On the complements of 3-dimensional convex polyhedra as polynomial images of \(\mathbb{R}^{3}\)
type: Item
instance of: scholarly article
zbMATH Open document ID: 1328.14088
DOI: 10.1142/S0129167X14500712
author: José F. Fernando
author: Carlos Ueno
published in: International Journal of Mathematics
publication date: 2014-09-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/1212.1815
Mathematics Subject Classification ID: 14P10
Mathematics Subject Classification ID: 52B10
Mathematics Subject Classification ID: 52B55
Mathematics Subject Classification ID: 90C26
zbMATH DE Number: 6340428
zbMATH Keywords: polynomial maps and images
zbMATH Keywords: convex polyhedron and its complement
zbMATH Keywords: first and second trimming position
MaRDI profile type: Publication
arXiv ID: 1212.1815
cites work: Geometry. I, II. Transl. from the French by M. Cole and S. Levy
cites work: On the one dimensional polynomial and regular images of \(\mathbb R^n\)
cites work: Polynomial images of \(\mathbb{R}^n\)
cites work: Polynomial and regular images of \(\mathbb R^n\)
cites work: On convex polyhedra as regular images of ℝ n
cites work: Q4235027
cites work: A note on boundaries of open polynomial images of \(\mathbb R^2\)
cites work: On convex polygons and their complements as images of regular and polynomial maps of \(\mathbb R^2\)
cites work: Lectures on Polytopes
title: On the complements of 3-dimensional convex polyhedra as polynomial images of \(\mathbb{R}^{3}\)
review text: The paper under review continues the study of polynomial and regular images of Euclidean spaces initiated in [\textit{J. F. Fernando} and \textit{J. M. Gamboa}, J. Pure Appl. Algebra 179, No. 3, 241--254 (2003; Zbl 1042.14035)] and continued in a long series of articles: [\textit{J. F. Fernando} and \textit{J. M. Gamboa}, Isr. J. Math. 153, 61--92 (2006; Zbl 1213.14109)]; [\textit{C. Ueno}, Rev. Mat. Iberoam. 24, No. 3, 981--988 (2008; Zbl 1156.14330)]; [\textit{J. F. Fernando} et al., Proc. Lond. Math. Soc. (3) 103, No. 5, 847--878 (2011; Zbl 1282.14101)]; [\textit{C. Ueno}, J. Pure Appl. Algebra 216, No. 11, 2436--2448 (2012; Zbl 1283.14024)] and [\textit{J. F. Fernando}, J. Pure Appl. Algebra 218, No. 9, 1745--1753 (2014; Zbl 1295.14052)].NEWLINENEWLINEA map \(f:\mathbb R^n\to\mathbb R^m\) is said to be polynomial if there exist \(f_1,\dots,f_m\in\mathbb R[{\mathtt x}_1,\dots,{\mathtt x}_n]\) such that \(f(x)=(f_1(x),\dots,f_m(x))\) for every point \(x\in\mathbb R^n\). It is a challenge problem to characterize geometrically the semialgebraic subsets \({\mathcal S}\subset\mathbb R^m\) that are polynomial images of \(\mathbb R^n\) for some positive integer \(n\). Indeed, just the case of one dimensional \({\mathcal S}\) is completely understood; see the last quoted paper by J. F. Fernando.NEWLINENEWLINEMotivation for the study of these sets comes from the remark that important problems in real algebraic geometry concerning semialgebraic sets \({\mathcal S}\) such as optimization or computation of trajectories can be reduced to the case \({\mathcal S}=\mathbb R^n\) if \({\mathcal S}\) is a polynomial image of an Euclidean space.NEWLINENEWLINEMost known polynomial images of Euclidean spaces have piecewise linear boundary, and this is also the case in the article under review. Let \({\mathcal K}\subset\mathbb R^n\) be a closed convex and possibly unbounded \(n\)-dimensional polyhedron and denote \({\mathcal S}:=\mathbb R^n\setminus{\mathcal K}\). Let \({\mathcal {\ove…
reviewed by: Jose Manuel Gamboa
description: scientific article; zbMATH DE number 6340428