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Nash sets and global equations. (English) Zbl 0735.14037
An 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.