{
  "id": "math/0411369",
  "version": "v3",
  "published": "2004-11-16T23:22:33.000Z",
  "updated": "2007-07-03T21:31:30.000Z",
  "title": "Power-free values, large deviations, and integer points on irrational curves",
  "authors": [
    "H. A. Helfgott"
  ],
  "comment": "39 pages; rather major revision, with strengthened and generalized statements",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $f\\in \\mathbb{Z}\\lbrack x\\rbrack$ be a polynomial of degree $d\\geq 3$ without roots of multiplicity $d$ or $(d-1)$. Erd\\H{o}s conjectured that, if $f$ satisfies the necessary local conditions, then $f(p)$ is free of $(d-1)$th powers for infinitely many primes $p$. This is proved here for all $f$ with sufficiently high entropy. The proof serves to demonstrate two innovations: a strong repulsion principle for integer points on curves of positive genus, and a number-theoretical analogue of Sanov's theorem from the theory of large deviations.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2007-07-03T21:31:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N32",
      "11D45",
      "11G05",
      "11G30",
      "11N25"
    ],
    "keywords": [
      "large deviations",
      "integer points",
      "irrational curves",
      "power-free values",
      "necessary local conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math.....11369H"
    }
  }
}