On convergence sets of divergent power series

• Autorzy: Buma L. Fridman , Daowei Ma , Tejinder S. Neelon
• Języki publikacji: 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 $φ_{s}(t,x)=st$.

Kategorie tematyczne

• 32A05: Power series, series of functions
• 32D15: Continuation of analytic objects
• 31A15: Potentials and capacity, harmonic measure, extremal length
• 31C15: Potentials and capacities

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 2012
• Tom: 106
• Numer: 1
• Strony: 193-198

Daty

wydano

2012

Twórcy

autor

Buma L. Fridman

• Department of Mathematics, Wichita State University, Wichita, KS 67260-0033, U.S.A.

autor

Daowei Ma

• Department of Mathematics, Wichita State University, Wichita, KS 67260-0033, U.S.A.

autor

Tejinder S. Neelon

• Department of Mathematics, California State University San Marcos, San Marcos, CA 92096-0001, U.S.A.

Identyfikatory

DOI

10.4064/ap106-0-14