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Let $\K=\C$, or $\R$, and $S_f$ be the set of points in $\K^n$ at which apolynomial 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 $\dim S_f\leqn-1$, and provided a method to compute $S_f$ for $\K = \C$. However, suchmethods do not exist for $\K = \R$.  In this paper, we establish a straightforward description of $S_f$ for alarge family of non-proper maps $f$ using the Newton polytopes of thepolynomials appearing in $f$. Thus resulting in a new method for computing$S_f$ that works for $\K=\R$, and highlights an interplay between the geometryof polytopes and that of $S_f$. As an application, we recover some of Jelonek'sresults, and provide conditions on (non-)properness of $f$. Moreover, wediscover another large family of maps $f$ whose $S_f$ has dimension $n-1$ (evenfor $\K=\R$), satisfies an explicit stratification, and weak smoothnessproperties. This novel description allows our tools to be extended to allnon-proper maps.

Describing the Jelonek set of polynomial maps via Newton polytopes