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We introduce novel mathematical and computational tools to develop a completealgorithm for computing the set of non-properness of polynomials maps in theplane. In particular, this set, which we call \emph{the Jelonek set}, is asubset of $\mathbb{K}^2$ where a dominant polynomial map$f:\mathbb{K}^2\to\mathbb{K}^2$ is not proper; $\mathbb{K}$ could be either$\mathbb{C}$ or $\mathbb{R}$. Unlike all the previously known approaches wemake no assumptions on $f$ whenever $\mathbb{K} = \mathbb{R}$; this is thefirst algorithm with this property. The algorithm takes into account the Newtonpolytopes of the polynomials. As a byproduct we provide a finer representationof the set of non-properness as a union of semi-algebraic curves, thatcorrespond to edges of the Newton polytopes, which is of independent interest.Finally, we present a precise Boolean complexity analysis of the algorithm anda prototype implementation in Maple.

Computing the non-properness set of real polynomial maps in the plane