{
  "id": "2005.11835",
  "version": "v1",
  "published": "2020-05-24T20:21:35.000Z",
  "updated": "2020-05-24T20:21:35.000Z",
  "title": "Average Bateman--Horn for Kummer polynomials",
  "authors": [
    "Francesca Balestrieri",
    "Nick Rome"
  ],
  "comment": "25 pages",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "For any $r \\in \\mathbb{N}$ and almost all $k \\in \\mathbb{N}$, we show that the polynomial $f(n) = n^r + k$ takes infinitely many prime values. As a consequence, we deduce statements concerning variants of the Hasse principle and of the integral Hasse principle for certain open varieties defined by equations of the form $N_{K/\\mathbb{Q}}(\\mathbf{z}) = t^r +k \\neq 0$ where $K/\\mathbb{Q}$ is a quadratic extension. A key ingredient in our proof is a new large sieve inequality for Dirichlet characters of exact order $r$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-24T20:21:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N32",
      "14G05",
      "11D85"
    ],
    "keywords": [
      "kummer polynomials",
      "average bateman-horn",
      "integral hasse principle",
      "large sieve inequality",
      "deduce statements concerning variants"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}