{
  "id": "math/0401356",
  "version": "v1",
  "published": "2004-01-26T19:41:36.000Z",
  "updated": "2004-01-26T19:41:36.000Z",
  "title": "Common divisors of a^n-1 and b^n-1 over function fields",
  "authors": [
    "Joseph H. Silverman"
  ],
  "journal": "New York Journal of Math. (electronic) 10 (2004), 37--43",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Ailon and Rudnick have shown that if $a,b \\in C[T]$ are multiplicatively independent polynomials, then $\\deg(\\gcd(a^n-1,b^n-1))$ is bounded for all $n\\ge1$. We show that if instead $a,b \\in F[T]$ for a finite field $F$ of characteristic $p$, then $\\deg(\\gcd(a^n-1,b^n-1))$ is larger than $Cn$ for a constant $C=C(a,b)>0$ and for infinitely many $n$, even if $n$ is restricted in various reasonable ways (e.g., $gec(n,p)=1$).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-01-26T19:41:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11T55",
      "11R58",
      "11D61"
    ],
    "keywords": [
      "function fields",
      "common divisors",
      "multiplicatively independent polynomials",
      "finite field"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......1356S"
    }
  }
}