About certain subsets of the Euclidean space
type: Item
instance of: scholarly article
title: About certain subsets of the Euclidean space
zbMATH Open document ID: 0744.14036
author: Jean-Yves Charbonnel
publication date: 1992-09-26T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/74934
review text: Let \(\tilde{\mathcal A}_ m\) be the algebra of functions on \(\mathbb{R}^ n\) generated by polynomial functions and exponentials of linear forms. The subset \(S\) in \(\mathbb{R}^ n\) belongs to \({\mathcal P}_ n\) if and only if there exist \(m\) and \(F\) in \(\tilde{\mathcal A}_{n+m}\) for which \(S\) is the image of the zerosubset of \(F\) by the canonical projection of \(\mathbb{R}^{n+m}\) onto \(\mathbb{R}^ n\). Let \(\tilde{\mathcal P}_ n\) be the smallest subset of parts in \(\mathbb{R}^ n\) which contains \({\mathcal P}_ n\), their closures and the images by the canonical projection of the elements in \(\tilde{\mathcal P}_{n+m}\). --- This family of sets is defined by an induction in two steps. The main goal of this article is to prove that \(\tilde{\mathcal P}_{n+m}\) contains the complementary part of each element in \(\tilde{\mathcal P}_{n+m}\), the union and the intersection of every finite family in \(\tilde{\mathcal P}_{n+m}\). The key results for the proof are theorems by A. G. Khovanskij on the set of the solutions of a Pfaff system on a Pfaff manifold.
Mathematics Subject Classification ID: 14P20
Mathematics Subject Classification ID: 46E99
Mathematics Subject Classification ID: 58A17
Mathematics Subject Classification ID: 58A07
Mathematics Subject Classification ID: 14A05
zbMATH DE Number: 55932
zbMATH Keywords: polynomial functions
zbMATH Keywords: exponential
zbMATH Keywords: zeroset of a function
zbMATH Keywords: projection
zbMATH Keywords: A. G. Khovanskij's theorems
zbMATH Keywords: Pfaff system
zbMATH Keywords: Pfaff manifold
MaRDI profile type: Publication
OpenAlex ID: W2048048049
cites work: Q5339415
cites work: ASYMPTOTIC PROPERTIES OF POLYNOMIALS AND ALGEBRAIC FUNCTIONS OF SEVERAL VARIABLES
cites work: Q4773813
cites work: Q4724607
DOI: 10.5802/AIF.1270
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 55932