# zbMATH — the first resource for mathematics

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.

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