We are interested on families of formal power series in $\text{(}{ℂ}^{},0\text{)}$ parameterized by ${ℂ}^n$ ($\widehat{f}={\sum }_{m=0}^{\infty }{P}_m(x_1,\cdots ,x_n)x^m$). If every $P_m$ is a polynomial whose degree is bounded by a linear function ($deg{P}_m\le Am+B$ for some $A\>0$ and $B\ge 0$) then the family is either convergent or the series $\widehat{f}(c_1,\cdots ,c_n,x)\notin ℂ\left\{x\right\}$ for all $(c_1,\cdots ,c_n)\in {ℂ}^n$ except a pluri-polar set. Generalizations of these results are provided for formal objects associated to germs of diffeomorphism (formal power series, formal meromorphic functions, etc.). We are interested on describing the nature of the set of parameters where...

Let $ℱ$ be a holomorphic foliation by curves on ${\mathbb{P}}^3$. We treat the case where the set $Sing\left(ℱ\right)$ consists of disjoint regular curves and some isolated points outside of them. In this situation, using Baum-Bott’s formula and Porteuos’theorem, we determine the number of isolated singularities, counted with multiplicities, in terms of the degree of $ℱ$, the multiplicity of $ℱ$ along the curves and the degree and genus of the curves.