Dixmier Conjecture

The Dixmier conjecture (Dixmier 1968) asserts that every algebra endomorphism of a Weyl algebra over a field of field characteristic 0 is an automorphism. In the differential-operator realization, an endomorphism chooses new polynomial differential operators obeying the same commutation relations. The conjecture requires that these operators generate the whole algebra.

The all-dimensional conjecture is equivalent to the all-dimensional Jacobian conjecture, although the reductions can change the dimension (Adjamagbo and van den Essen 2007). It is also closely related to the Poisson conjecture. In the same dimension over a field of field characteristic 0, the Dixmier conjecture implies the Jacobian conjecture (Adjamagbo and van den Essen 2007).

The three-variable Jacobian conjecture counterexample discussed by Speyer (2026) also gives a nonautomorphic endomorphism of . If is that polynomial map, its constant nonzero Jacobian makes a matrix with polynomial entries. The resulting derivations with commute, so sending the generators to and preserves the Weyl algebra relations. The image contains a multiplication operator only when its coefficient is in , which is a proper subring because is not injective. Thus the conjecture fails for and, by adjoining unchanged generators, for every .

Blore (2026b) formalized the same-dimension implication and the refutation for every over any field of field characteristic 0, using the three-variable counterexample and adjoining unchanged generators. The formalization was developed with Claude Code assistance. Palomar Registry (2026b) records acceptance by the Lean, nanoda, and con-ron kernels.

Long (2026) reports an explicit nonautomorphic endomorphism of in an appendix to the rank-two Poisson counterexample. The construction sends the four coordinate generators to multiplication by the four output polynomials , , , and of the Poisson map. Writing for its Poisson bracket, the corresponding derivative generators are sent to the derivations , , , and . The Poisson bracket identities imply that these eight differential operators satisfy the Weyl algebra relations. Thus counts four multiplication-derivation pairs, although the underlying Poisson map has two canonical pairs. The proof uses the differential-order filtration to show that surjectivity would make the original polynomial map invertible. The rank-three construction above already implies failure in rank 4.

Long's construction was developed with AI assistance and checked by its author. Blore's (2026a) Lean proof concerns the Poisson refutation (Palomar Registry 2026a). Long's specific fourth Weyl algebra construction is separate from both formalizations. Ranks 1 and 2 remain open.