Fix a polynomial mapping $\mathbb R^n\overset{f}{\to} \mathbb R$. This answer shows that if the top degree homogeneous component of $f$ is zero only at the origin, then $f$ is proper. Intuitively, this condition eventually forces the fibers to be "closed contours" about the origin.

Can this be generalized to a condition on the coefficients of a power series to ensure properness of an *analytic mapping* $\mathbb R^n\overset{f}{\to} \mathbb R$?