Structure theorems for Power Series in Several Complex Variables

G. P. Balakumar

Published 2021-03-25Version 1

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 decomposition into elementary power series i.e., $g$ 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 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 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 $g$.