In one dimension, a cubic polynomial mapping $\Bbb{R}$ to $\Bbb{R}$
$$y = A + Bx + Cx^2 + Dx^3$$
is one-to-one and onto when its derivative $y'(x) = B + 2Cx + 3Dx^2$ has less than two zeros, i.e., $4C^2 - 12BD \leq 0$. Then $y(x)$ is monotonic almost everywhere, one-to-one, and onto.
How do things work for $\Bbb{R^n}$ to $\Bbb{R^n}$? Suppose (after some coordinate transformation)
$$y_i = x_i + \sum_{jk} A_{ijk} x_j x_k + \sum_{jkm} B_{ijkm} x_j x_k x_m$$
It's easy to take the Jacobian, but what is the precise condition for $y(x)$ to be one-to-one and onto from $\Bbb{R^n}$ to $\Bbb{R^n}$?
I have a guess, but I imagine this is a pretty well-known result in algebraic geometry. A source would also be appreciated.