{
  "id": "math/9808021",
  "version": "v1",
  "published": "1998-08-05T02:00:19.000Z",
  "updated": "1998-08-05T02:00:19.000Z",
  "title": "Reducibility of polynomials $f(x,y)$ modulo $p$",
  "authors": [
    "Wolfgang M. Ruppert"
  ],
  "comment": "Latex, 7 pages",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "We consider absolutely irreducible polynomials $f \\in Z[x,y]$ with $\\deg_x(f)=m$, $\\deg_y(f)=n$ and height $H$. We show that for any prime $p$ with $p>c_{mn} H^{2mn+n-1}$ the reduction $f \\bmod p$ is also absolutely irreducible. Furthermore if the Bouniakowsky conjecture is true we show that there are infinitely many absolutely irreducible polynomials $f \\in Z[x,y]$ which are reducible mod $p$ where $p$ is a prime with $p>H^{2m}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1998-08-05T02:00:19.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "reducibility",
      "absolutely irreducible polynomials",
      "bouniakowsky conjecture",
      "reducible mod"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1998math......8021R"
    }
  }
}