Holomorphic extensions of formal objects

Javier Ribón[1]

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

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2004)

  • Volume: 3, Issue: 4, page 657-680
  • ISSN: 0391-173X

Abstract

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We are interested on families of formal power series in ( ℂ , 0 ) parameterized by ℂ 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 ( d e g P m ≤ A m + B for some A > 0 and B ≥ 0 ) then the family is either convergent or the series f ^ ( c 1 , ⋯ , c n , x ) ∉ ℂ { x } for all ( c 1 , ⋯ , c n ) ∈ 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 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.

How to cite

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Ribón, Javier. "Holomorphic extensions of formal objects." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 3.4 (2004): 657-680. <http://eudml.org/doc/84545>.

@article{Ribón2004,
abstract = {We are interested on families of formal power series in $\text\{($\{\mathbb \{C\}\}^\{\},0$)\}$ parameterized by $\{\mathbb \{C\}\}^\{n\}$ ($\hat\{f\} = \sum _\{m=0\}^\{\infty \} \{P_\{m\}(x_\{1\},\hdots ,x_\{n\}) \{x\}^\{m\}\}$). If every $P_\{m\}$ is a polynomial whose degree is bounded by a linear function ($deg P_\{m\} \le A m + B$ for some $A&gt;0$ and $B \ge 0$) then the family is either convergent or the series $\hat\{f\}(c_\{1\},\hdots ,c_\{n\},x) \notin \{\mathbb \{C\}\} \lbrace x \rbrace $ for all $(c_\{1\},\hdots ,c_\{n\}) \in \{\mathbb \{C\}\}^\{n\}$ 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 $\hat\{f\} = \sum _\{m=0\}^\{\infty \} \{P_\{m\}(x_\{1\},\hdots ,x_\{n\}) \{x\}^\{m\}\}$ converges. We prove that in dimension $n=1$ the sets of convergence of the divergent power series are exactly the $F_\{\sigma \}$ polar sets.},
affiliation = {UCLA, Dept. of Mathematics 405 Hilgard Ave., Los Angeles CA90095-1555, USA},
author = {Ribón, Javier},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {4},
pages = {657-680},
publisher = {Scuola Normale Superiore, Pisa},
title = {Holomorphic extensions of formal objects},
url = {http://eudml.org/doc/84545},
volume = {3},
year = {2004},
}

TY - JOUR
AU - Ribón, Javier
TI - Holomorphic extensions of formal objects
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2004
PB - Scuola Normale Superiore, Pisa
VL - 3
IS - 4
SP - 657
EP - 680
AB - We are interested on families of formal power series in $\text{(${\mathbb {C}}^{},0$)}$ parameterized by ${\mathbb {C}}^{n}$ ($\hat{f} = \sum _{m=0}^{\infty } {P_{m}(x_{1},\hdots ,x_{n}) {x}^{m}}$). If every $P_{m}$ is a polynomial whose degree is bounded by a linear function ($deg P_{m} \le A m + B$ for some $A&gt;0$ and $B \ge 0$) then the family is either convergent or the series $\hat{f}(c_{1},\hdots ,c_{n},x) \notin {\mathbb {C}} \lbrace x \rbrace $ for all $(c_{1},\hdots ,c_{n}) \in {\mathbb {C}}^{n}$ 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 $\hat{f} = \sum _{m=0}^{\infty } {P_{m}(x_{1},\hdots ,x_{n}) {x}^{m}}$ converges. We prove that in dimension $n=1$ the sets of convergence of the divergent power series are exactly the $F_{\sigma }$ polar sets.
LA - eng
UR - http://eudml.org/doc/84545
ER -

References

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  1. [1] J. Ecalle, Théorie itérative: introduction à la théorie des invariants holomorphes, J. Math. Pures Appl. 54 (1975), 183–258. Zbl0285.26010MR499882
  2. [2] M. K. Klimek, “Pluripotential theory”, Oxford, 1991. Zbl0742.31001
  3. [3] R. Pérez Marco, Convergence or generic divergence of Birkhoff normal form, preprint, UCLA, 2000. http://xxx.lanl.gov/abs/math.DS/0009028. MR1973055
  4. [4] R. Pérez Marco, Linearization of holomorphic germs with resonant linear part, preprint, UCLA, 2000. http://xxx.lanl.gov/abs/math.DS/0009030. 
  5. [5] R. Pérez Marco, A note on holomorphic extensions, preprint, UCLA, 2000. http://xxx.lanl.gov/abs/math.DS/0009031. 
  6. [6] R. Pérez Marco, Total convergence or general divergence in Small Divisors, Comm. Math. Phys. (223) 3 (2001), 451–464. Zbl1161.37331MR1866162
  7. [7] T. Ransford, “Potential theory in the complex plane”, London Mathematical Society, Student Texts 28, Cambridge University Press, 1995. Zbl0828.31001MR1334766
  8. [8] A. Seidenberg, Reduction of singularities of the differential equation A d y = B d x , Amer. J. Math. 90 (1968), 248–269. Zbl0159.33303MR220710
  9. [9] M. Tsuji, “Potential theory in modern function theory”, Maruzen, Tokyo, 1959. Zbl0087.28401MR114894

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