2021/01/13 by Boulos El Hilany, Hilany, Boulos El, Elias Tsigaridas +1
Mathematics · Computer Science · #Advanced Differential Equations and Dynamical Systems #Polynomial and algebraic computation #Commutative Algebra and Its Applications
paper · pdf · doi:10.48550/arxiv.2101.05245
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 \mathbbK2 where a dominant polynomial map f:\mathbbK2→\mathbbK2 is not proper; \mathbbK could be either ℂ or ℝ. Unlike all the previously known approaches we make no assumptions on f whenever \mathbbK = 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.