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The Weierstrass approximation theorem for maps between real algebraic varieties. (English) Zbl 0945.14032

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.

MSC:

14P05 Real algebraic sets
14P25 Topology of real algebraic varieties
41A05 Interpolation in approximation theory
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