Nonlinear Convergence Sets of Divergent Power Series

Abstract

A nonlinear generalization of convergence sets of formal power series, in the sense of Abhyankar-Moh, is introduced. Given a family y=\phi_{s}(t,x)=sb_{1}(x)t+b_{2}(x)t^{2}+... of analytic curves in C\timesC^{n} passing through the origin, Conv_{\phi}(f) of a formal power series f(y,t,x)\inC[[y,t,x]] is defined to be the set of all s\inC for which the power series f(\phi_{s}(t,x),t,x) converges as a series in (t,x). We prove that for a subset E\subsetC there exists a divergent formal power series f(y,t,x)\inC[[y,t,x]] such that E=Conv_{\phi}(f) if and only if E is a F_{{\sigma}} set of zero capacity. This generalizes the results of P. Lelong and A. Sathaye for the linear case \phi_{s}(t,x)=st.