic geometryimportance · minorImportance 0.30, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
In the 2026 Jacobian-counterexample construction, the complement of the ramification divisor and hyperplane preimage in P¹×P² is isomorphic to affine three-space.
VerifiedThe claim traces to reliable primary sources through a clear chain of evidence.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Algebraic geometry2026 Jacobian counterexample constructionimportance · minorImportance 0.25, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
Non-invertible constant-Jacobian polynomial self-maps of ℂ³ exist in every geometric degree at least three
SupportedEvidence favors the claim, but the chain is incomplete or the sources are secondary.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Algebraic geometryAlpöge's ℂ³ mapimportance · minorImportance 0.35, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
Gao's explicit four- and five-dimensional Keller maps F4, F5, F6, F7 have constant Jacobian determinant and generic fiber sizes 5, 10, 6 and 12
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Gao tangent-sweep constructionimportance · settledImportance 0.20, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The four-dimensional Hessian conjecture implies the two-dimensional Jacobian conjecture
VerifiedThe claim traces to reliable primary sources through a clear chain of evidence.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Hessian conjectureAlgebraic geometryimportance · minorImportance 0.30, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The 2026 Jacobian conjecture counterexample introduces no essentially new techniques beyond Vitushkin's 1999 work
ContestedCredible evidence or argument exists on multiple sides.constitution →evaluativeA judgment of worth or quality against some standard: good, fair, effective.constitution →Alpöge's ℂ³ mapAlgebraic geometryimportance · minorImportance 0.35, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
A counterexample to the Jacobian conjecture in three variables was announced in July 2026 and independently verified
VerifiedThe claim traces to reliable primary sources through a clear chain of evidence.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Algebraic geometryimportance · minorImportance 0.30, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The Jacobian conjecture counterexample follows straightforwardly from non-injectivity of multiplying a linear and a quadratic homogeneous polynomial.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →mathematicalA proposition of mathematics: true or false by proof rather than by observation. Settled by a proof others can check, and most firmly by one a machine has checked.constitution →Algebraic geometryimportance · minorImportance 0.30, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
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