{
  "id": "1810.08813",
  "version": "v1",
  "published": "2018-10-20T14:31:43.000Z",
  "updated": "2018-10-20T14:31:43.000Z",
  "title": "On the semigroup ring of holomorphic Artin L-functions",
  "authors": [
    "Mircea Cimpoeas"
  ],
  "comment": "11 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $K/\\mathbb Q$ be a finite Galois extension and let $\\chi_1,\\ldots,\\chi_r$ be the irreducible characters of the Galois group $G:=Gal(K/\\mathbb Q)$. Let $f_1:=L(s,\\chi_1),\\ldots,f_r:=L(s,\\chi_r)$ be their associated Artin L-functions. For $s_0\\in \\mathbb C\\setminus\\{1\\}$, we denote $Hol(s_0)$ the semigroup of Artin $L$-functions, holomorphic at $s_0$. Let $\\mathbb F$ be a field with $\\mathbb C \\subseteq \\mathbb F \\subseteq \\mathcal M_{<1}:=$ the field of meromorphic functions of order $<1$. We note that the semigroup ring $\\mathbb F[Hol(s_0)]$ is isomorphic to a toric ring $\\mathbb F[H(s_0)]\\subseteq \\mathbb F[x_1,\\ldots,x_r]$, where $H(s_0)$ is an affine subsemigroup of $\\mathbb N^r$ minimally generated by at least $r$ elements, and we describe $\\mathbb F[H(s_0)]$ when the toric ideal $I_{H(s_0)}=(0)$. Also, we describe $\\mathbb F[H(s_0)]$ and $I_{H(s_0)}$ when $f_1,\\ldots,f_r$ have only simple zeros and simple poles at $s_0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-20T14:31:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R42",
      "16S36"
    ],
    "keywords": [
      "holomorphic artin l-functions",
      "semigroup ring",
      "finite galois extension",
      "associated artin l-functions",
      "simple zeros"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}