{
  "id": "2108.12577",
  "version": "v1",
  "published": "2021-08-28T05:16:38.000Z",
  "updated": "2021-08-28T05:16:38.000Z",
  "title": "Generic Newton polygons for $L$-functions of $(A,B)$-exponential sums",
  "authors": [
    "Liping Yang",
    "Hao Zhang"
  ],
  "comment": "16 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper, we consider the following $(A, B)$-polynomial $f$ over finite field: $$f(x_0,x_1,\\cdots,x_n)=x_0^Ah(x_1,\\cdots,x_n)+g(x_1,\\cdots,x_n)+P_B(1/x_0),$$ where $h$ is a Deligne polynomial of degree $d$, $g$ is an arbitrary polynomial of degree $< dB/(A+B)$ and $P_B(y)$ is a one-variable polynomial of degree $\\le B$. Let $\\Delta$ be the Newton polyhedron of $f$ at infinity. We show that $\\Delta$ is generically ordinary if $p\\equiv 1 \\mod D$, where $D$ is a constant only determined by $\\Delta$. In other words, we prove that the Adolphson--Sperber conjecture is true for $\\Delta$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-28T05:16:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11T06",
      "11T23",
      "11S40"
    ],
    "keywords": [
      "generic newton polygons",
      "exponential sums",
      "newton polyhedron",
      "finite field",
      "arbitrary polynomial"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}