Jacobian Conjecture Lab

Why the 1939 conjecture fell in 2026 — drag a polynomial map and feel its Jacobian field

P(x,y) =

Q(x,y) =

J =

det(J) =

Preset morph: Drużkowski

Inspector

Status

constant Jacobian, invertible everywhere — 2D safe (Drużkowski, 1983).

Diagnostics

How to use

Dragamber handle

Slider5 presets

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Alpöge's C³ Counterexample (2026)

The 1939 Jacobian Conjecture held in 2D but fell in 3D. Levent Alpöge posted this exact polynomial map on July 19, 2026 — it has constant Jacobian 1 yet is not invertible.

First component: (1+xy)³z + y²(1+xy)(4+3xy)
Second component: y + 3x(1+xy)²z + 3xy²(4+3xy)
Third component: 2x − 3x²y − x³z

A 2×2×2 cube becomes tangled under Alpöge's map — the conjecture fails above dimension 2.