Jacobian conjecture

The **Jacobian Conjecture**: any regular function (i.e. vector valued polynomial function from) kⁿ → kᵐ whose Jacobian is a non-zero constant has an inverse that is given by a regular function, where k is a field of characteristic 0.

This is false: F has Jacobian determinant 1 but identifies two distinct points, so it admits no inverse. This counterexample works in all characteristics.