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

TL;DR

This paper presents a new algorithm for computing the Jelonek set, the set of non-proper points of polynomial maps in the plane, applicable over real and complex fields without restrictions.

Contribution

It introduces a novel, assumption-free algorithm for the Jelonek set in the real case, utilizing Newton polytopes and providing a detailed complexity analysis.

Findings

Algorithm computes the Jelonek set for polynomial maps in the plane.

Provides a semi-algebraic curve decomposition of the non-properness set.

Includes a prototype implementation and complexity analysis.

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 \emph{the Jelonek set}, is a subset 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 we make no assumptions on $f$ whenever $\mathbb{K} = \mathbb{R}$; 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.