Claims
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KindShowingImportancePrizesTopicAlpöge's ℂ³ map
The polynomial map (a,b,c) on ℂ³ presented by Alpöge, with constant Jacobian determinant −2. Claims about its generic mapping degree, injectivity, reduction modulo primes, or its status as a putative Jacobian conjecture counterexample belong here; general Jacobian conjecture claims that do not concern this construction do not.
Non-invertible constant-Jacobian polynomial self-maps of ℂ³ exist in every geometric degree at least three
SupportedThe evidence favors the claim, but the examination is incomplete or the evidence is indirect.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Jacobian 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 →
The 2026 Jacobian conjecture counterexample introduces no essentially new techniques beyond Vitushkin's 1999 work
DoubtfulThe evidence weighs against the claim, but the examination is incomplete or the evidence is indirect.constitution →evaluativeA judgment of worth or quality against some standard: good, fair, effective.constitution →Jacobian 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 →
Vitushkin's construction method faces an obstruction in two dimensions that the three-dimensional polynomial counterexample circumvents
SupportedThe evidence favors the claim, but the examination is incomplete or the evidence is indirect.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Algebraic geometryJacobian conjectureVitushkin rational constructionimportance · 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 →
Alpöge's polynomial map on ℂ³, whose Jacobian determinant is the constant −2, has no polynomial inverse.
VerifiedThe evidence, examined directly, establishes 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 geometryJacobian 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 →
Alpöge's map (a,b,c) on ℂ³ has constant Jacobian determinant −2 yet is generically three-to-one
VerifiedThe evidence, examined directly, establishes 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 geometryJacobian 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 2026 Jacobian conjecture counterexample closely parallels Vitushkin's 1999 rational construction lifted to three dimensions
VerifiedThe evidence, examined directly, establishes the claim.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Jacobian conjectureAlgebraic geometryVitushkin rational 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 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 →Algebraic geometryJacobian 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 →
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 verificationAlgebraic geometryJacobian 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 →Algebraic geometryJacobian 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 →Algebraic geometryJacobian 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 P¹×P^{d−1} construction yields an affine-space source only in the case d equals three.
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 · 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 →
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