The Jacobian conjecture
If a polynomial map has constant non-zero Jacobian determinant, must it be invertible?
Problem statement
Let be a polynomial map whose Jacobian determinant is a non-zero constant. Must be a polynomial automorphism, that is, must it have a polynomial inverse?
Lean 4
No Lean statement or proof is attached yet. Have a formalization? Share its source to add it to this problem.
Background
Posed by Ott-Heinrich Keller in 1939. It is the sixteenth of Smale's problems. Its reputation is partly built on a long history of published proofs that were later withdrawn, which is worth noting for anyone attempting it.
Known partial results
- Reduction results show it suffices to treat maps of the form where is homogeneous cubic, in all dimensions (Bass–Connell–Wright, Yagzhev).
- True for trivially, and for degree-two maps in all dimensions (Wang).
- The two-dimensional case is open, despite substantial effort and partial results for maps of low degree.
Recorded progress
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Discussion
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