{
  "id": "2501.19193",
  "version": "v1",
  "published": "2025-01-31T15:01:11.000Z",
  "updated": "2025-01-31T15:01:11.000Z",
  "title": "On the convex hull of integer points above the hyperbola",
  "authors": [
    "David Alcántara",
    "Mónica Blanco",
    "Francisco Criado",
    "Francisco Santos"
  ],
  "comment": "26 pages",
  "categories": [
    "math.CO",
    "cs.CG"
  ],
  "abstract": "We show that the polyhedron defined as the convex hull of the lattice points above the hyperbola $\\left\\{xy = n\\right\\}$ has between $\\Omega(n^{1/3})$ and $O(n^{1/3} \\log n)$ vertices. The same bounds apply to any hyperbola with rational slopes except that instead of $n$ we have $n/\\Delta$ in the lower bound and by $\\max\\left\\{\\Delta, n/\\Delta\\right\\}$ in the upper bound, where $\\Delta \\in \\mathbb{Z}_{>0}$ is the discriminant. We also give an algorithm that enumerates the vertices of these convex hulls in logarithmic time per vertex. One motivation for such an algorithm is the deterministic factorization of integers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-01-31T15:01:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52C05",
      "52C45",
      "68R99",
      "52-08",
      "I.3.5"
    ],
    "keywords": [
      "convex hull",
      "integer points",
      "lattice points",
      "logarithmic time",
      "rational slopes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}