Nash sets and global equations.

*(English)*Zbl 0735.14037An affine Nash manifold \(M\) is an analytic submanifold of \(\mathbb{R}^ n\) which is also a semialgebraic set; a Nash function on an open set \(U\) of \(M\) is an analytic function whose graph is semialgebraic. Then a subset \(X\) of \(M\) is called:

(a) Nash, if for every point \(x\in M\) there is a Nash function \(f:U\to R\) defined in a neighborhood \(U\) of \(x\) in \(M\) such that \(X\cap U=\{x\in U\mid f(x)=0\}\), and

(b) global Nash, if there is a global Nash function \(g:M\to R\) such that \(X=\{x\in M\mid g(x)=0\}\). It is known that a Nash set need not be global Nash [M. Shiota, “Nash manifolds”, Lect. Notes Math. 1269 (1987; Zbl 0629.58002){]} and this paper discusses the problem of characterizing when that happens. The main result is that for a Nash set \(X\) of the \(n\)-sphere \(M=S^ n\) the following assertions are equivalent:

(i) \(X\) is global Nash,

(ii) any function on \(X\) that extends to a Nash function on some neighborhood of \(X\) extends to a Nash function on \(S^ n\),

(iii) there is a Nash blowing-down of \(S^ n\) onto a Nash set \(Y\subset \mathbb{R}^ m\) that collapses \(X\) to a point \(y\) and is a Nash diffeomorphism from \(S^ n\backslash X\) onto \(Y\backslash\{y\}\).

In case \(X\) is a Nash coherent hypersurface, it defines a line bundle \(F_ X\to S^ n\), and the three conditions above are equivalent to:

(iv) some power \(F_ X^{\otimes p}\) of \(F_ X\) can be Nash embedded in a trivial bundle. The paper ends with an example of a Nash subset \(X\) of \(\mathbb{R}^ 4\) which cannot be Nash embedded as a global Nash set in any affine space. The techniques used in the proofs are two standard ones in the field: separation of semialgebraic sets by Nash functions and complexification.

(a) Nash, if for every point \(x\in M\) there is a Nash function \(f:U\to R\) defined in a neighborhood \(U\) of \(x\) in \(M\) such that \(X\cap U=\{x\in U\mid f(x)=0\}\), and

(b) global Nash, if there is a global Nash function \(g:M\to R\) such that \(X=\{x\in M\mid g(x)=0\}\). It is known that a Nash set need not be global Nash [M. Shiota, “Nash manifolds”, Lect. Notes Math. 1269 (1987; Zbl 0629.58002){]} and this paper discusses the problem of characterizing when that happens. The main result is that for a Nash set \(X\) of the \(n\)-sphere \(M=S^ n\) the following assertions are equivalent:

(i) \(X\) is global Nash,

(ii) any function on \(X\) that extends to a Nash function on some neighborhood of \(X\) extends to a Nash function on \(S^ n\),

(iii) there is a Nash blowing-down of \(S^ n\) onto a Nash set \(Y\subset \mathbb{R}^ m\) that collapses \(X\) to a point \(y\) and is a Nash diffeomorphism from \(S^ n\backslash X\) onto \(Y\backslash\{y\}\).

In case \(X\) is a Nash coherent hypersurface, it defines a line bundle \(F_ X\to S^ n\), and the three conditions above are equivalent to:

(iv) some power \(F_ X^{\otimes p}\) of \(F_ X\) can be Nash embedded in a trivial bundle. The paper ends with an example of a Nash subset \(X\) of \(\mathbb{R}^ 4\) which cannot be Nash embedded as a global Nash set in any affine space. The techniques used in the proofs are two standard ones in the field: separation of semialgebraic sets by Nash functions and complexification.

Reviewer: J.M.Ruiz (Madrid)