{
  "id": "1408.3224",
  "version": "v2",
  "published": "2014-08-14T08:58:09.000Z",
  "updated": "2015-02-09T17:51:41.000Z",
  "title": "On a theorem of Ax and Katz",
  "authors": [
    "Hui June Zhu"
  ],
  "comment": "11 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "The well-known theorem of Ax and Katz gives a p-divisibility bound for the number of rational points on an algebraic variety V over a finite field of characteristic p in terms of the degree and number of variables of defining polynomials of V. It was strengthened by Adolphson-Sperber in terms of Newton polytope of the support set G of V. In this paper we prove that for every generic algebraic variety over a number field supported on G the Adolphson-Sperber bound can be achieved on special fibre at p for a set of prime p of positive density in SpecZ. Moreover we show that if G has certain combinatorial conditional number nonzero then the above bound is achieved at special fiber at p for all but finitely many primes p.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-08-14T08:58:09.000Z",
      "abstract": "The well-known theorem of Ax and Katz gives a p-divisibility bound for the number of rational points on an algebraic variety V over a finite field of characteristic p in terms of the degree and number of variables of defining polynomials of V . This paper gives a p-divisibility bound in terms of the supporting coefficient sets of the algebraic variety that refines the existing bounds; given prescribed sets of nonzero coefficient supports, suppose their combinatorial conditional number is nonzero, we prove that any algebraic variety over the rationals supported on these sets achieves our p-divisibility bound at special fibers at a set of primes p of positive density.",
      "comment": "12 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-02-09T17:51:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "algebraic variety",
      "p-divisibility bound",
      "combinatorial conditional number",
      "nonzero coefficient supports",
      "well-known theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1408.3224Z"
    }
  }
}