On Convergence Sets of Formal Power Series

TL;DR

This paper characterizes the convergence sets of divergent formal power series in multiple variables, showing they can be any countable union of closed complete pluripolar sets, and extends previous results to higher dimensions.

Contribution

It proves that any countable union of closed complete pluripolar sets can be realized as a convergence set of some divergent series, generalizing earlier higher-dimensional results.

Findings

Every countable union of closed complete pluripolar sets is a convergence set.

Convergence sets of formal power series with polynomial coefficients are characterized.

The results extend and generalize prior work by Sathaye, Lelong, Levenberg, Molzon, and Ribon.

Abstract

The (projective) convergence set of a divergent formal power series $f(x_{1},...,x_{n})$ is defined to be the image in $\PP^{n-1}$ of the set of all $x\in \mathbb{C}^{n}$ such that $f(x_{1}t,...,x_{n}t)$, as a series in $t$, converges absolutely near $t=0$. We prove that every countable union of closed complete pluripolar sets in $\PP^{n-1}$ is the convergence set of some divergent series $f$. The (affine) convergence sets of formal power series with polynomial coefficients are also studied. The higher-dimensional results of A. Sathaye, P. Lelong, N. Levenberg and R.E. Molzon, and of J. Rib\'{o}n are thus generalized.