{
  "id": "2103.13986",
  "version": "v1",
  "published": "2021-03-25T17:18:14.000Z",
  "updated": "2021-03-25T17:18:14.000Z",
  "title": "Structure theorems for Power Series in Several Complex Variables",
  "authors": [
    "G. P. Balakumar"
  ],
  "categories": [
    "math.CV"
  ],
  "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 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 $g$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-25T17:18:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "32A05",
      "32A07"
    ],
    "keywords": [
      "complex variables",
      "structure theorems",
      "elementary power series",
      "logarithmically convex complete reinhardt domains",
      "convergence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}