Computing efficiently the non-properness set of polynomial maps on the plane∗

We introduce novel mathematical and computational tools to develop a complete and efficient algorithm for computing the set of non-properness of polynomial maps on the plane. This is a subset of K where a dominant polynomial map f : K → K is not proper; K could be either C or R. Unlike previously known approaches we make no assumptions on f . The algorithm takes into account the sparsity of polynomials as it depends on (the Minkowski sum of) the Newton polytopes of f . As a byproduct we provide a finer representation of the set of non-properness as a union of algebraic or semi-algebraic sets, that correspond to edges of the Newton polytope, which is of independent interest. Finally, we present a precise bit complexity analysis of the algorithm and a prototype implementation in maple.

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