$\mathcal{A}$-topological triviality of map germs and Newton filtrations

Abstract

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.