Abstract
Let ℳ be a smooth closed manifold in ℝn. The Nash-Tognoli theorem says that M can be arbitrarily well approximated (in the Cr-topology with r < ∞) in ℝn by a nonsingular real algebraic set under the condition that dim ℳ<(n-1)/2 There is a familiar conjecture, going back at least to Nash, that the restriction on dim ℳ in the Nash-Tognoli theorem is unnecessary. However, up to now in unstable dimensions [i.e., for dim ℳ⩾(n-1)/2 ] the possibility of approximating was known only for orientable ℳ of codimension (in ℝn) 1 or 2. The goal of the paper is to prove the following theorem, relaxing the restriction on dim ℳ in the Nash-Tognoli theorem to dim M<(2n-1)/3. If ℳ is a smooth closed manifold in IK and dim M<(2n−1)/3, then ℳ can be arbitrarily well approximated in ℝn by a nonsingular real algebraic set.
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