{
  "id": "1607.05773",
  "version": "v1",
  "published": "2016-07-19T22:24:30.000Z",
  "updated": "2016-07-19T22:24:30.000Z",
  "title": "Almost prime solutions to diophantine systems of high rank",
  "authors": [
    "Akos Magyar",
    "Tatchai Titichetrakun"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\F$ be a family of $r$ integral forms of degree $k\\geq 2$ and $\\LL=(l_1,\\ldots,l_m)$ be a family of pairwise linearly independent linear forms in $n$ variables $\\x=(x_1,...,x_n)$. We study the number of solutions $\\x\\in[1,N]^n$ to the diophantine system $\\F(\\x)=\\vv$ under the restriction that $l_i(\\x)$ has a bounded number of prime factors for each $1\\leq i\\leq m$. We show that the system $\\F$ have the expected number of such \"almost prime\" solutions under similar conditions as was established for existence of integer solutions by Birch.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-07-19T22:24:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D72",
      "11P32"
    ],
    "keywords": [
      "diophantine system",
      "high rank",
      "prime solutions",
      "pairwise linearly independent linear forms",
      "integral forms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}