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VOL. 43 | 2006 $\mathcal{A}$-topological triviality of map germs and Newton filtrations
Marcelo José Saia, Liane Mendes Feitosa Soares

Editor(s) Shyuichi Izumiya, Goo Ishikawa, Hiroo Tokunaga, Ichiro Shimada, Takasi Sano


We apply the method of constructing controlled vector fields to give sufficient conditions for the $\mathcal{A}$-topological triviality of deformations of map germs $f_t : (\mathbb{C}^n, 0) \to (\mathbb{C}^p, 0)$ of type $f_t(x) = f(x) + th(x)$, with $n \ge p$ or $n \le 2p$. These conditions are given in terms of an appropriate choice of Newton filtrations for $\mathcal{O}_n$ and $\mathcal{O}_p$ and are for the $\mathcal{A}$-tangent space of the germ $f$.

For the case $n \ge p$, we follow the technique used by M. A. S. Ruas in her Ph.D. Thesis [7] and construct control functions in the target and in the source to obtain, via a partition of the unit, a unique control function. We use the control function of the target to give an estimate for the case $p \ge 2n$. Moreover, in this case we show that if the coordinates of the map germ satisfy a Newton non-degeneracy condition, deformations by terms of higher filtration are topologically trivial.

As an application we obtain for both cases, $n \ge p$ and $p \ge 2n$, the results of Damon in [3] for deformations of weighted homogeneous map germs.


Published: 1 January 2006
First available in Project Euclid: 3 January 2019

zbMATH: 1130.58022
MathSciNet: MR2325147

Digital Object Identifier: 10.2969/aspm/04310383

Primary: 32S15 , 32S50 , 58K15

Keywords: $\mathcal{A}$-topological triviality , controlled vector fields , Newton filtration , Newton non-degenerate map germs

Rights: Copyright © 2006 Mathematical Society of Japan


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