On convergence sets of divergent power series. (English) Zbl 1254.32001

Fridman, Buma L.; Ma, Daowei; Neelon, Tejinder S.

Ann. Pol. Math. 106, 193-198 (2012).

Summary: A nonlinear generalization of convergence sets of formal power series, in the sense of S. S. Abhyankar and T. T. Moh [J. Reine Angew. Math. 241, 27–33 (1970; Zbl 0191.04403)], is introduced. Given a family \( y=\varphi _{s}(t,x)=sb_{1}(x)t+b_{2}(x)t^{2}+\cdots\) of analytic curves in \(\mathbb C\times \mathbb C^n\) passing through the origin, \(\text{Conv}_\varphi(f)\) of a formal power series \(f(y,t,x)\in \mathbb C[[y,t,x]]\) is defined to be the set of all \(s\in \mathbb{C}\) for which the power series \(f\big(\varphi _{s}(t,x),t,x\big)\) converges as a series in \((t,x)\). We prove that for a subset \(E\subset \mathbb{C}\) there exists a divergent formal power series \(f(y,t,x)\in \mathbb C[[y,t,x]]\) such that \(E=\text{Conv}_\varphi(f)\) if and only if \(E\) is an \(F_\sigma\) set of zero capacity. This generalizes results of P. Lelong [Proc. Am. Math. Soc. 2, 12–19 (1951; Zbl 0043.29601)] and A. Sathaye [J. Reine Angew. Math. 283/284, 86–98 (1976; Zbl 0334.13010)] for the linear case \(\varphi_s(t,x)=st\).

MSC:

32A05	Power series, series of functions of several complex variables
31A15	Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
31C15	Potentials and capacities on other spaces
32D15	Continuation of analytic objects in several complex variables

Keywords:

convergence sets; analytic functions; formal power series; capacity

Citations:

Zbl 0191.04403; Zbl 0043.29601; Zbl 0334.13010