On complements of convex polyhedra as polynomial images of ${\mathbb R}^n$

TL;DR

This paper proves constructively that the complements of certain unbounded convex polyhedra and their interiors are polynomial images of Euclidean space, extending previous results to the non-compact case with new techniques.

Contribution

It provides a full characterization of complements of convex polyhedra as polynomial images of R^n, including unbounded cases, using advanced rational separation methods.

Findings

Complements of unbounded convex polyhedra are polynomial images of R^n.

Complements of their interiors are also polynomial images of R^n.

The results extend previous work on compact and small-dimensional cases.

Abstract

In this work we prove constructively that the complement ${\mathbb R}^n\setminus{\mathcal K}$ of an $n$-dimensional unbounded convex polyhedron ${\mathcal K}\subset{\mathbb R}^n$ and the complement ${\mathbb R}^n\setminus{\rm Int}({\mathcal K})$ of its interior are polynomial images of ${\mathbb R}^n$ whenever ${\mathcal K}$ does not disconnect ${\mathbb R}^n$. The compact case and the case of convex polyhedra of small dimension were approached by the authors in previous works. Consequently, the results of this article provide a full answer to the representation as polynomial images of Euclidean spaces of complements of convex polyhedra and its interiors. The techniques here are more sophisticated than those corresponding to the compact case and require a rational separation result for certain type of (non-compact) semialgebraic sets, that has interest by its own.