{
  "id": "1502.02714",
  "version": "v1",
  "published": "2015-02-09T22:28:56.000Z",
  "updated": "2015-02-09T22:28:56.000Z",
  "title": "On definitions of polynomials over function fields of positive characteristi",
  "authors": [
    "Alexandra Shlapentokh"
  ],
  "categories": [
    "math.NT",
    "math.LO"
  ],
  "abstract": "We consider the problem of defining polynomials over function fields of positive characteristic. Among other results, we show that the following assertions are true. 1. Let $\\G_p$ be an algebraic extension of a field of $p$ elements and assume $\\G_p$ is not algebraically closed. Let $t$ be transcendental over $\\G_p$, and let $K$ be a finite extension of $\\G_p(t)$. In this case $\\G_p[t]$ has a definition (with parameters) over $K$ of the form $\\forall \\exists \\ldots \\exists P$ with only one variable in the range of the universal quantifier and $P$ being a polynomial over $K$. 2. For any $q$, for all $p \\not=q$ and all function fields $K$ as above with $\\G_p$ having an extension of degree $q$ and a primitive $q$-th root of unity, there is a uniform in $p$ and $K$ definition (with parameters) of $\\G_p[t]$, of the form $\\exists \\ldots \\exists \\forall \\forall \\exists \\ldots \\exists P$ with only two variables in the range of universal quantifiers and $P$ being a finite collection of disjunction and conjunction of polynomial equations over $\\Z/p$. Further, for any finite collection $\\calS_K$ of primes of $K$ of fixed size $m$, there is a uniform in $K$ and $p$ definition of the ring of $\\calS_K$-integers of the form $\\forall\\forall\\exists \\ldots \\exists P$ with the range of universal quantifiers and $P$ as above. 3. Let $M$ be a function field of positive characteristic in one variable $t$ over an arbitrary constant field $H,$ and let $\\G_p$ be the algebraic closure of a finite field in $H$. Assume $\\G_p$ is not algebraically closed. In this case $\\G_p[t]$ is first-order definable over $M$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-02-09T22:28:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11U09"
    ],
    "keywords": [
      "function field",
      "polynomial",
      "definition",
      "universal quantifier",
      "finite collection"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}