{
  "id": "2207.09556",
  "version": "v1",
  "published": "2022-07-19T21:26:26.000Z",
  "updated": "2022-07-19T21:26:26.000Z",
  "title": "Solubility of Additive Forms of Twice Odd Degree over $\\mathbb{Q}_2(\\sqrt{5})$",
  "authors": [
    "Drew Duncan",
    "David B. Leep"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We prove that an additive form of degree $d=2m$, $m$ odd, $m\\ge3$, over the unramified quadratic extension $\\mathbb{Q}_2(\\sqrt{5})$ has a nontrivial zero if the number of variables $s$ satisifies $s \\ge 4d+1$. If $3 \\nmid d$, then there exists a nontrivial zero if $s \\ge \\frac{3}{2}d + 1$, this bound being optimal. We give examples of forms in $3d$ variables without a nontrivial zero in case that $3 \\mid d$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-19T21:26:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D72",
      "11D88",
      "11E76"
    ],
    "keywords": [
      "twice odd degree",
      "additive form",
      "nontrivial zero",
      "solubility",
      "unramified quadratic extension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}