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ShowingImportancePrizesTopicAlgebraic geometry
The study of solution sets of polynomial equations — affine and projective varieties, polynomial and regular maps, and automorphisms of polynomial rings — including affine algebraic geometry questions such as the structure of polynomial coordinate changes. Broad field tag; specific conjectures and named objects get their own tags alongside it, and complex-analytic or noncommutative-algebra claims
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 →Jacobian conjectureimportance · 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 →Jacobian conjectureGao 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 conjectureJacobian conjectureimportance · 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 →Jacobian conjectureVitushkin 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 →Jacobian conjectureimportance · 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 →
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 →Jacobian conjectureAI 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 →
Mochizuki's Key Principle of Inter-Universality requires arbitrary geometric base-points for tempered fundamental groups, which only algebraically closed perfectoid fields supply.
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 →Inter-universal Teichmüller theoryNumber theoryTempered fundamental groupsimportance · 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 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 →Jacobian conjectureDixmier conjectureimportance · 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 →Jacobian conjectureSeparable 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 →
The separable Jacobian conjecture holds: every étale polynomial self-map of affine space in positive characteristic with generic degree prime to the characteristic is an automorphism.
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 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 →
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 →Jacobian conjectureSeparable 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 →
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 →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 →
A polynomial map in positive characteristic with nonzero Jacobian determinant induces a separable algebraic extension of rational function fields.
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 function field extensionsimportance · settledImportance 0.12, 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 →Jacobian conjectureSeparable 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 constructionJacobian conjectureimportance · 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 mapJacobian 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 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 mapJacobian conjectureimportance · 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 →Jacobian conjectureimportance · 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 →Jacobian conjectureAlpöge'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 →Jacobian conjectureimportance · 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 →Poisson conjectureJacobian conjectureDixmier 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 →
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 →Alpöge's ℂ³ mapFormal proof verificationJacobian conjectureimportance · 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 →Alpöge's ℂ³ mapJacobian 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 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 mapJacobian 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 →
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 →Jacobian conjectureVitushkin 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 conjectureJacobian conjectureimportance · 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 →Jacobian conjectureimportance · 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 →Jacobian conjectureimportance · 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 →Jacobian conjectureimportance · 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 →Jacobian conjectureimportance · 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 →Jacobian conjecture2026 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 →Jacobian conjectureAlpöge'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 →
The monomial-twist side conditions in Gao's tangent-sweep construction are solvable for sweep data of arbitrarily high degree in every dimension at least four
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's 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 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 →Jacobian conjectureHessian conjectureimportance · 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 →Jacobian conjectureAlpöge'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 →
The Hessian conjecture is false in dimension five
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 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 →
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 →Jacobian conjectureimportance · 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 →Jacobian conjectureimportance · 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 →
Every finite flat group scheme of order n is annihilated by 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 →Group schemesimportance · 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 three-dimensional Jacobian conjecture counterexample is homogeneous for a mixed-sign integer grading of its coordinates
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 constructionJacobian conjectureimportance · settledImportance 0.12, 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 →
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