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Nash triviality in families of Nash manifolds. (English) Zbl 0801.14017
One 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.

##### MSC:
 14P20 Nash functions and manifolds
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##### References:
  [ABB] Aquistapace, F., Benedetti, R., Broglia, F.: Effectiveness-non effectiveness in semialgebraic and PL geometry. Invent. Math.102, 141–156 (1990) · Zbl 0729.14040 · doi:10.1007/BF01233424  [BCR] Bochnak, J., Coste, M., Roy, M-F.: Géometrie algébrique réelle. (Ergeb. Math. Grenzgeb., 3. Folge, vol. 12) Berlin Heidelberg New York: Springer 1987  [H] Hardt, R.M.: Semi-algebraic local-triviality in semi-algebraic mappings. Am. J. Math.102 (n. 2), 291–302 (1980) · Zbl 0465.14012 · doi:10.2307/2374240  [M] Milnor, J.: Morse theory. (Ann. Math. Stud. vol. 51) Princeton: Princeton University Press 1963 · Zbl 0108.10401  [N] Nabutovski, A.: Non-recursive functions in real algebraic geometry. In: Galbiati, M., Tognoli, A. (eds.) Real Analytic and Algebraic geometry. (Lect. Notes Math., vol. 1420) Berlin Heidelberg New York: Springer 1990  [R] Ramanakoraisina, R.: Complexité des fonctions de Nash. Commun. Algebra17 (n. 6), 1395–1406 (1989) · Zbl 0684.14008 · doi:10.1080/00927878908823796  [S] Shiota, M.: Nash manifolds. (Lect. Notes Math., vol. 1269) Berlin Heidelberg New York: Springer 1987 · Zbl 0629.58002
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