The Dixmier conjecture, stated by Jacques Dixmier in 1968, originally asked whether any endomorphism of the first Weyl algebra over a field of characteristic zero is an automorphism. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementEvery endomorphism of the n-th Weyl algebra over a field of characteristic zero is an automorphism; Dixmier originally asked this for the first Weyl algebra. 1 Proposed Year Progress Toward ResolutionStably equivalent to the Jacobian conjecture. Since a counterexample to the Jacobian conjecture in three variables was found in July 2026, the Dixmier conjecture is false for the n-th Weyl algebra for every n of at least 3. It remains open for the first and second Weyl algebras, since the Jacobian conjecture is still open in two variables. 1 Classification
Resolution Status Prize Status
Prize Status (category) Sources
1. Dixmier Conjecture (Wikipedia)
Wikimedia FoundationLead section
stated by Jacques Dixmier in 1968
Lead section, statement sentence
originally asked whether any endomorphism of the first Weyl algebra
Current status paragraph
The conjecture remains open for the first and second Weyl algebras, since the Jacobian conjecture is still open in two variables
Generalization section
The analogous statement for the n-th Weyl algebra
Equivalence with the Jacobian conjecture paragraph
showed that the Dixmier conjecture is stably equivalent to the Jacobian conjecture
Recent developments paragraph
In July 2026, a counterexample to the Jacobian conjecture in three variables was found
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