On complements of convex polyhedra as polynomial and regular images of \(\mathbb{R}^{n}\)
A map \(f:\mathbb R^n\to\mathbb R^m\) is regular if there exist \(f_1,\dots,f_m, g\in\mathbb R[{\mathtt x}_1,\dots,{\mathtt x}_n]\) such that \(g^{-1}(0)\) is empty and for every point \(x\in\mathbb R^n\), NEWLINE\[NEWLINE f(x)=\Big(\frac{f_1(x)}{g(x)},\dots,\frac{f_m(x)}{g(x)}\Big). NEWLINE\]NEWLINE The map \(f\) is polynomial if \(g\) can be chosen to be constant. We are far from achieving a geometric characterization of the semialgebraic subsets \(\mathcal S\subset\mathbb R^m\) that are either a polynomial or a regular image of some \(\mathbb R^n\), that is, that can be represented as \(\mathcal S=f(\mathbb R^n)\) for some polynomial or regular map \(f:\mathbb R^n\to\mathbb R^m\). Motivation for such a characterization comes from the fact that important problems in real algebraic geometry involving semialgebraic sets \(\mathcal S\) (say, optimization, Positivstellensatz or computation of trajectories) can be reduced somehow to the case \(\mathcal S=\mathbb R^n\) if \(\mathcal S\) is either a polynomial or a regular image of an Euclidean space.NEWLINENEWLINEThe authors of the work under review have contributed significantly in the last years to answer this question for a subclass of the class of semialgebraic sets with piecewise linear boundary. Let \(\mathcal P\subset\mathbb R^n\) be a convex polyhedron and denote \({\text{Int}}(\mathcal P)\) its interior as a manifold with boundary. In a previous work by the second author, [\textit{C. Ueno}, J. Pure Appl. Algebra 216, No. 11, 2436--2448 (2012; Zbl 1283.14024)] it is proved that for every convex polygon \(\mathcal P\subset\mathbb R^2\) that is neither a line nor a band, the semialgebraic sets \(\mathbb R^2\setminus\mathcal P\) and \(\mathbb R^2\setminus{\text{Int}}(\mathcal P)\) are polynomial images of \(\mathbb R^2\). A band is the closed convex polygon determined by two parallel lines. In addition, it is shown that both \(\mathcal P\) and \({\text{Int}}(\mathcal P)\) are regular images of \(\mathbb R^2\).NEWLINENEWLINELater on it was proved in [\textit{J. F. Fernando} and \textit{C. Ueno}, Int. J. Math. 25, No. 7, Article ID 1450071, 18 p. (2014; Zbl 1328.14088)] that for every convex polyhedron \(\mathcal P\subset\mathbb R^3\) which is neither a plane nor a layer both \(\mathbb R^3\setminus\mathcal P\) and \(\mathbb R^3\setminus{\text{Int}}(\mathcal P)\) are polynomial images of \(\mathbb R^3\). A layer is the closed convex polyhedron determined by two parallel planes. In addition, they showed that for \(n\geq4\) the involved constructions do not work further to represent either \(\mathbb R^n\setminus\mathcal P\) or \(\mathbb R^n\setminus{\text{Int}}(\mathcal P)\) as polynomial images of \(\mathbb R^n\) for a general convex polyhedron \(\mathcal P\subset\mathbb R^n\).NEWLINENEWLINEIn the article under review the authors go a step further. The main result is Theorem 1.1 where they prove that for arbitrary \(n\geq2\) and for every convex polyhedron \(\mathcal P\subset\mathbb R^n\) that is neither a hyperplane nor a layer the semialgebraic sets \(\mathbb R^n\setminus\mathcal P\) and \(\mathbb R^n\setminus{\text{Int}}(\mathcal P)\) are polynomial images of \(\mathbb R^n\) in case \(\mathcal P\) is bounded and otherwise they are regular images of \(\mathbb R^n\). As far as the reviewer knows it remains open to characterize geometrically for \(n\geq 4\) the unbounded polyhedra \(\mathcal P\subset\mathbb R^n\) for which either \(\mathbb R^n\setminus\mathcal P\) or \(\mathbb R^n\setminus{\text{Int}}(\mathcal P)\) are polynomial images of \(\mathbb R^n\).NEWLINENEWLINEThe proof of the main result of the work under review is easier if \(\dim(\mathcal P)<n\). Indeed, it follows straightforwardly from a more general result, which is Theorem 3.1 in the article and has its own interest: for every proper basic semialgebraic set \(\mathcal S\subsetneq\mathbb R^n\), the set \(\mathbb R^{n+1}\setminus(\mathcal S\times\{0\})\) is a polynomial image of \(\mathbb R^{n+1}\). The hardest part is to deal with \(n\)-dimensional polyhedra \(\mathcal P\subset\mathbb R^n\). To attack it the authors use successfully a technique previously introduced in their joint work quoted above: to place the polyhedron in what they call first or second trimming position and they work by induction on the number of facets of the polyhedron if it is bounded and by double induction on the dimension and the number of facets of \(\mathcal P\) if it is unbounded.NEWLINENEWLINEIn the last section of this work the authors substitute \(\mathcal P\) by the closed ball \(\mathcal B_n\subset\mathbb R^n\) (which can be understood as the convex polyhedron with infinitely many facets). It was proved in [\textit{J. F. Fernando} and \textit{J. M. Gamboa}, Isr. J. Math. 153, 61--92 (2006; Zbl 1213.14109)] that \(\mathbb R^2\setminus\mathcal B_2\) is a polynomial image of \(\mathbb R^3\) but it is not a polynomial image of \(\mathbb R^2\). Moreover, \(\mathbb R^2\setminus{\text{Int}}(\mathcal B_2)\) is a polynomial image of \(\mathbb R^2\). The proofs of the latter results are specific of the two-dimensional case. The authors have developed a completely different approach to show that for arbitrary \(n\geq2\) the set \(\mathbb R^n\setminus\mathcal B_n\) is a polynomial image of \(\mathbb R^{n+1}\) but it is not a polynomial image of \(\mathbb R^n\). Moreover, they show that \(\mathbb R^n\setminus{\text{Int}}(\mathcal B_n)\) is a polynomial image of \(\mathbb R^n\).NEWLINENEWLINEAll proofs in this work are clearly written and have a strong geometrical flavor, illustrated with enlightening pictures.
scientific article; zbMATH DE number 6365885