Nash triviality in families of Nash manifolds.

*(English)*Zbl 0801.14017One of the most important theorems in semialgebraic geometry is the semialgebraic local triviality in semialgebraic maps [cf. R. M. Hardt, Am. J. Math. 102, 291-302 (1980; Zbl 0465.14012)]. We want to give here a Nash (i.e. \({\mathcal C}^ \infty\) and semialgebraic, which is the same as analytic and semialgebraic) analog of this result.

Theorem A. Let \(B\) be a semialgebraic set and let \(\Pi : \mathbb{R}^ n \times B \to B\) denote the projection. Let \(X\) be a semialgebraic subset of \(\mathbb{R}^ n \times B\) such that for any \(b \in B\), \(X_ b = \{x \in \mathbb{R}^ n; (x,b) \in X\}\) is a Nash submanifold of \(\mathbb{R}^ n\). Then there is a finite partition of \(B\) into Nash submanifolds \(M^ i\), and for any \(i\) there are an affine Nash manifold \(F^ i \subset \mathbb{R}^ n\) and a Nash diffeomorphism \(h^ i : F^ i \times M^ i \to X \cap \Pi^{ - 1} (M^ i)\) compatible with the projections onto \(M^ i\).

Theorem A above has consequences on finiteness and effectiveness:

Theorem B. Given integers \(n\) and \(c\), there are integers \(s\) and \(d\) and Nash submanifolds \(X^ 1, \dots, X^ s\) of \(\mathbb{R}^ n\) of degree \(\leq c\), such that for any Nash submanifold \(X\) of \(\mathbb{R}^ n\) of degree \(\leq c\), there is a Nash isotopy of degree \(\leq d\) connecting \(X\) to one of the \(X^ i\) through Nash submanifolds of \(\mathbb{R}^ n\) of degree \(\leq c\). Moreover, \(s\) and \(d\) are bounded by recursive functions of \(n\) and \(c\).

The tools that we need (approximation and real spectrum) are briefly reviewed in section 1. – Hardt’s theorem (loc. cit.) is a theorem about simultaneous triangulation in families. For Nash manifolds, the triangulation is inappropriate, but there is something similar: the existence of a Nash diffeomorphism onto a Nash manifold which is defined “without parameter”, i.e. on the field of real algebraic numbers \(\mathbb{R}_{\text{alg}}\), the smallest real closed field. In section 2 we prove the equivalence of this property with theorem A. We also prove the equivalence in the semialgebraic \({\mathcal C}^ 1\) category.

The third section is devoted to the proof of theorem A. The short fourth section is an example of how theorem A can replace integration of vector fields. Finally, in section 5, we give the proof of theorem B.

Theorem A. Let \(B\) be a semialgebraic set and let \(\Pi : \mathbb{R}^ n \times B \to B\) denote the projection. Let \(X\) be a semialgebraic subset of \(\mathbb{R}^ n \times B\) such that for any \(b \in B\), \(X_ b = \{x \in \mathbb{R}^ n; (x,b) \in X\}\) is a Nash submanifold of \(\mathbb{R}^ n\). Then there is a finite partition of \(B\) into Nash submanifolds \(M^ i\), and for any \(i\) there are an affine Nash manifold \(F^ i \subset \mathbb{R}^ n\) and a Nash diffeomorphism \(h^ i : F^ i \times M^ i \to X \cap \Pi^{ - 1} (M^ i)\) compatible with the projections onto \(M^ i\).

Theorem A above has consequences on finiteness and effectiveness:

Theorem B. Given integers \(n\) and \(c\), there are integers \(s\) and \(d\) and Nash submanifolds \(X^ 1, \dots, X^ s\) of \(\mathbb{R}^ n\) of degree \(\leq c\), such that for any Nash submanifold \(X\) of \(\mathbb{R}^ n\) of degree \(\leq c\), there is a Nash isotopy of degree \(\leq d\) connecting \(X\) to one of the \(X^ i\) through Nash submanifolds of \(\mathbb{R}^ n\) of degree \(\leq c\). Moreover, \(s\) and \(d\) are bounded by recursive functions of \(n\) and \(c\).

The tools that we need (approximation and real spectrum) are briefly reviewed in section 1. – Hardt’s theorem (loc. cit.) is a theorem about simultaneous triangulation in families. For Nash manifolds, the triangulation is inappropriate, but there is something similar: the existence of a Nash diffeomorphism onto a Nash manifold which is defined “without parameter”, i.e. on the field of real algebraic numbers \(\mathbb{R}_{\text{alg}}\), the smallest real closed field. In section 2 we prove the equivalence of this property with theorem A. We also prove the equivalence in the semialgebraic \({\mathcal C}^ 1\) category.

The third section is devoted to the proof of theorem A. The short fourth section is an example of how theorem A can replace integration of vector fields. Finally, in section 5, we give the proof of theorem B.

##### MSC:

14P20 | Nash functions and manifolds |

##### References:

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