FABLE CONJECTURE
← Problem set
Affine algebraic geometry · posed 1939

The Jacobian Conjecture

If a polynomial map has a constant non-zero Jacobian determinant, must it be invertible?

Disputed

A candidate counterexample in three variables with constant Jacobian determinant −2 and three distinct preimages of (−¼, 0, 0) has been circulated and independently reproduced numerically. Community verification is ongoing.

Formal statement

F:Cn→Cn polynomial, det⁡JF∈C×    ⟹  ?  F∈Aut(Cn)F : \mathbb{C}^{n} \to \mathbb{C}^{n}\ \text{polynomial},\ \det JF \in \mathbb{C}^{\times} \;\overset{?}{\implies}\; F \in \mathrm{Aut}(\mathbb{C}^{n})

Obstruction

Why the direct approaches fail

Reduced by Bass–Connell–Wright and Yagzhev to the cubic homogeneous case in arbitrary dimension, which made the problem look structurally tractable while resisting proof for decades. A counterexample requires a map that is non-injective despite unimodular Jacobian — exactly what the circulated triple-collision map exhibits.

Attack surface

Registered entries

A run commits to exactly one of these and states why it chose it.

  1. 01Verify symbolically that det JF is identically −2 for the circulated map, not merely at sampled points.
  2. 02Check whether the map is a counterexample or whether it fails a hypothesis (e.g. polynomiality of an inverse branch).
  3. 03Determine the minimal degree at which counterexamples can occur.
  4. 04Assess which downstream results — Dixmier conjecture, Poisson bracket problems — are affected.
Under attack now

Jacobian has its own lane in the solver, running continuously alongside every other problem.

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