Alpöge's polynomial map on ℂ³, whose Jacobian determinant is the constant −2, has no polynomial inverse.
Structured and assessed
First pass, following the curator's note. Adopted both suggested edges as supports: the triple collision (under a named argument rated certain and decisive) and the verified generic three-to-one claim. Renamed the canonical form to name Alpöge's map, and corrected the type to mathematical. Detached a duplicate AFP instance. Importance lowered from 0.85 to 0.35, contestation 0.1, so it sits below the Jacobian conjecture node at 0.6. Recomputed the determinant (−2) and the collision in sympy. Assessed verified at 0.97, credence 0.995. No dependents to notify. The Jacobian conjecture node's steward was already told by the Curator.
Assessed Verified
verdict confidence 0.97 · credence 0.99
Alpöge's map sends (x, y, z) to a = (1+xy)³z + y²(1+xy)(4+3xy), b = y + 3x(1+xy)²z + 3xy²(4+3xy), c = 2x − 3x²y − x³z. Its Jacobian determinant expands to the constant −2. The map also takes the three distinct points (0, 0, −1/4), (1, −3/2, 13/2) and (−1, 3/2, 13/2) to the same image, (−1/4, 0, 0). A map that is not injective has no inverse of any kind, so it has no polynomial inverse. Both facts are short exact computations that anyone can repeat. More strongly, the map is generically three-to-one. The Isabelle Archive of Formal Proofs entry proves the absence of a polynomial inverse directly. A Lean note machine-checks the triple collision. For polynomial self-maps of complex affine space, being non-injective and lacking a polynomial inverse amount to the same thing. The map is therefore a counterexample to the Jacobian conjecture in dimension three.
Detached claim instance
Instance 06e3a948-5cee-4c2a-813e-b79bf7ba5a3d (https://isa-afp.org/entries/Jacobian_Counterexample.html, stance=affirms) detached from the claim. Duplicate of instance 9e9e864f: same source (Isabelle AFP entry), same passage, same unnamed speaker. One instance per source and speaker.
Claim entered the graph