{
  "id": "1701.06158",
  "version": "v1",
  "published": "2017-01-22T11:44:43.000Z",
  "updated": "2017-01-22T11:44:43.000Z",
  "title": "A Note on Value Sets of Polynomials over Finite Fields",
  "authors": [
    "Leyla Işık",
    "Alev Topuzoğlu"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Most results on the value sets $V_f$ of polynomials $f \\in \\mathbb{F}_q[x]$ relate the cardinality $|V_f|$ to the degree of $f$. In particular, the structure of the spectrum of the class of polynomials of a fixed degree $d$ is rather well known. We consider a class $\\mathcal{F}_{q,n}$ of polynomials, which we obtain by modifying linear permutations at $n$ points. The study of the spectrum of $\\mathcal{F}_{q,n}$ enables us to obtain a simple description of polynomials $F \\in \\mathcal{F}_{q,n}$ with prescribed $V_F$, especially those avoiding a given set, like cosets of subgroups of the multiplicative group $\\mathbb{F}_q^*$. The value set count for such $F$ can also be determined. This yields polynomials with evenly distributed values, which have small maximum count.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-22T11:44:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite fields",
      "small maximum count",
      "value set count",
      "modifying linear permutations",
      "yields polynomials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}