Holomorphic extensions of formal objects
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 3 (2004) no. 4, p. 657-680

We are interested on families of formal power series in $\text{(}{ℂ}^{},0\text{)}$ parameterized by ${ℂ}^{n}$ ($\stackrel{^}{f}={\sum }_{m=0}^{\infty }{P}_{m}\left({x}_{1},\cdots ,{x}_{n}\right){x}^{m}$). 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 $\stackrel{^}{f}\left({c}_{1},\cdots ,{c}_{n},x\right)\notin ℂ\left\{x\right\}$ for all $\left({c}_{1},\cdots ,{c}_{n}\right)\in {ℂ}^{n}$ 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 $\stackrel{^}{f}={\sum }_{m=0}^{\infty }{P}_{m}\left({x}_{1},\cdots ,{x}_{n}\right){x}^{m}$ converges. We prove that in dimension $n=1$ the sets of convergence of the divergent power series are exactly the ${F}_{\sigma }$ polar sets.

Classification:  32D15,  31A15,  32S65,  40A05
@article{ASNSP_2004_5_3_4_657_0,
author = {Rib\'on, Javier},
title = {Holomorphic extensions of formal objects},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 3},
number = {4},
year = {2004},
pages = {657-680},
zbl = {1170.32307},
mrnumber = {2124584},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2004_5_3_4_657_0}
}
Ribón, Javier. Holomorphic extensions of formal objects. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 3 (2004) no. 4, pp. 657-680. http://www.numdam.org/item/ASNSP_2004_5_3_4_657_0/

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