Neste trabalho utilizamos a tecnica de construcao de campos de vetores controlados para obter estimativas do valor da filtracao de uma aplicacao polinomial $Theta:R^n,0 ightarrowR^p,0$ para que a familia $f_t=f+tTheta$ seja $C^ell$-$mathcal{G}$-trivial, bi-lipschitz trivial ou topologicamente trivial, onde $ellgeq 1$, $mathcal{G}=mathcal{R}$, $mathcal{C}$ ou $mathcal{K}$ e $f:R^n,0 ightarrow R^p,0$ e um germe de aplicacao polinomial satisfazendo uma condicao de nao-degeneracao com relacao a algum poliedro de Newton. Obtemos tambem resultados sobre a trivializacao $C^ell$-modificada para familias de aplicacoes semi-quase-homogeneas de classe $C^{ell + 1}$, e familias de funcoes Newton nao-degeneradas de classe $C^{ell + 1}$.

In this work we use controlled vector fields to obtain estimates for the filtration of a polynomial map-germ $Theta:R^n,0 ightarrowR^p,0$ such that the family $f_t=f+tTheta$ is $C^ell$-$mathcal{G}$-trivial, bi-Lipschitz trivial, or topologicaly trivial, where $ellgeq 1$, $mathcal{G}=mathcal{R}$, $mathcal{C}$ or $mathcal{K}$ and $f:R^n,0 ightarrowR^p,0$ is a polynomial map-germ satisfying a non-degeneracy condition. Results are also obtained on the modified $C^ell$-trivialization for families of semi-wheighted homogeneous maps of class $C^{ell+1}$ with an isolated sigularity at the origin, and families of Newton non-degenerate functions of class $C^{ell+1}$.