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ShowingImportancePrizesTopicSeparable Jacobian conjecture
The variant of the Jacobian conjecture asking whether a separable polynomial map over a field whose Jacobian determinant is a nonzero constant has a polynomial inverse (the Keller conjecture fails for inseparable maps in positive characteristic). Claims about its truth in given dimensions or characteristics, counterexamples (e.g. Mondello's in characteristic two), and proofs belong here; the class
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 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 →
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 →Algebraic 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 →
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 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 →
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 conjectureAlgebraic geometryimportance · 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 →
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 →Algebraic geometryJacobian 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 →
Huq-Kuruvilla's map (x+x²y, y+xz+x²yz, z+x²z²) in characteristic two has Jacobian determinant 1 and generic degree 3 yet 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 →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Algebraic geometryHuq–Kuruvilla mapJacobian 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 →
Mondello's plane map (P,Q) over the algebraic closure of F₂ has Jacobian determinant 1 and generic degree 3 yet 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 →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.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 →
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