Resumen de Holomorphic extensions of formal objects
We are interested on families of formal power series in (C, 0) parameterized by Cn ( fˆ = ∞ m=0 Pm(x1, . . . , xn)xm). If every Pm is a polynomial whose degree is bounded by a linear function (degPm ≤ Am + B for some A > 0 and B ≥ 0) then the family is either convergent or the series fˆ(c1, . . . , cn, x) ∈ C{x} for all (c1, . . . , cn) ∈ Cn 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 fˆ = ∞ m=0 Pm(x1, . . . , xn)xm converges. We prove that in dimension n = 1 the sets of convergence of the divergent power series are exactly the Fσ polar sets.
