{
  "id": "0707.1837",
  "version": "v2",
  "published": "2007-07-12T17:44:03.000Z",
  "updated": "2008-05-08T22:49:48.000Z",
  "title": "A new family of exceptional polynomials in characteristic two",
  "authors": [
    "Robert M. Guralnick",
    "Joel E. Rosenberg",
    "Michael E. Zieve"
  ],
  "comment": "30 pages; changed notation, fixed some minor errors, added Corollary 1.3, and removed the perfectness hypothesis in Theorem 1.2",
  "journal": "Annals of Math. 172 (2010) 1367-1396",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "We produce a new family of polynomials f(x) over fields K of characteristic 2 which are exceptional, in the sense that f(x)-f(y) has no absolutely irreducible factors in K[x,y] besides the scalar multiples of x-y; when K is finite, this condition is equivalent to saying there are infinitely many finite extensions L/K for which the map c --> f(c) is bijective on L. Our polynomials have degree (2^e-1)*2^(e-1), where e is odd. Combined with our previous paper arxiv:0707.1835, this completes the classification of indecomposable exceptional polynomials of degree not a power of the characteristic. The strategy of our proof is to identify the curves that can arise as the Galois closure of the branched cover P^1 --> P^1 induced by an exceptional polynomial f. In this case, the curves turn out to be x^(q+1)+y^(q+1)=a+T(xy), where T(z)=z^(q/2)+z^(q/4)+...+z. Our proofs rely on new properties of ramification in Galois covers of curves, as well as the computation of the automorphism groups of all curves in a certain 2-parameter family.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-05-08T22:49:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "12F12",
      "14G27"
    ],
    "keywords": [
      "characteristic",
      "finite extensions l/k",
      "scalar multiples",
      "indecomposable exceptional polynomials",
      "galois closure"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Princeton University and the Institute for Advanced Study",
      "journal": "Ann. Math."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0707.1837G"
    }
  }
}