On convex polygons and their complements as images of regular and polynomial maps of \(\mathbb R^2\)
type: Item
instance of: scholarly article
title: On convex polygons and their complements as images of regular and polynomial maps of \(\mathbb R^2\)
zbMATH Open document ID: 1283.14024
author: Carlos Ueno
published in: Journal of Pure and Applied Algebra
publication date: 2013-02-21T00:00:00Z
review text: 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 \(\…
Mathematics Subject Classification ID: 14P10
Mathematics Subject Classification ID: 52B11
zbMATH DE Number: 6138024
zbMATH Keywords: polynomial image of \(\mathbb R^n\)
zbMATH Keywords: regular image of \(\mathbb R^n\)
zbMATH Keywords: convex polyhedra
reviewed by: Jose Manuel Gamboa
MaRDI profile type: Publication
full work available at URL: j.jpaa.2012.03.016
OpenAlex ID: W1980589511
cites work: Geometry. I, II. Transl. from the French by M. Cole and S. Levy
cites work: Polynomial images of \(\mathbb{R}^n\)
cites work: Polynomial and regular images of \(\mathbb R^n\)
cites work: Geometry of real polynomial mappings
cites work: Q4954118
cites work: A counterexample to the strong real Jacobian conjecture
cites work: A note on boundaries of open polynomial images of \(\mathbb R^2\)
DOI: 10.1016/J.JPAA.2012.03.016
description: scientific article; zbMATH DE number 6138024