  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 $g$.
