On the complements of 3-dimensional convex polyhedra as polynomial images of ℝ³

Abstract

Let [Formula: see text] be a convex polyhedron of dimension n. Denote [Formula: see text] and let [Formula: see text] be its closure. We prove that for n = 3 the semialgebraic sets [Formula: see text] and [Formula: see text] are polynomial images of ℝ 3 . The former techniques cannot be extended in general to represent the semialgebraic sets [Formula: see text] and [Formula: see text] as polynomial images of ℝ n if n ≥ 4.