Structure theorems for Power Series in Several Complex Variables

TL;DR

This paper characterizes the convergence domains of power series in several complex variables through a decomposition into elementary series, providing a new constructive proof of classical domain characterization and revealing structural properties of such series.

Contribution

It introduces a novel decomposition of power series into elementary series with domains as half-spaces or wedges, offering a new constructive proof of classical domain characterization.

Findings

Every power series can be decomposed into elementary power series.

Domains of convergence are characterized as intersections of supporting half-spaces.

Existence of singular points in each fiber of the boundary restriction of the absolute map.

Abstract

It is a classical fact that domains of convergence of power series of several complex variables are characterized as logarithmically convex complete Reinhardt domains; let $D \subsetneq \mathbb{C}^N$ be such a domain. We show that a necessary as well as sufficient condition for a power series $g$ to have $D$ as its domain of convergence is that it admits a certain decomposition into elementary power series; specifically, $g$ can be expressed as a sum of a sequence of power series $g_n$ with the property that each of the logarithmic images $G_n$ of their domains of convergence are half-spaces, all containing the logarithmic image $G$ of $D$ and such that the largest open subset of $\mathbb{C}^N$ on which all the $g_n$'s and $g$ converge absolutely is $D$. In short, every power series admits a decomposition into elementary power series. The proof of this leads to a new way of arriving at a constructive proof of the aforementioned classical fact. This proof inturn leads to another decomposition result in which the $G_n$'s are now wedges formed by intersections of pairs of {\it supporting} half-spaces of $G$. Along the way, we also show that in each fiber of the restriction of the absolute map to the boundary of the domain of convergence of $g$, there exists a singular point of $g$.