- The Jacobian conjecture asserts that a polynomial map with non-zero constant Jacobian is globally invertible.
- A counterexample in three dimensions (and higher) has been found, disproving the conjecture for dimensions ≥3.
- The counterexample is constructed using a multiplication map of homogeneous polynomials in two variables.
- Local injectivity is achieved by normalizing via resultants and using the PSL(2,C) action.
- Global non-injectivity arises from the fact that a cubic polynomial can factor into linear and quadratic parts in three ways.
- A three-dimensional slice of a four-dimensional variety is shown to be isomorphic to affine space, yielding the desired polynomial map.