On complements of convex polyhedra as polynomial and regular images of \(\mathbb{R}^{n}\)
type: Item
instance of: scholarly article
zbMATH Open document ID: 1328.14089
author: José F. Fernando
author: Carlos Ueno
publication date: 2014-11-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/1212.1813
Mathematics Subject Classification ID: 14P10
Mathematics Subject Classification ID: 52B05
Mathematics Subject Classification ID: 52B10
zbMATH DE Number: 6365885
zbMATH Keywords: polynomial and regular image
zbMATH Keywords: convex polyhedron
zbMATH Keywords: first and second trimming positions
MaRDI profile type: Publication
OpenAlex ID: W3099049298
arXiv ID: 1212.1813
DOI: 10.1093/IMRN/RNT112
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published in: IMRN. International Mathematics Research Notices
title: On complements of convex polyhedra as polynomial and regular images of \(\mathbb{R}^{n}\)
review text: 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 \(\mathbb R^2\).NEWLI…
reviewed by: Jose Manuel Gamboa
description: scientific article; zbMATH DE number 6365885