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

Abstract

Abstract 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 $$\mathbb {K}^2$$ K 2 , where a dominant polynomial map $$f: \mathbb {K}^2 \rightarrow \mathbb {K}^2$$ f : K 2 → K 2 is not proper; $$\mathbb {K}$$ K could be either $$\mathbb {C}$$ C or $$\mathbb {R}$$ R . Unlike all the previously known approaches we make no assumptions on f whenever $$\mathbb {K} = \mathbb {R}$$ 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 .