$C^{\ell}$-$G$-triviality of map germs and Newton polyhedra, $G = \mathcal R$, $\mathcal C$ and $\mathcal K$

Abstract

We provide estimates for the $C^{\ell}$-$G$-triviality, for $ 0 \leq \ell < \infty$ and $G$ is one of Mather's groups ${\mathcal R}$, ${\mathcal C}$ or ${\mathcal K}$, of deformations of analytic map germs $f: (\mathbb{R}^n,0) \to (\mathbb{R}^p,0)$ of type $f_t(x)=f(x)+θ(x,t)$ which satisfy a non-degeneracy condition with respect to some Newton polyhedron. We apply the method of construction of controlled vector fields and, for each group $G$, the control function is determined from the choice of a convenient {\it Newton filtration } in the ring of real analytic germs. The results are given in terms of the filtration of the coordinate function germs $f_1, \ldots , f_p$ of $f$.