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 ( ,0) parameterized by n (f ^= m=0 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 B0) then the family is either convergent or the series f ^(c 1 ,,c n ,x){x} for all (c 1 ,,c n ) 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 f ^= m=0 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 σ 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/

[1] J. Ecalle, Théorie itérative: introduction à la théorie des invariants holomorphes, J. Math. Pures Appl. 54 (1975), 183-258. | MR 499882 | Zbl 0285.26010

[2] M. K. Klimek, “Pluripotential theory”, Oxford, 1991. | Zbl 0742.31001

[3] R. Pérez Marco, Convergence or generic divergence of Birkhoff normal form, preprint, UCLA, 2000. http://xxx.lanl.gov/abs/math.DS/0009028. | MR 1973055

[4] R. Pérez Marco, Linearization of holomorphic germs with resonant linear part, preprint, UCLA, 2000. http://xxx.lanl.gov/abs/math.DS/0009030.

[5] R. Pérez Marco, A note on holomorphic extensions, preprint, UCLA, 2000. http://xxx.lanl.gov/abs/math.DS/0009031.

[6] R. Pérez Marco, Total convergence or general divergence in Small Divisors, Comm. Math. Phys. (223) 3 (2001), 451-464. | MR 1866162 | Zbl 1161.37331

[7] T. Ransford, “Potential theory in the complex plane”, London Mathematical Society, Student Texts 28, Cambridge University Press, 1995. | MR 1334766 | Zbl 0828.31001

[8] A. Seidenberg, Reduction of singularities of the differential equation Ady=Bdx, Amer. J. Math. 90 (1968), 248-269. | MR 220710 | Zbl 0159.33303

[9] M. Tsuji, “Potential theory in modern function theory”, Maruzen, Tokyo, 1959. | MR 114894 | Zbl 0087.28401