{
  "id": "2010.06833",
  "version": "v1",
  "published": "2020-10-12T21:08:56.000Z",
  "updated": "2020-10-12T21:08:56.000Z",
  "title": "Solubility of Additive Forms of Twice Odd Degree over Ramified Quadratic Extensions of $\\mathbb{Q}_2$",
  "authors": [
    "Drew Duncan",
    "David B. Leep"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:2005.09770",
  "categories": [
    "math.NT"
  ],
  "abstract": "We determine the minimal number of variables $\\Gamma^*(d, K)$ which guarantees a nontrivial solution for every additive form of degree $d=2m$, $m$ odd, $m \\ge 3$ over the six ramified quadratic extensions of $\\mathbb{Q}_2$. We prove that if $K$ is one of $\\{\\mathbb{Q}_2(\\sqrt{2}), \\mathbb{Q}_2(\\sqrt{10}), \\mathbb{Q}_2(\\sqrt{-2}), \\mathbb{Q}_2(\\sqrt{-10})\\}$, $\\Gamma^*(d,K) = \\frac{3}{2}d$, and if $K$ is one of $\\{\\mathbb{Q}_2(\\sqrt{-1}), \\mathbb{Q}_2(\\sqrt{-5})\\}$, $\\Gamma^*(d,K) = d+1$. The case $d=6$ was previously known.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-12T21:08:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D72",
      "11D88",
      "11E76"
    ],
    "keywords": [
      "ramified quadratic extensions",
      "twice odd degree",
      "additive form",
      "solubility",
      "minimal number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}