Complements of Unbounded Convex Polyhedra as Polynomial Images of $${{\mathbb {R}}}^n$$ R n

Abstract

We prove constructively that: The complement $${{\mathbb {R}}}^n{\setminus }\mathcal {K}$$ of ann-dimensional unbounded convex polyhedron $$\mathcal {K}\subset {{\mathbb {R}}}^n$$ and the complement $${{\mathbb {R}}}^n{\setminus }{\text {Int}}(\mathcal {K})$$ of its interior are polynomial images of $${{\mathbb {R}}}^n$$ whenever $$\mathcal {K}$$ does not disconnect $${{\mathbb {R}}}^n$$ . The case of a compact convex polyhedron and the case of convex polyhedra of small dimension were approached by the authors in previous works. The techniques here are more sophisticated than those corresponding to the compact case and require rational separation results for tuples of variables, which have interest by their own and can be applied to separate certain types of (non-compact) semialgebraic sets.