# 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

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topBuma 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 -

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