Complements of unbounded convex polyhedra as polynomial images of \({{\mathbb{R}}}^n\) (Q2316794)

In this paper, the authors address the question of which semialgebraic subsets of \(\mathbb{R}^n\) are polynomial (or regular) images of \(\mathbb{R}^n\). Namely, they prove constructively the following results: Theorem 1.4. Let \(n \ge 1\) and let \(K\) be an \(n\)-dimensional unbounded convex polyhedron in \(\mathbb{R}^n\) that is not a layer. Then the semialgebraic set \( \mathbb{R}^n \setminus \mathrm{Int}(K)\) is a polynomial image of \(\mathbb{R}^n\), where \(\mathrm{Int}(K)\) stands for the interior of \(K\). Theorem 1.5. Let \(n \ge 2\) and let \(K\) be an \(n\)-dimensional unbounded convex polyhedron in \(\mathbb{R}^n\) that is not a layer. Then the semialgebraic set \( \mathbb{R}^n \setminus K\) is a polynomial image of \(\mathbb{R}^n\). (A \textit{layer} is a convex polyhedron of \(\mathbb{R}^n\) affinely equivalent to \([-a, a] \times \mathbb{R}^{n - 1}\) with \(a > 0\)). The paper is very clear, well written and quite interesting.