{
  "id": "1912.07223",
  "version": "v1",
  "published": "2019-12-16T07:18:17.000Z",
  "updated": "2019-12-16T07:18:17.000Z",
  "title": "Counting mod n in pseudofinite fields",
  "authors": [
    "Will Johnson"
  ],
  "comment": "Expanded version of thesis chapter; 45 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "We show that in an ultraproduct of finite fields, the mod-$n$ nonstandard size of definable sets varies definably in families. Moreover, if $K$ is any pseudofinite field, then one can assign \"nonstandard sizes mod $n$\" to definable sets in $K$. As $n$ varies, these nonstandard sizes assemble into a definable strong Euler characteristic on $K$, taking values in the profinite completion $\\hat{\\mathbb{Z}}$ of the integers. The strong Euler characteristic is not canonical, but depends on the choice of a nonstandard Frobenius. When $\\operatorname{Abs}(K)$ is finite, the Euler characteristic has some funny properties for two choices of the nonstandard Frobenius. Additionally, we show that the theory of finite fields remains decidable when first-order logic is expanded with parity quantifiers. However, the proof depends on a computational algebraic geometry statement whose proof is deferred to a later paper.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-16T07:18:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "pseudofinite field",
      "counting mod",
      "nonstandard frobenius",
      "computational algebraic geometry statement",
      "nonstandard sizes mod"
    ],
    "tags": [
      "dissertation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}