Let $F \colon \mathbb{C}^n \to \mathbb{C}^n$ be a polynomial map, i.e., $$F(x_1,\dots,x_n) = (f_1(x_1,\dots,x_n), \dots,f_n(x_1,\dots,x_n))$$ for certain polynomials $f_i \in \mathbb{C}[X_1,\dots,X_n]$

If $F$ is invertible, then its Jacobi determinant $\det(\partial f_i/\partial x_j)$ which is a polynomial over $\mathbb{C}$ vanishes nowhere and hence must be a non-zero constant.

The Jacobian conjecture asserts the converse: every polynomial map $\mathbb{C}^n \to \mathbb{C}^n$ whose Jacobi determinant is a non-zero constant is invertible.