Holomorphic extensions of formal objects

Ribón, Javier

Ribón, Javier ¹

¹ UCLA, Dept. of Mathematics 405 Hilgard Ave., Los Angeles CA90095-1555, USA

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 3 (2004) no. 4, pp. 657-680.

Résumé

We are interested on families of formal power series in $(ℂ^{}, 0)$ parameterized by $ℂ^{n}$ ( $\hat{f} = \sum_{m = 0}^{\infty} P_{m} (x_{1}, \dots, x_{n}) x^{m}$ ). If every $P_{m}$ is a polynomial whose degree is bounded by a linear function ( $d e g P_{m} \leq A m + B$ for some $A > 0$ and $B \geq 0$ ) then the family is either convergent or the series $\hat{f} (c_{1}, \dots, c_{n}, x) \notin ℂ {x}$ for all $(c_{1}, \dots, c_{n}) \in ℂ^{n}$ 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 $\hat{f} = \sum_{m = 0}^{\infty} P_{m} (x_{1}, \dots, x_{n}) x^{m}$ converges. We prove that in dimension $n = 1$ the sets of convergence of the divergent power series are exactly the $F_{σ}$ polar sets.

MR Zbl

Classification : 32D15, 31A15, 32S65, 40A05

Affiliations des auteurs :

Ribón, Javier ¹

¹ UCLA, Dept. of Mathematics 405 Hilgard Ave., Los Angeles CA90095-1555, USA

@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},
     pages = {657--680},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 3},
     number = {4},
     year = {2004},
     mrnumber = {2124584},
     zbl = {1170.32307},
     language = {en},
     url = {http://archive.numdam.org/item/ASNSP_2004_5_3_4_657_0/}
}

TY  - JOUR
AU  - Ribón, Javier
TI  - Holomorphic extensions of formal objects
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2004
SP  - 657
EP  - 680
VL  - 3
IS  - 4
PB  - Scuola Normale Superiore, Pisa
UR  - http://archive.numdam.org/item/ASNSP_2004_5_3_4_657_0/
LA  - en
ID  - ASNSP_2004_5_3_4_657_0
ER  -

%0 Journal Article
%A Ribón, Javier
%T Holomorphic extensions of formal objects
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2004
%P 657-680
%V 3
%N 4
%I Scuola Normale Superiore, Pisa
%U http://archive.numdam.org/item/ASNSP_2004_5_3_4_657_0/
%G en
%F ASNSP_2004_5_3_4_657_0

Ribón, Javier. Holomorphic extensions of formal objects. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 3 (2004) no. 4, pp. 657-680. http://archive.numdam.org/item/ASNSP_2004_5_3_4_657_0/

Bibliographie
Cité par

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

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

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

[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 | Zbl

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

[8] A. Seidenberg, Reduction of singularities of the differential equation $A d y = B d x$ , Amer. J. Math. 90 (1968), 248-269. | MR | Zbl

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