  Let $\K=\C$, or $\R$, and $S_f$ be the set of points in $\K^n$ at which a
polynomial map $f:\K^n\rightarrow\K^n$ is non-proper. Jelonek proved that $S_f$
is a semi-algebraic set that is ruled by polynomial curves, with $\dim S_f\leq
n-1$, and provided a method to compute $S_f$ for $\K = \C$. However, such
methods do not exist for $\K = \R$.
  In this paper, we establish a straightforward description of $S_f$ for a
large family of non-proper maps $f$ using the Newton polytopes of the
polynomials appearing in $f$. Thus resulting in a new method for computing
$S_f$ that works for $\K=\R$, and highlights an interplay between the geometry
of polytopes and that of $S_f$. As an application, we recover some of Jelonek's
results, and provide conditions on (non-)properness of $f$. Moreover, we
discover another large family of maps $f$ whose $S_f$ has dimension $n-1$ (even
for $\K=\R$), satisfies an explicit stratification, and weak smoothness
properties. This novel description allows our tools to be extended to all
non-proper maps.

Algebraic Geometry;Combinatorics