Holomorphic extensions of formal objects

We are interested in families of formal power series in (C, 0) parameterized by C-n ((f) over cap = Sigma(m=0)(infinity) P-m(x(1),...,x(n))x(m)). If every P-m is a polynomial whose degree is bounded by a linear function (degP(m) < Am + B for some A > 0 and B > 0) then either the family is convergent or the series (f) over cap (C-1,...,c(n),x) is not an element of C{x} for all (C-1,...,c(n)) is an element of 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 in describing the nature of the set of parameters where (f) over cap = Sigma(m=0)(infinity) P-m (x(1),...,x(n))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.