Computing the Non-properness Set of Real Polynomial Maps in the Plane

We introduce novel mathematical and computational tools to develop a complete algorithm for computing the set of non-properness of polynomials maps in the plane. In particular, this set, which we call the Jelonek set , is a subset of 𝕂^2 , where a dominant polynomial map f: 𝕂^2 →𝕂^2 is not proper; 𝕂 could be either ℂ or ℝ . Unlike all the previously known approaches we make no assumptions on f whenever 𝕂 = ℝ ; this is the first algorithm with this property. The algorithm takes into account the Newton polytopes of the polynomials. As a byproduct we provide a finer representation of the set of non-properness as a union of semi-algebraic curves, that correspond to edges of the Newton polytopes, which is of independent interest. Finally, we present a precise Boolean complexity analysis of the algorithm and a prototype implementation in maple .