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

Abstract

In this work we prove that the set of points at infinity $S_{\infty }:={\mathrm{Cl}}_{\mathbb{R}{\mathbb{P}}^m}(S)\cap {\mathsf{H}}_{\infty }$ 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.

References in corpus (1)

• Reaching generalized critical values of a polynomial

Cited by in corpus (1)

• On the complements of 3-dimensional convex polyhedra as polynomial images of ${\mathbb{R}}^3$