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

Abstract

We prove that the complement $\mathcal{S}:={\mathbb{R}}^3\setminus \mathcal{K}$ of a 3-dimensional convex polyhedron $\mathcal{K}\subset {\mathbb{R}}^3$ and its closure $\overline{\mathcal{S}}$ are polynomial images of ${\mathbb{R}}^3$. The former techniques cannot be extended in general to represent such semialgebraic sets $\mathcal{S}$ and $\overline{\mathcal{S}}$ as polynomial images of ${\mathbb{R}}^n$ if $n\ge 4$.

arXiv comment12 pages, 1 figure

References in corpus (2)

• Unbounded convex polygons as polynomial images of the plane
• On the set of points at infinity of a polynomial image of ${\mathbb{R}}^n$