On the set of points at infinity of a polynomial image of ${\mathbb R}^n$

TL;DR

This paper proves that the set of points at infinity of a polynomial image of Euclidean space is connected, extending known results and identifying conditions under which this property holds.

Contribution

It establishes the connectedness of points at infinity for polynomial images and introduces quasi-polynomial maps where this property persists.

Findings

Connectedness of points at infinity for polynomial images.

Extension of the result to quasi-polynomial maps.

Counterexamples for regular maps in general.

Abstract

In this work we prove that the set of points at infinity $S_\infty:={\rm Cl}_{{\mathbb R}{\mathbb P}^m}(S)\cap\mathsf{H}_\infty$ of a semialgebraic set $S\subset{\mathbb R}^m$ which is the image of a polynomial map $f:{\mathbb R}^n\to{\mathbb R}^m$ is connected. This result is no further true in general if $f$ is a regular map, although it still works for a large family of regular maps that we call quasi-polynomial maps.