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 $C^{ell+1}$.

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 $C^{ell+1}$.

The thesis deals with stochastic and algebraic aspects of the integer convex hull. In the first part, the intrinsic volumes of the randomized integer convex hull are investigated. In particular, we obtained an exact asymptotic order of the expected intrinsic volumes difference in a smooth convex body and a tight inequality for the expected mean width difference. In the algebraic part, an exact formula for the Bhattacharya function of complete primary monomial ideas in two variables is given. As a consequence, we derive an effective characterization for complete monomial ideals in two variables.