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.

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 .

Long (2026) reports an explicit nonautomorphic endomorphism of in an appendix to the rank-two Poisson counterexample. This separate construction was developed with AI assistance and checked by its author. These constructions do not settle ranks 1 and 2.