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• # Artykuł - szczegóły

## Annales Polonici Mathematici

2012 | 106 | 1 | 193-198

## On convergence sets of divergent power series

EN

### Abstrakty

EN
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 $φ_{s}(t,x)=st$.

193-198

wydano
2012

### Twórcy

autor
• Department of Mathematics, Wichita State University, Wichita, KS 67260-0033, U.S.A.
autor
• Department of Mathematics, Wichita State University, Wichita, KS 67260-0033, U.S.A.
autor
• Department of Mathematics, California State University San Marcos, San Marcos, CA 92096-0001, U.S.A.