On convex polygons and their complements as images of regular and polynomial maps of \(\mathbb R^2\)
Recall that a map \(f:\mathbb R^n\to\mathbb R^m\) is said to be regular if there exist polynomials \(f_1,\dots,f_m,g\in\mathbb R[{\mathtt x}_1,\dots,{\mathtt x}_n]\) such that \(g^{-1}(0)=\emptyset\) and for every point \(x\in\mathbb R^n\), \[ g(x)f(x)=(f_1(x),\dots,f_m(x)). \] If \(g\) is a constant polynomial the map \(f\) is said to be polynomial. It is a challenge problem to characterize geometrically the semialgebraic subsets \(S\subset\mathbb R^m\) that are either polynomial or regular images of \(\mathbb R^n\) for some positive integer \(n\). Just the case of one dimensional \(S\) is completely understood, see [\textit{J. F. Fernando}, ``On the one dimensional polynomial and regular images of \(\mathbb R^n\)'', to appear in J. Pure Appl. Algebra (2014)]. After some pioneering work by \textit{J. F. Fernando} and the reviewer, [J. Pure Appl. Algebra 179, No. 3, 241--254 (2003; Zbl 1042.14035)] and [Isr. J. Math. 153, 61--92 (2006; Zbl 1213.14109)], the first conclusive results in the bidimensional case were obtained by Ueno in the article under review. Throughout this paper a convex polygon is the intersection of a finite family of closed half-planes with nonempty interior in \(\mathbb R^2\), and a band is a convex polygon in \(\mathbb R^2\) bounded by two parallel lines. The main results of this article are the following: \noindent Theorem 1. Let \(K\subset\mathbb R^2\) be a convex polygon which is not a band. Then, \(\mathbb R^2\setminus K\) and \(\mathbb R^2\setminus\text{Int}(K)\) are polynomial images of \(\mathbb R^2\). \noindent Theorem 2. Each convex polygon and its interior are regular images of \(\mathbb R^2\). Both results are optimal, in the following sense. First, the complement of a band is not connected, so it cannot be a continuous image of \(\mathbb R^2\). Secondly, singletons are the unique bounded polynomial images of \(\mathbb R^n\). Hence, regular mappings are needed, at least to realize bounded polygons and their interiors as images of \(\mathbb R^2\). The article is based on a part of the doctoral dissertation of the author, written under the supervision of J.F. Fernando. It is transparently written and introduces very original techniques. In fact, its results are interesting by themselves and they have an extra value because they have been a fundamental seminal work for further progress. In [\textit{J. F. Fernando} and \textit{C. Ueno}, ``On complements of convex polyhedra as polynomial and regular images of \(\mathbb R^n\)'', Int. Math. Res. Not. (2014) to appear] the authors extend Theorem 1 to convex polyhedra \(K\) of \(\mathbb R^n\). They prove that if \(K\) is moreover bounded, both \(\mathbb R^n\setminus K\) and \(\mathbb R^n\setminus\text{Int}(K)\) are polynomial images of \(\mathbb R^n\). Indeed, the boundedness condition can be removed for \(n=3\), as it is proved by Fernando and Ueno (preprint). Concerning Theorem 2, it was extended to convex polyhedra of \(\mathbb R^n\) in [\textit{J. F. Fernando} et al., Proc. Lond. Math. Soc. (3) 103, No. 5, 847--878 (2011; Zbl 1282.14101)] generalizing the techniques introduced by Ueno in the article under review.
scientific article; zbMATH DE number 6138024