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On convergence sets of divergent power series

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Fridman B., Ma D., Neelon T.

Annales Polonici Mathematici

2012

tom 106

nr 1

193-198

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 $φ_{s}(t,x)=st$.

Ograniczanie wyników

1 Annales Polonici Mathematici

1 Fridman B.

1 Ma D.

1 Neelon T.

1 2012

Wyniki wyszukiwania

On convergence sets of divergent power series