{
  "id": "2008.08962",
  "version": "v1",
  "published": "2020-08-20T13:31:36.000Z",
  "updated": "2020-08-20T13:31:36.000Z",
  "title": "The density of rational lines on hypersurfaces: A bihomogeneous perspective",
  "authors": [
    "Julia Brandes"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $F$ be a non-singular homogeneous polynomial of degree $d$ in $n$ variables. We give an asymptotic formula of the pairs of integer points $(\\mathbf x, \\mathbf y)$ with $|\\mathbf x| \\le X$ and $|\\mathbf y| \\le Y$ which generate a line lying in the hypersurface defined by $F$, provided that $n > 2^{d-1}d^4(d+1)(d+2)$. In particular, by restricting to Zariski-open subsets we are able to avoid imposing any conditions on the relative sizes of $X$ and $Y$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-20T13:31:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D72",
      "11P55",
      "11E76",
      "14G05"
    ],
    "keywords": [
      "rational lines",
      "bihomogeneous perspective",
      "hypersurface",
      "integer points",
      "non-singular homogeneous polynomial"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}