The Weierstrass approximation theorem for maps between real algebraic varieties
type: Item
instance of: scholarly article
title: The Weierstrass approximation theorem for maps between real algebraic varieties
zbMATH Open document ID: 0945.14032
author: Jacek Bochnak
author: Wojciech Kucharz
published in: Mathematische Annalen
publication date: 2000-06-18T00:00:00Z
review text: The two main results of the paper can be summarized as follows. Theorem. Let \(X\) and \(Y\) be real algebraic varieties. (1) If \(X\) is compact and nonsingular with dim \(X = 1\) and \(Y\) is rational and nonsingular, then the set \(R(X,Y)\) of regular maps from \(X\) into \(Y\) is dense in the space \(C(X,Y)\) of all \(C\) maps from \(X\) into \(Y\) endowed with the \(C\) compact-open topology. (2) If dim \(X\) is positive, \(Y\) is compact and nonsingular and \(R(X,Y)\) is dense in \(C(X, Y)\), then the first Betti number \(b_1(Y;C)\) of any projective nonsingular complexification of \(Y\) equals \(0\). The first part is theorem 1.1 of the paper, which is proved in a stronger version (cf. theorem 2.5). Other more technical results are proved where the compactness assumption on X is relaxed (cf. theorems 2.6 and 2.7). The second part is theorem 1.2. Condition \(b_1(Y;C) = 0\) is by far too weak to insure the validity of theorem 1.1. The authors show that if \(Y\) is a `general' surface in \({\mathbb{P}}^3(\mathbb{R})\) and \(Y\) is not homeomorphic to \({\mathbb{P}}^2(\mathbb{R})\) or the unit 2-sphere, then \(b_1(Y;C) = 0\), but \(R(X,Y)\) is not dense in \(C(X,Y)\). The simplest challenge to prove or disprove the validity of theorem 1.1 is the Fermat surface of degree \(4,\) \(Y = {(x,y,z)\in {\mathbb{R}}^3 : x^4 + y^4 + z^4 = 1}.\) The paper is very clear and the elegant and deep proofs combine classical arguments in approximation results, with a clever use of the real Jacobian variety.
reviewed by: Jose Manuel Gamboa
Mathematics Subject Classification ID: 14P05
Mathematics Subject Classification ID: 14P25
Mathematics Subject Classification ID: 41A05
zbMATH DE Number: 1336998
zbMATH Keywords: Weierstrass approximation theorem
zbMATH Keywords: real algebraic varieties
zbMATH Keywords: real Jacobian variety
MaRDI profile type: Publication
full work available at URL: s002080050309
OpenAlex ID: W2082858708
DOI: 10.1007/S002080050309
description: scientific article; zbMATH DE number 1336998