On Convergence Sets of Power Series with Holomorphic Coefficients

TL;DR

This paper characterizes convergence sets of power series with holomorphic coefficients as exactly the $\sigma$-convex sets, linking complex analysis and geometric properties of sets.

Contribution

It establishes a precise equivalence between convergence sets of such power series and $\sigma$-convex sets in the complex plane.

Findings

Convergence sets are exactly the $\sigma$-convex sets.

$\sigma$-convexity characterizes convergence sets.

The result bridges complex analysis and convex geometry.

Abstract

We consider convergence sets of formal power series of the form $f(z,t)=\sum_{n=0}^{\infty} f_n(z)t^n$, where $f_n(z)$ are holomorphic functions on a domain $\Omega$ in $\mathbb{C}$. A subset $E$ of $\Omega$ is said to be a convergence set in $\Omega$ if there is a series $f(z,t)$ such that $E$ is exactly the set of points $z$ for which $f(z,t)$ converges as a power series in a single variable $t$ in some neighborhood of the origin. A $\sigma$-convex set is defined to be the union of a countable collection of polynomially convex compact subsets. We prove that a subset of $\mathbb{C}$ is a convergence set if and only if it is $\sigma$-convex.