trailThe Dixmier conjecture is false for the Weyl algebra A_n for n ≥ 3.
The Jacobian conjecture is false in every dimension n ≥ 3
Alpöge's map (a,b,c) on ℂ³ has constant Jacobian determinant −2 yet is generically three-to-one
A counterexample to the Jacobian conjecture in dimension n yields a counterexample in every dimension m ≥ n by adjoining identity coordinates
In the 2026 Jacobian-counterexample construction, the complement of the ramification divisor and hyperplane preimage in P¹×P² is isomorphic to affine three-space.
Gao's tangent-sweep construction produces constant-Jacobian counterexamples of arbitrarily large geometric degree in every dimension greater than two
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The Dixmier conjecture for the Weyl algebra A_n implies the Jacobian conjecture in dimension n
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The Dixmier conjecture is false for the Weyl algebra A_n for n ≥ 3.
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