Let be a polynomial map, i.e.,
If is invertible, then its Jacobi determinant , which is a polynomial over , vanishes nowhere and hence must be a non-zero constant.
The Jacobian conjecture asserts the converse: every polynomial map whose Jacobi determinant is a non-zero constant is invertible.
![$ f_i \in \mathbb{C}[X_1,\dots,X_n]$](http://images.planetmath.org:8080/cache/objects/3935/l2h/img3.png "$ f_i \in \mathbb{C}[X_1,\dots,X_n]$"){kind=small}
