On convergence sets of divergent power series

Buma L. Fridman; Daowei Ma; Tejinder S. Neelon

Annales Polonici Mathematici (2012)

  • Volume: 106, Issue: 1, page 193-198
  • ISSN: 0066-2216

Abstract

top
A nonlinear generalization of convergence sets of formal power series, in the sense of Abhyankar-Moh [J. Reine Angew. Math. 241 (1970)], is introduced. Given a family y = φ s ( t , x ) = s b ( x ) t + b ( x ) t ² + of analytic curves in ℂ × ℂⁿ passing through the origin, C o n v φ ( f ) of a formal power series f(y,t,x) ∈ ℂ[[y,t,x]] is defined to be the set of all s ∈ ℂ for which the power series f ( φ s ( t , x ) , t , x ) converges as a series in (t,x). We prove that for a subset E ⊂ ℂ there exists a divergent formal power series f(y,t,x) ∈ ℂ[[y,t,x]] such that E = C o n v φ ( f ) if and only if E is an F σ set of zero capacity. This generalizes the results of P. Lelong and A. Sathaye for the linear case φ s ( t , x ) = s t .

How to cite

top

Buma L. Fridman, Daowei Ma, and Tejinder S. Neelon. "On convergence sets of divergent power series." Annales Polonici Mathematici 106.1 (2012): 193-198. <http://eudml.org/doc/280514>.

@article{BumaL2012,
abstract = {A nonlinear generalization of convergence sets of formal power series, in the sense of Abhyankar-Moh [J. Reine Angew. Math. 241 (1970)], is introduced. Given a family $y = φ_\{s\}(t,x) = sb₁(x)t + b₂(x)t² + ⋯ $ of analytic curves in ℂ × ℂⁿ passing through the origin, $Conv_\{φ\}(f)$ of a formal power series f(y,t,x) ∈ ℂ[[y,t,x]] is defined to be the set of all s ∈ ℂ for which the power series $f(φ_\{s\}(t,x),t,x)$ converges as a series in (t,x). We prove that for a subset E ⊂ ℂ there exists a divergent formal power series f(y,t,x) ∈ ℂ[[y,t,x]] such that $E = Conv_\{φ\}(f)$ if and only if E is an $F_\{σ\}$ set of zero capacity. This generalizes the results of P. Lelong and A. Sathaye for the linear case $φ_\{s\}(t,x)=st$.},
author = {Buma L. Fridman, Daowei Ma, Tejinder S. Neelon},
journal = {Annales Polonici Mathematici},
keywords = {convergence sets; analytic functions; formal power series; capacity},
language = {eng},
number = {1},
pages = {193-198},
title = {On convergence sets of divergent power series},
url = {http://eudml.org/doc/280514},
volume = {106},
year = {2012},
}

TY - JOUR
AU - Buma L. Fridman
AU - Daowei Ma
AU - Tejinder S. Neelon
TI - On convergence sets of divergent power series
JO - Annales Polonici Mathematici
PY - 2012
VL - 106
IS - 1
SP - 193
EP - 198
AB - A nonlinear generalization of convergence sets of formal power series, in the sense of Abhyankar-Moh [J. Reine Angew. Math. 241 (1970)], is introduced. Given a family $y = φ_{s}(t,x) = sb₁(x)t + b₂(x)t² + ⋯ $ of analytic curves in ℂ × ℂⁿ passing through the origin, $Conv_{φ}(f)$ of a formal power series f(y,t,x) ∈ ℂ[[y,t,x]] is defined to be the set of all s ∈ ℂ for which the power series $f(φ_{s}(t,x),t,x)$ converges as a series in (t,x). We prove that for a subset E ⊂ ℂ there exists a divergent formal power series f(y,t,x) ∈ ℂ[[y,t,x]] such that $E = Conv_{φ}(f)$ if and only if E is an $F_{σ}$ set of zero capacity. This generalizes the results of P. Lelong and A. Sathaye for the linear case $φ_{s}(t,x)=st$.
LA - eng
KW - convergence sets; analytic functions; formal power series; capacity
UR - http://eudml.org/doc/280514
ER -

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.