Nonlinear Convergence Sets of Divergent Power Series

TL;DR

This paper introduces a nonlinear extension of convergence sets for formal power series, characterizing which sets can be realized as convergence sets through complex analysis and capacity theory.

Contribution

It generalizes previous linear results by characterizing convergence sets for nonlinear families of curves using capacity and set theory.

Findings

Existence of divergent power series with prescribed convergence sets

Characterization of convergence sets as F_sigma sets of zero capacity

Extension of linear convergence set results to nonlinear families

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.