Modified Nash triviality of a family of zero-sets of weighted homogeneous polynomial mappings
Let \(J\) be an open interval and \(f_t: (\mathbb{R}^n, 0)\to (\mathbb{R}^p, 0)\) be a polynomial mapping where each \(f_{t, i}\) is weighted homogeneous of type \((\alpha_1, \dots, \alpha_n; L_i)\), \(i=1, \dots, p\), for any \(t\in J\). Assume that \(F: (\mathbb{R}^n \times J, \{0\})\to (\mathbb{R}^p, 0)\) defined by \(F(x, t)= f_t (x)\) is a polynomial mapping. It is well-known that if \(f^{-1}_t (0)\cap \Sigma f_t= \{0\}\) for any \(t\in J\), where \(\Sigma f_t\) denotes the singular locus of \(f_t\), then \((\mathbb{R}^n \times J, F^{-1} (0))\) is topologically trivial. The author shows, under the same assumptions, a stronger triviality called the modified Nash triviality which is induced by a \(t\)-level preserving Nash diffeomorphism. He also discusses the relation between the modified Nash triviality and the strong \(C^0\)- triviality introduced by himself in [J. Math. Soc. Japan 45, No. 2, 313- 320 (1993; Zbl 0788.32024)].
scientific article; zbMATH DE number 729235