{
  "id": "1710.00137",
  "version": "v1",
  "published": "2017-09-30T02:23:45.000Z",
  "updated": "2017-09-30T02:23:45.000Z",
  "title": "Generic Newton polygon for exponential sums in $n$ variables with parallelotope base",
  "authors": [
    "Rufei Ren"
  ],
  "comment": "36 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $p$ be a prime number. Every $n$-variable polynomial $f(\\underline x)$ over a finite field of characteristic $p$ defines an Artin--Schreier--Witt tower of varieties whose Galois group is isomorphic to $\\mathbb{Z}_p$. Our goal of this paper is to study the Newton polygon of the $L$-function associated to a finite character of $\\mathbb{Z}_p$ and a generic polynomial whose convex hull is an $n$-dimensional paralleltope $\\Delta$. We denote this polygon by $\\mathrm{GNP}(\\Delta)$. We prove a lower bound of $\\mathrm{GNP}(\\Delta)$, which is called the improved Hodge polygon $\\mathrm{IHP}(\\Delta)$. We show that $\\mathrm{IHP}(\\Delta)$ lies above the usual Hodge polygon $\\mathrm{HP}(\\Delta)$ at certain infinitely many points, and when $p$ is larger than a fixed number determined by $\\Delta$, it coincides with $\\mathrm{GNP}(\\Delta)$ at these points. As a corollary, we roughly determine the distribution of the slopes of $\\mathrm{GNP}(\\Delta)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-30T02:23:45.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "generic newton polygon",
      "exponential sums",
      "parallelotope base",
      "usual hodge polygon",
      "prime number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}