zbMATH — the first resource for mathematics

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.

14P20 Nash functions and manifolds
Full Text: DOI EuDML
[1] [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
[2] [BCR] Bochnak, J., Coste, M., Roy, M-F.: Géometrie algébrique réelle. (Ergeb. Math. Grenzgeb., 3. Folge, vol. 12) Berlin Heidelberg New York: Springer 1987
[3] [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
[4] [M] Milnor, J.: Morse theory. (Ann. Math. Stud. vol. 51) Princeton: Princeton University Press 1963 · Zbl 0108.10401
[5] [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
[6] [R] Ramanakoraisina, R.: Complexité des fonctions de Nash. Commun. Algebra17 (n. 6), 1395–1406 (1989) · Zbl 0684.14008 · doi:10.1080/00927878908823796
[7] [S] Shiota, M.: Nash manifolds. (Lect. Notes Math., vol. 1269) Berlin Heidelberg New York: Springer 1987 · Zbl 0629.58002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.