{
  "id": "0707.1835",
  "version": "v2",
  "published": "2007-07-12T17:50:56.000Z",
  "updated": "2008-05-08T22:45:29.000Z",
  "title": "Polynomials with PSL(2) monodromy",
  "authors": [
    "Robert M. Guralnick",
    "Michael E. Zieve"
  ],
  "comment": "44 pages; changed notation throughout and made various minor changes",
  "journal": "Annals of Math. 172 (2010), 1321-1365",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "Let K be a field of characteristic p>0, and let q be a power of p. We determine all polynomials f in K[t]\\K[t^p] of degree q(q-1)/2 such that the Galois group of f(t)-u over K(u) has a transitive normal subgroup isomorphic to PSL_2(q), subject to a certain ramification hypothesis. As a consequence, we describe all polynomials f in K[t], of degree not a power of p, such that f is functionally indecomposable over K but f decomposes over an extension of K. Moreover, except for one ramification setup (which is treated in the companion paper arxiv:0707.1837), we describe all indecomposable polynomials f in K[t] of non-p-power degree which are exceptional, in the sense that x-y is the only absolutely irreducible factor of f(x)-f(y) which lies in K[x,y]. It is known that, when K is finite, a polynomial f is exceptional if and only if it induces a bijection on infinitely many finite extensions of K.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-05-08T22:45:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "12F12",
      "14G27"
    ],
    "keywords": [
      "polynomial",
      "transitive normal subgroup isomorphic",
      "finite extensions",
      "non-p-power degree",
      "galois group"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Princeton University and the Institute for Advanced Study",
      "journal": "Ann. Math."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0707.1835G"
    }
  }
}