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Modified Nash triviality theorem for a family of zero-sets of weighted homogeneous polynomial mappings. (English) Zbl 0916.58005
Let \(f_t:({\mathbb{R}}^n,0)\to ({\mathbb{R}}^p,0)\) be a family of weighted homogeneous polynomial maps, where \(t\in J\) (an open interval of \({\mathbb{R}}\)). We define \(F:({\mathbb{R}}^n\times J,\{0\}\times J) \to ({\mathbb{R}}^p,0)\) by \(F(x,t)=f_t(x)\) and we suppose that it is also polynomial.
The main result of this paper is that if \(f_t^{-1}(0)\cap \Sigma f_t=\{0\}\) for any \(t\in J\), then \(({\mathbb{R}}^n\times J,F^{-1}(0))\) admits a \(\pi_\alpha\)-modified Nash trivialization. This means that there exists a \(t\)-level preserving Nash diffeomorphism \(\phi:(E\times J,E_0\times J)\to (E\times J,E_0\times J)\) which induces a \(t\)-level preserving homeomorphism \(\widetilde\phi:({\mathbb{R}}^n\times J,\{0\}\times J)\to ({\mathbb{R}}^n\times J,\{0\}\times J)\) such that \[ \widetilde\phi({\mathbb{R}}^n\times J,F^{-1}(0))= ({\mathbb{R}}^n\times J,f_{t_0}^{-1}(0)\times J). \] Here, \(E\) is a Nash manifold, \(E_0\) is a Nash submanifold and \(\pi_\alpha:(E,E_0)\to({\mathbb{R}}^n,0)\) is a finite modification. As a consequence, it follows not only the well-known fact that the family is topologically trivial, but that it is modified analytically trivial in the sense of T.-C. Kuo [J. Math. Soc. Japan 32, 605-614 (1980; Zbl 0509.58007)]. It is also shown that in the homogeneous case (that is, when the weights are equal to 1), it also implies strong \(C^0\) triviality.

58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory
57R45 Singularities of differentiable mappings in differential topology
14B07 Deformations of singularities
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