On convergence sets of divergent power series
Volume 106 / 2012
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=\varphi _{s}(t,x)=sb_{1}(x)t+b_{2}(x)t^{2}+\cdots$ of analytic curves in $% \mathbb{C\times C}^{n}$ passing through the origin, $\mathop{\rm Conv} _{\varphi }(f)$ 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 $f(\varphi _{s}(t,x),t,x)$ 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 $E=\mathop{\rm Conv}_{\varphi }(f)$ if and only if $E$ is an $F_{\sigma }$ set of zero capacity. This generalizes the results of P. Lelong and A. Sathaye for the linear case $\varphi _{s}(t,x)=st.$
