On convergence sets of divergent power series

@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 -