Abstract

We are interested on families of formal power series in $\text{}$ parameterized by ${\mathbb{C}}^{n}$ {x}^{m}}$). If every $P_{m}$ is a polynomial whose degree is bounded by a linear function then the family is either convergent or the series $\hat{f} \notin {\mathbb{C}} \lbrace x \rbrace $ for all $ \in {\mathbb{C}}^{n}$ except a pluri-polar set. Generalizations of these results are provided for formal objects associated to germs of diffeomorphism. We are interested on describing the nature of the set of parameters where $\hat{f} = \sum _{m=0}^{\infty } {P_{m} {x}^{m}}$ converges. We prove that in dimension $n=1$ the sets of convergence of the divergent power series are exactly the $F_{\sigma }$ polar sets